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2412.02046v1 | http://arxiv.org/abs/2412.02046v1 | Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results | \documentclass[a4paper, 10pt, twoside, notitlepage]{amsart}
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\title[Inverse problems for damped nonlocal wave equations]{Optimal Runge approximation for damped nonlocal wave equations and simultaneous determination results}
\author[P. Zimmermann]{Philipp Zimmermann}
\address{Departament de Matem\`atiques i Inform\`atica, Universitat de Barcelona, Barcelona, Spain}
\email{[email protected]}
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\begin{document}
\maketitle
\begin{abstract}
The main purpose of this article is to establish new uniqueness results for Calder\'on type inverse problems related to damped nonlocal wave equations. To achieve this goal we extend the theory of very weak solutions to our setting, which allows to deduce an optimal Runge approximation theorem. With this result at our disposal, we can prove simultaneous determination results in the linear and semilinear regime.
\medskip
\noindent{\bf Keywords.} Fractional Laplacian, wave equations, nonlinear PDEs, inverse problems, Runge approximation, very weak solutions.
\noindent{\bf Mathematics Subject Classification (2020)}: Primary 35R30; secondary 26A33, 42B37
\end{abstract}
\tableofcontents
\section{Introduction}
\label{sec: introduction}
In recent years, inverse problems for nonlocal partial differential equations (PDEs) of elliptic, parabolic and hyperbolic type have been studied. This line of research was initiated by Ghosh, Salo and Uhlman \cite{GSU20}, in which they have considered the (partial data) Calder\'on problem related to the \emph{fractional Schr\"odinger equation}
\begin{equation}
\label{eq: fractional Schroedinger equation}
\begin{cases}
((-\Delta)^s+q)u=0&\text{ in }\Omega,\\
u =\varphi & \text{ in } \Omega_e,
\end{cases}
\end{equation}
where $\Omega\subset\R^n$ is a bounded domain, $\Omega_e=\R^n\setminus \overline{\Omega}$, $0<s<1$, $q$ is a suitable potential and $(-\Delta)^s$ is the \emph{fractional Laplacian} which is the operator with Fourier symbol $|\xi|^{2s}$. In this problem one asks whether the knowledge of the \emph{(partial) Dirichlet to Neumann (DN) map}
\begin{equation}
\label{eq: partial DN map Schroeding}
\Lambda_q \varphi= (-\Delta)^s u_\varphi|_{W_2}, \quad \varphi\in C_c^{\infty}(W_1),
\end{equation}
where $W_1,W_2\subset\Omega_e$ are given measurement sets (i.e.~nonempty open sets) and $u_\varphi$ denotes the unique solution to \eqref{eq: fractional Schroedinger equation}, uniquely determines the potential $q$. The overall strategy to establish unique determination results for the above Calder\'on problem is as follows (see \cite{GSU20,RS17,RZ-unbounded}):
\begin{enumerate}[(i)]
\item\label{item: integral identity} \emph{Integral identity:} Assume that the potentials $q_j$ are suitably regular, then one can write
\begin{equation}
\label{eq: integral identity schroeding}
\langle (\Lambda_{q_1}-\Lambda_{q_2})\varphi_1,\varphi_2\rangle=\int_{\Omega} (q_1-q_2)(u_{\varphi_1}-\varphi_1),(u_{\varphi_2}-\varphi_2)\,dx,
\end{equation}
when the right hand side is interpreted accordingly.
\item\label{item: Runge approx} Establish one of the following \emph{Runge approximation theorems}:
\begin{enumerate}[(I)]
\item\label{L2 Runge} $\mathcal{R}_W =\{u_f|_{\Omega}\,;\,f\in C_c^{\infty}(W)\}$ is dense in $L^2(\Omega)$ (see \cite{GSU20} for $q\in L^{\infty}(\Omega)$).
\item\label{Hs Runge} $\mathscr{R}_W =\{u_f-f\,;\,f\in C_c^{\infty}(W)\}$ is dense in $\widetilde{H}^s(\Omega)$ (see \cite{RS17} for Sobolev multipliers $q$ or \cite{RZ-unbounded} for local, bounded bilinear forms).
\end{enumerate}
\item\label{item: Conclusion} If the potentials $q_j$ for $j=1,2$ have suitable continuity properties, then $\Lambda_{q_1}=\Lambda_{q_2}$ together with \ref{item: Runge approx} ensure that there holds $q_1=q_2$ in $\Omega$.
\end{enumerate}
In \ref{Hs Runge}, the space $\widetilde{H}^s(\Omega)$ is the closure of $C_c^{\infty}(\Omega)$ in the energy space
\[
H^s(\R^n)=\{u\in\tempered(\R^n)\,;\,\|u\|_{H^s(\R^n)}\vcentcolon = \|\langle D\rangle^s u\|_{L^2(\R^n)}<\infty\},
\]
where $\langle D\rangle^s$ is the Bessel potential operator.
Observe the similarity of the above strategy to the one of \cite{SU87} for showing unique determination for the classical Calder\'on problem, where instead of the Runge approximation theorem suitable geometric optics solutions are used. Moreover, the Runge approximation \ref{item: Runge approx} relies on a Hahn--Banach argument and the \emph{unique continuation property (UCP)} of the fractional Laplacian $(-\Delta)^s$. For more results on Calder\'on problems for elliptic nonlocal PDEs, we refer the interested reader to \cite{GLX,cekic2020calderon,CLL2017simultaneously,LL2020inverse,LL2022inverse, LZ2023unique,KLZ-2022,KLW2022,LRZ2022calder,Semilinear-nonlocal-wave-paper,LLU2023calder,CGRU2023reduction,LLU2023calder,RZ-unbounded,RZ2022LowReg,CRTZ-2022,LZ2024uniqueness,feizmohammadi2021fractional,feizmohammadi2021fractional_closed,FKU24,Trans-anisotropic-LNZ} and the references therein.
\subsection{Mathematical model and main results}
\label{subsec: mathematical model and main results}
Recently, the above approach for solving elliptic nonlocal inverse problems has also been adapted to deduce uniqueness results for the Calder\'on problem of nonlocal hyperbolic equations. Let us next describe some of these results in more detail and for this purpose consider the problem
\begin{equation}
\label{eq: discussion existing results}
\begin{cases}
\partial_t^2u+\lambda (-\Delta)^s \partial_t u+(-\Delta)^s u +f(u)= 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega,
\end{cases}
\end{equation}
where $\lambda\in\R$ and $f\colon \Omega\times\R\to\R$ is a possibly nonlinear function. If the problem \eqref{eq: discussion existing results} is well-posed in the energy class $H^s(\R^n)$, then for any two given measurement sets $W_1,W_2\subset\Omega_e$ we may introduce the DN map $\Lambda^{\lambda}_{f}$ via
\[
\Lambda^{\lambda}_{f}\varphi=\left.(\lambda (-\Delta)^s\partial_t u_\varphi+(-\Delta)^s u_\varphi)\right|_{(W_2)_T},
\]
whenever $\varphi$ is supported in $(W_1)_T$ and $u_\varphi$ is the solution of \eqref{eq: discussion existing results}. The \emph{Calder\'on problem} for \eqref{eq: discussion existing results} reads as follows:
\begin{question}
\label{question: calderon discussion}
Does the DN map $\Lambda^{\lambda}_{f}$ uniquely determine the function $f$?
\end{question}
A suitable class of nonlinearities are the so-called weak nonlinearities, which are defined next.
\begin{definition}\label{main assumptions on nonlinearities}
We call a Carath\'eodory function $f\colon \Omega\times \R\to\R$ \emph{weak nonlinearity}, if it satisfies the following conditions:
\begin{enumerate}[(i)]
\item\label{prop f} $f$ has partial derivative $\partial_{\tau}f$, which is a Carath\'eodory function, and there exists $a\in L^p(\Omega)$ such that
\begin{equation}
\label{eq: bound on derivative}
\left|\partial_\tau f(x,\tau)\right|\lesssim a(x)+|\tau|^r
\end{equation}
for all $\tau\in\R$ and a.e. $x\in\Omega$. Here the exponents $p$ and $r$ satisfy the restrictions
\begin{equation}
\label{eq: restrictions on p}
\begin{cases}
n/s\leq p\leq \infty, &\, \text{if }\, 2s< n,\\
2<p\leq \infty, &\, \text{if }\, 2s= n,\\
2\leq p\leq \infty, &\, \text{if }\, 2s\geq n.
\end{cases}
\end{equation}
and
\begin{equation}
\label{eq: cond on r}
\begin{cases}
0\leq r<\infty, &\, \text{if }\, 2s\geq n,\\
0\leq r\leq \frac{2s}{n-2s}, &\, \text{if }\, 2s< n,
\end{cases}
\end{equation}
respectively. Moreover, $f$ fulfills the integrability condition $f(\cdot,0)\in L^2(\Omega)$.
\item\label{prop F} The function $F\colon \Omega\times\R\to\R$ defined via
\[
F(x,\tau)=\int_0^\tau f(x,\rho)\,d\rho
\]
satisfies $F(x,\tau)\geq 0$ for all $\tau\in\R$ and $x\in\Omega$.
\end{enumerate}
\end{definition}
Let us note that $f(x,\tau)=q(x)|\tau|^r\tau$ with $r$ satisfying \eqref{eq: cond on r} and $0\leq q\in L^{\infty}(\Omega)$ is a weak nonlinearity. Using the above notions, we can now discuss some of the existing results.
\begin{enumerate}[(a)]
\item\label{item: basic nonlocal wave} The article \cite{KLW2022} gives a positive answer for $\lambda=0, f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(\Omega)$. Their proof relied on the observation that the related nonlocal wave equation \eqref{eq: discussion existing results} satisfies an $L^2(\Omega_T)$ Runge approximation theorem.
\item\label{item: viscous nonlocal wave} The work \cite{zimmermann2024calderon} deals on the one hand with the linear case $\lambda=1$, $f(x,\tau)=q(x)\tau$ with $q\in L^{\infty}(0,T;L^p(\Omega))$, where $p$ satisfies the restrictions \eqref{eq: restrictions on p}, and $q$ is weakly continuous in $t$ and on the other hand with the nonlinear case $\lambda=1$ and $f$ is a $r+1$ homogeneous, weak nonlinearity. The uniqueness proofs use substantially that due to presence of the viscosity term $(-\Delta)^s\partial_t$ solutions $u$ to \eqref{eq: discussion existing results} satisfy $\partial_t u\in L^2(0,T;H^s(\R^n))$ and as a consequence the linearized equations have the Runge approximation property in $L^2(0,T;\widetilde{H}^s(\Omega))$.
\item\label{item: semilinear nonlocal wave} In \cite{Semilinear-nonlocal-wave-eq}, uniqueness is proved in the case $\lambda=0$ and $f$ satisfies the same properties as in \ref{item: viscous nonlocal wave}, but with the additional restriction $r\leq 1$. This article only uses an $L^2(\Omega_T)$ Runge approximation result for the linearized nonlocal wave equation.
\item\label{item: optimal runge} By establishing a theory for very weak solutions of linear nonlocal wave equations with $\lambda=0$, the authors of \cite{Optimal-Runge-nonlocal-wave} could deduce an optimal $L^2(0,T;\widetilde{H}^s(\Omega))$ Runge approximation theorem for these equations. This allowed to extend the results in \ref{item: semilinear nonlocal wave} to the cases $r>1$ and additionally showed that one can recover any linear perturbation $q\in L^p(\Omega)$ with $p$ satisfying the restrictions \eqref{eq: restrictions on p}. Furthermore, by this improved Runge approximation theorem the authors could also treat the case of serially or asymptotically polyhomogeneous nonlinearities (see \cite[Theorem 1.5]{Optimal-Runge-nonlocal-wave}).
\end{enumerate}
In this context, let us also mention the recent article \cite{fu2024wellposednessinverseproblemsnonlocal} which deals with the Calder\'on problem for a third order semilinear, nonlocal, viscous wave equation.
The goal of this paper us to present an extension of the models described in \ref{item: viscous nonlocal wave} and \ref{item: semilinear nonlocal wave}, which we discuss next. Let $0<s<1$ and suppose that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^p(\Omega)$, where $0<s<\alpha\leq 1$ and $1\leq p\leq \infty$ satisfies the restrictions in \eqref{eq: restrictions on p}.
Then we define the following \emph{damped, nonlocal wave operator}
\begin{equation}
\label{eq: damped, nonlocal wave operator}
L_{\gamma}\vcentcolon = \partial_t^2+\gamma\partial_t +(-\Delta)^s
\end{equation}
and consider the problem
\begin{equation}
\label{eq: wave problem}
\begin{cases}
L_{\gamma}u +f(u)= F & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
where $f$ is a weak nonlinearity or $f(u)=q(x)u$. In fact, this is a possibly nonlinear generalization of the model (G2) with $s_1=0$ in \cite[Section 1.1]{zimmermann2024calderon}. By \cite[Proposition 3.7]{Semilinear-nonlocal-wave-eq} (see Section \ref{subsec: weak solutions} for the linear case), we know that the problem \eqref{eq: wave problem} is well-posed, whenever the source $F$, exterior condition $\varphi$ and initial conditions $u_0,u_1$ are sufficiently regular. Thus, we can introduce the related (partial) DN map via
\begin{equation}
\label{eq: formal DN map}
\Lambda_{\gamma,f}\varphi\vcentcolon = \left.(-\Delta)^s u_{\varphi}\right|_{(W_2)_T},
\end{equation}
where $W_1,W_2\subset \Omega_e$ are some measurement sets, $\varphi$ is supported in $(W_1)_T$ and $u_\varphi$ is the unique solution of \eqref{eq: wave problem} with $u_0=u_1=0$. Then we ask the following question:
\begin{question}
\label{question: Caldeorn problem of this work}
Does the partial DN map $\Lambda_{\gamma,f}$ uniquely determine the damping coefficient $\gamma$ and the function $f$?
\end{question}
In this work we establish the following affirmative answers to this question, whereas the first result discusses the linear case and the second one the semilinear perturbations.
\begin{theorem}[Uniqueness for linear perturbations]
\label{thm: uniqueness linear}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that for $j=1,2$ we have given coefficients $(\gamma_j,q_j)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$ and let $\Lambda_{\gamma_j,q_j}$ be the DN map associated to the problem
\begin{equation}
\label{eq: PDEs uniqueness theorem}
\begin{cases}
(L_{\gamma_j}+q_j)u = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
for $j=1,2$. If $W_1,W_2\subset\Omega_e$ are two measurement sets such that
\begin{equation}
\label{eq: equality of DN maps}
\left.\Lambda_{\gamma_1,q_1}\varphi\right|_{(W_2)_T}=\left.\Lambda_{\gamma_2,q_2}\varphi\right|_{(W_2)_T}
\end{equation}
for all $\varphi\in C_c^{\infty}((W_1)_T)$, then there holds
\begin{equation}
\label{eq: equal coefficients}
\gamma_1=\gamma_2\text{ and }q_1=q_2 \text{ in }\Omega.
\end{equation}
\end{theorem}
\begin{theorem}[Uniqueness for semilinear perturbations]
\label{thm: uniqueness semilinear}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$ and $0<s<\alpha\leq 1$. Assume that for $j=1,2$ we have given coefficients $\gamma_j\in C^{0,\alpha}(\R^n)$ and $r+1$ homogeneous, weak nonlinearities $f_j$, where $r>0$ satisfies \eqref{eq: cond on r}. Let $\Lambda_{\gamma_j,f_j}$ be the DN map associated to the problem
\begin{equation}
\label{eq: PDEs uniqueness theorem}
\begin{cases}
L_{\gamma_j}u +f_j(u) = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
for $j=1,2$. If $W_1,W_2\subset\Omega_e$ are two measurement sets such that
\begin{equation}
\label{eq: equality of DN maps semilinear}
\left.\Lambda_{\gamma_1,f_1}\varphi\right|_{(W_2)_T}=\left.\Lambda_{\gamma_2,f_2}\varphi\right|_{(W_2)_T}
\end{equation}
for all $\varphi\in C_c^{\infty}((W_1)_T)$, then there holds
\begin{equation}
\label{eq: equal coefficients}
\gamma_1=\gamma_2\text{ in }\Omega\text{ and }f_1=f_2 \text{ in }\Omega\times \R.
\end{equation}
\end{theorem}
\begin{remark}
For simplicity we restrict our attention to homogeneous nonlinearities $f$, but the unique determination remains valid in some polyhomogeneous cases as described in \cite{Optimal-Runge-nonlocal-wave} for $\gamma=0$.
\end{remark}
\subsection{Organization of the article} The rest of this article is structured as follows. In Section \ref{sec: Very weak solutions to damped, nonlocal wave equations}, we establish the existence of unique weak and very weak solutions to damped nonlocal wave equations. In Section \ref{sec: inverse problem} we then move on to the inverse problem part of this work. First, in Section \ref{sec: Runge approx} we establish the optimal Runge approximation theorem. Afterwards, in Section \ref{sec: integral identity} we prove a suitable integral identity that allows us to recover simultaneously the damping coefficient $\gamma$ and potential $q$ in Section \ref{sec: linear uniqueness}. Finally, Section \ref{sec: semilinear uniqueness} contains the proof of the simultaneous determination of the damping coefficient $\gamma$ and the homogeneous nonlinearity $f$.
\section{Weak and very weak solutions to damped, nonlocal wave equations}
\label{sec: Very weak solutions to damped, nonlocal wave equations}
The main purpose of this section is to show existence of unique weak and very weak solutions to \emph{damped, nonlocal wave equations (DNWEQ)}
\begin{equation}
\label{eq: damped nonlocal wave equations well-posedness}
\begin{cases}
(L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
where $L_\gamma$ is given by \eqref{eq: damped, nonlocal wave operator} and only the case $\varphi=0$ is considered for very weak solutions.
\subsection{Weak solutions}
\label{subsec: weak solutions}
This section deals with the well-posedness of \eqref{eq: damped nonlocal wave equations well-posedness} for regular sources, exterior conditions and initial data. We also prove well-posedness for the case when instead of initial values the values at $t=T$ are specified, which will be needed for the development of the theory of very weak solutions.
\begin{theorem}[Weak solutions to homogeneous DNWEQ]
\label{thm: weak sol to hom DNWEQ}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;\widetilde{L}^2(\Omega))$\footnote{Here and below we set $\widetilde{L}^2(\Omega)\vcentcolon =\widetilde{H}^0(\Omega)$.} and initial conditions $(u_0,u_1)\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, there exists a unique weak solution $u\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ of
\begin{equation}
\label{eq: damped nonlocal wave equations well-posedness weak sol}
\begin{cases}
(L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
which means that $(u(0),\partial_tu(0))=(u_0,u_1)$ in $\widetilde{H}^s(\Omega)\times \widetilde{L}^2(\Omega)$ and there holds
\begin{equation}
\label{eq: weak formulation damped nonlocal wave eq}
\begin{split}
&\frac{d}{dt} \langle \partial_t u,v\rangle_{L^2(\Omega)}+\langle \gamma\partial_t u,v\rangle_{L^2(\Omega)}+\langle (-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^2(\R^n)}+\langle qu,v\rangle_{L^2(\Omega)}\\
&=\langle F,v\rangle_{L^2(\Omega)}
\end{split}
\end{equation}
for all $v\in \widetilde{H}^s(\Omega)$ in the sense of distributions on $(0,T)$. Moreover, the unique solution $u$ obeys the energy identity
\begin{equation}
\label{eq: energy identity}
\begin{split}
&\|\partial_t u(t)\|_{L^2(\Omega)}^2+\|(-\Delta)^{s/2}u(t)\|_{L^2(\R^n)}^2+2 \int_0^t \langle \gamma\partial_t u(\tau)+qu(\tau),\partial_t u(\tau)\rangle_{L^2(\Omega)}\,d\tau\\
&=\|(-\Delta)^{s/2}u_0\|_{L^2(\R^n)}^2+\|u_1\|_{L^2(\Omega)}^2+2 \int_0^t\langle F(\tau),\partial_tu (\tau)\rangle_{L^2(\Omega)}\,d\tau
\end{split}
\end{equation}
which implies
\begin{equation}
\label{eq: energy estimate}
\begin{split}
&\|\partial_t u(t)\|_{L^2(\Omega)}+\|(-\Delta)^{s/2}u(t)\|_{L^2(\R^n)}\\
&\leq C(\|u_1\|_{L^2(\Omega)}+\|(-\Delta)^{s/2}u_0\|_{L^2(\R^n)}+\|F\|_{L^2(0,t;L^2(\Omega))})
\end{split}
\end{equation}
for all $0\leq t\leq T$ and some $C>0$ only depending on $\|q\|_{L^p(\Omega)}$, $\|\gamma\|_{L^{{\infty}(\Omega)}}$ and $T>0$.
\end{theorem}
\begin{proof}
Throughout the proof, we endow $\widetilde{H}^s(\Omega)$ with the equivalent norm $\|u\|_{\widetilde{H}^s(\Omega)}=\|(-\Delta)^{s/2}u\|_{L^2(\R^n)}$ (see \cite[Lemma 2.3]{Semilinear-nonlocal-wave-eq}) and we introduce the following continuous sesquilinear forms
\begin{equation}
\label{eq: sesquilinear forms}
a_0(u,v)=\langle (-\Delta)^{s/2}u,(-\Delta)^{s/2}v\rangle_{L^2(\R^n)},\quad a_1(u,v)= \langle qu,v\rangle_{L^2(\Omega)}
\end{equation}
for $u,v\in \widetilde{H}^s(\Omega)$ and
\begin{equation}
b(u,v)=\langle \gamma u,v\rangle_{L^2(\Omega)}
\end{equation}
for $u,v\in \widetilde{L}^2(\Omega)$. Next, recall that by \cite[eq.~(3.7)]{Semilinear-nonlocal-wave-eq} one has
\begin{equation}
\label{eq: L2 estimate potential}
\|qu\|_{L^2(\Omega)}\leq C\|q\|_{L^p(\Omega)}\|u\|_{\widetilde{H}^s(\Omega)}
\end{equation}
for all $u\in \widetilde{H}^s(\Omega)$. It is not hard to see that we can invoke the existence and uniqueness results \cite[Chapter XVIII, \S 5, Theorem~3 \& 4]{DautrayLionsVol5} (see \cite[p.~571]{DautrayLionsVol5}), which ensure the existence of a unique, real-valued solution $u\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ to \eqref{eq: damped nonlocal wave equations well-posedness weak sol}. Furthermore, by \cite[Chapter XVIII, \S 5, Lemma~7]{DautrayLionsVol5} the solution $ u$ satisfies the following energy identity
\begin{equation}
\label{eq: energy identity proof}
\begin{split}
&\|\partial_t u(t)\|_{L^2(\Omega)}^2+\|(-\Delta)^{s/2}u(t)\|_{L^2(\R^n)}^2+2 \int_0^t \langle \gamma\partial_t u(\tau)+qu(\tau),\partial_t u(\tau)\rangle_{L^2(\Omega)}\,d\tau\\
&=\|(-\Delta)^{s/2}u_0\|_{L^2(\R^n)}^2+\|u_1\|_{L^2(\Omega)}^2+2 \int_0^t\langle F(\tau),\partial_tu (\tau)\rangle_{L^2(\Omega)}\,d\tau
\end{split}
\end{equation}
for $0\leq t\leq T$. Hence, we have shown the identity \eqref{eq: energy identity}. Let us define $\Psi \in C([0,T])$ by
\[
\Psi(t)\vcentcolon = \|\partial_t u(t)\|_{L^2(\Omega)}^2+\|(-\Delta)^{s/2}u(t)\|_{L^2(\R^n)}^2
\]
for $0\leq t\leq T$. Using \eqref{eq: L2 estimate potential}, \eqref{eq: energy identity proof} and $\gamma\in L^{\infty}(\Omega)$, we get
\[
\begin{split}
&\Psi(t)\leq \Psi(0)+\int_0^t \|F(\tau)\|_{L^2(\Omega)}^2\,d\tau+ C\int_0^t (1+\|\gamma\|_{L^{\infty}(\Omega)}+\|q\|^2_{L^p(\Omega)})\Psi(\tau)\,d\tau
\end{split}
\]
and via Gronwall's inequality we deduce the energy estimate
\begin{equation}
\begin{split}
&\Psi(t)\leq C(\Psi(0)+\|F\|_{L^2(0,t;L^2(\Omega))}^2)
\end{split}
\end{equation}
for all $0\leq t\leq T$ and some $C>0$ only depending on $\|q\|_{L^p(\Omega)}$, $\|\gamma\|_{L^{\infty}(\Omega)}$ and $T>0$. This establishes the estimate \eqref{eq: energy estimate}.
\end{proof}
As a consequence we have the following result:
\begin{proposition}[Weak solutions to inhomogeneous DNWEQ]
\label{prop: Weak solutions to inhomogeneous DNWEQ}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;\widetilde{L}^2(\Omega))$, exterior condition $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and initial conditions $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(0)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(0)\in \widetilde{L}^2(\Omega)$, there exists a unique weak solution $u\in C([0,T];H^s(\R^n))\cap C^1([0,T];L^2(\R^n))$ of
\begin{equation}
\label{eq: damped nonlocal wave equations well-posedness weak sol inhom}
\begin{cases}
(L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
which means that $u$ satisfies \eqref{eq: weak formulation damped nonlocal wave eq}, the $(u(0),\partial_t u(0))=(u_0,u_1)$ in $H^s(\R^n)\times L^2(\R^n)$ and $u=\varphi$ in $(\Omega_e)_T$ means that $u(t)=\varphi(t)$ a.e. in $\Omega_e$ for any $0<t<T$. Furthermore, the following energy estimate holds
\begin{equation}
\label{eq: energy estimate inhomogeneous}
\begin{split}
&\|\partial_t u(t)\|_{L^2(\R^n)}+\|(-\Delta)^{s/2}u(t)\|_{L^2(\R^n)}\\
&\leq C(\|u_1\|_{L^2(\R^n)}+\|(-\Delta)^{s/2}u_0\|_{L^2(\R^n)}+\|\varphi\|_{C^2([0,t];H^{2s}(\R^n))}+\|F\|_{L^2(0,t;L^2(\Omega))})
\end{split}
\end{equation}
for any $0\leq t\leq T$.
\end{proposition}
\begin{proof}
Observe, under the current regularity assumptions and compatibility conditions, that $u$ solves \eqref{eq: damped nonlocal wave equations well-posedness weak sol inhom} if and only if $w\vcentcolon = u-\varphi$ solves
\begin{equation}
\label{eq: Usual cauchy homogeneous}
\begin{cases}
(L_{\gamma}+q)w = F-(L_\gamma+q) \varphi & \text{ in } \Omega_T,\\
w =0 & \text{ on } (\Omega_e)_T,\\
w(0) = u_0-\varphi(0),\, \partial_{t}w(0) = u_1-\partial_t\varphi(0) & \text{ on } \Omega.
\end{cases}
\end{equation}
The only fact to keep in mind is that if $u\in C([0,T];H^s(\R^n))$, then the condition $u(t)=\varphi(t)$ a.e. in $\Omega_e$ is equivalent to $u(t)-\varphi(t)\in \widetilde{H}^s(\Omega)$ as $\Omega\subset\R^n$ is a bounded Lipschitz domain. So, the assertions of Propsition \ref{prop: Weak solutions to inhomogeneous DNWEQ} follow immediately from Theorem \ref{thm: weak sol to hom DNWEQ}.
\end{proof}
Next, let us define for any $g\in L^1_{loc}(V_T)$, $V\subset\R^n$ open, its \emph{time-reversal}
\begin{equation}
\label{eq: time reversal}
g^\star(x,t)=g(x,T-t).
\end{equation}
Then, we have the following lemma.
\begin{lemma}
\label{lemma: time reversal of solution}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$. Let $F\in L^2(0,T;\widetilde{L}^2(\Omega))$, $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(0)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(0)\in \widetilde{L}^2(\Omega)$. Then $u$ solves
\begin{equation}
\label{eq: Usual cauchy}
\begin{cases}
(L_{\gamma}+q)u = F & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
if and only if $u^\star$ solves
\begin{equation}
\label{eq: time reversed cauchy}
\begin{cases}
(L_{-\gamma}+q)v = F^\star & \text{ in } \Omega_T,\\
v =\varphi^\star & \text{ on } (\Omega_e)_T,\\
v(T) = u_0,\, \partial_{t}v(T) = u_1 & \text{ on } \Omega.
\end{cases}
\end{equation}
In particular, for any $F\in L^2(\Omega_T)$, $\varphi\in C^2([0,T];H^{2s}(\R^n))$ and $(u_0,u_1)\in H^s(\R^n)\times L^2(\R^n)$ satisfying the compatibility conditions $u_0-\varphi(T)\in \widetilde{H}^s(\Omega)$ and $u_1-\partial_t\varphi(T)\in \widetilde{L}^2(\Omega)$, there exists a unique solution $u^\star$ of
\begin{equation}
\label{eq: backwards damped, nonlocal wave equation}
\begin{cases}
(L_{-\gamma}+q)v = F & \text{ in } \Omega_T,\\
v =\varphi & \text{ on } (\Omega_e)_T,\\
v(T) = u_0,\, \partial_{t}v(T) = u_1 & \text{ on } \Omega.
\end{cases}
\end{equation}
\end{lemma}
\begin{proof}
First, note that by the proof of Proposition \ref{prop: Weak solutions to inhomogeneous DNWEQ}, we can assume without loss of generality that $\varphi=0$. Secondly, one easily sees that $\partial_t u^\star=-(\partial_t u)^\star$ and thus a change of variables in \eqref{eq: weak formulation damped nonlocal wave eq} gives the asserted equivalence. The unique solvability of \eqref{eq: backwards damped, nonlocal wave equation} follows from the equivalence and Theorem \ref{thm: weak sol to hom DNWEQ}.
\end{proof}
\subsection{Very weak solutions}
Let us start by making some simple observations. Suppose that $u$ and $v$ are smooth solutions of the problems
\begin{equation}
\label{eq: motivation very weak solutions 1}
\begin{cases}
(L_\gamma+q) u = F & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
and
\begin{equation}
\label{eq: motivation very weak solutions 2}
\begin{cases}
(L_{-\gamma}+q) v = G & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega,
\end{cases}
\end{equation}
respectively. If we multiply the PDE \eqref{eq: motivation very weak solutions 1} by $v$ and integrate over $\Omega_T$, then we get
\begin{equation}
\label{eq: formal computation}
\begin{split}
&\int_{\Omega_T}Fv\,dxdt=\int_{\Omega_T}[(L_{\gamma}+q)u]v\,dxdt\\
&= \int_{\Omega}u_0\partial_t v(0)\,dx-\int_{\Omega}u_1v(0)\,dx-\int_\Omega \gamma u_0v(0)\,dx+\int_{\Omega_T}u(L_{-\gamma}+q)v\,dxdt\\
&= \int_{\Omega}u_0\partial_t v(0)\,dx-\int_{\Omega}u_1v(0)\,dx-\int_\Omega \gamma u_0v(0)\,dx+\int_{\Omega_T}Gu\,dxdt.
\end{split}
\end{equation}
Notice that if $G\in L^2(0,T;\widetilde{L}^2(\Omega))$, $v\in L^2(0,T;\widetilde{H}^s(\Omega))$ and $(v(0),\partial_t v(0))\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, then one can make sense of the first integral and the last line in \eqref{eq: formal computation}, even in the case $F\in L^2(0,T;H^{-s}(\Omega))$, $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$ and $u\in L^2(0,T;\widetilde{L}^2(\Omega))$. Here, $H^{-s}(\Omega)\subset\distr(\Omega)$ is defined by
\[
H^{-s}(\Omega)=\{u|_\Omega\,;\,u\in H^{-s}(\R^n)\}
\]
and it can be identified with the dual space of $\widetilde{H}^s(\Omega)$, when $\Omega$ is Lipschitz. The previous computation motivates the following definition.
\begin{definition}[Very weak solutions]
\label{def: very weak solutions}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in L^{\infty}(\Omega)\times L^{p}(\Omega)$, source $F\in L^2(0,T;H^{-s}(\Omega))$ and initial conditions $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$. Then we say that $u\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is a \emph{very weak solution} of
\begin{equation}
\label{eq: def very weak}
\begin{cases}
(L_\gamma+q) u = F & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
whenever there holds\footnote{Here and below we sometimes write $\langle \cdot,\cdot\rangle$ to denote the duality pairing between $H^{-s}(\Omega)\times \widetilde{H}^s(\Omega)$.}
\begin{equation}
\label{eq: weak formulation of very weak sols}
\int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle F,v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v(0)\rangle
\end{equation}
for all $G\in L^2(0,T;\widetilde{L}^2(\Omega))$, where $v\in C([0,T];\widetilde{H}^s(\Omega))\cap C^1([0,T];\widetilde{L}^2(\Omega))$ is the unique weak solution of the adjoint equation
\begin{equation}
\label{eq: adjoint eq of def very weak}
\begin{cases}
(L_{-\gamma}+q) v = G & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega,
\end{cases}
\end{equation}
(see Theorem \ref{thm: weak sol to hom DNWEQ}).
\end{definition}
Next, let us recall the following well-posedness result of very weak solutions.
\begin{theorem}[{Very weak solutions for $\gamma=q=0$,\cite[Theorem 3.6]{Optimal-Runge-nonlocal-wave}}]
\label{thm: well-posedness very weak without damping}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$ and $0<s< 1$. Then for any given source $F\in L^2(0,T;H^{-s}(\Omega))$ and initial conditions $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$, there exists a unique solution to
\begin{equation}
\label{eq: very weak without damping and potential}
\begin{cases}
(\partial_t^2+(-\Delta)^s) u = F & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega
\end{cases}
\end{equation}
and it satisfies the following energy estimate
\begin{equation}
\label{eq: energy estimate very weak without damping and potential}
\|u(t)\|_{L^2(\Omega)}+\|\partial_t u(t)\|_{H^{-s}(\Omega)}\leq C(\|u_0\|_{L^2(\Omega)}+\|u_1\|_{H^{-s}(\Omega)}+\|F\|_{L^2(0,t;H^{-s}(\Omega))})
\end{equation}
for all $0\leq t\leq T$.
\end{theorem}
Hence, we have a well-defined solution map.
\begin{proposition}[Solution map]
\label{prop: solution map}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and let $X_s\vcentcolon =\widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$ be endowed with the usual product norm
\[
\|(u,w)\|_{X_s}\vcentcolon = (\|u\|_{L^2(\Omega)}^2+\|w\|_{H^{-s}(\Omega)}^2)^{1/2}.
\]
Then the \emph{solution map} $S\colon L^2(0,T;H^{-s}(\Omega))\to C([0,T];X_s)$ defined by
\begin{equation}
\label{eq: solution map}
S(F)\vcentcolon = (u,\partial_t u),
\end{equation}
where $u\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is the unique solution of \eqref{eq: very weak without damping and potential} with $(u_0,u_1)=0$. Moreover, the solution map is continuous and satisfies the estimate
\begin{equation}
\label{eq: continuity estimate solution map}
\|S(F)(t)\|_{X_s}\leq C\|F\|_{L^2(0,t;H^{-s}(\Omega))}
\end{equation}
for any $0\leq t\leq T$.
\end{proposition}
\begin{proof}
First of all note that the solution map $S$ is well-defined by Theorem \ref{thm: well-posedness very weak without damping}. The estimate \eqref{eq: continuity estimate solution map} follows from \eqref{eq: energy estimate very weak without damping and potential}, which together with the linearity of $S$ gives the continuity of $S$. Observe that the linearity of $S$ is a direct consequence of the unique solvability of \eqref{eq: very weak without damping and potential} and the fact that the PDE is linear.
\end{proof}
\begin{theorem}
\label{thm: general well-posedness result of very weak solutions}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s< 1$ and suppose that $\mathcal{F}\colon C([0,T];X_s)\to L^2(0,T;H^{-s}(\Omega))$ satisfies the Lipschitz estimate
\begin{equation}
\label{eq: Lipschitz estimate nonlinearity}
\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\|_{H^{-s}(\Omega)}\leq C\|U(t)-V(t)\|_{X_s}
\end{equation}
for a.e.~$0\leq t\leq T$ and $U,V\in C([0,T];X_s)$. Then for all $(u_0,u_1)\in X_s$, there exists a unique solution $u$ of
\begin{equation}
\label{eq: well-posedness nonlinear very weak}
\begin{cases}
(\partial_t^2+(-\Delta)^s) u = \mathcal{F}(u,\partial_t u) & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
that is the formula \eqref{eq: weak formulation of very weak sols} holds with $F$ replaced by $\mathcal{F}(u,\partial_t u)$ in which we test against every weak solution $v$ of the adjoint equation
\begin{equation}
\label{eq: adjoint eq nonlinear}
\begin{cases}
(\partial_t^2+(-\Delta)^s) v = G & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
with $G\in L^2(0,T;\widetilde{L}^2(\Omega))$.
\end{theorem}
\begin{proof}[Proof of Theorem \ref{thm: general well-posedness result of very weak solutions}]
Let $u_h\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ be the unique solution to
\begin{equation}
\label{eq: homogeneous part}
\begin{cases}
(\partial_t^2+(-\Delta)^s) u = 0 & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega
\end{cases}
\end{equation}
and let us set $U_h\vcentcolon =(u_h,\partial_t u_h)\in C([0,T];X_s)$. Furthermore, we define the operator $\mathcal{T}\colon C([0,T];X_s)\to C([0,T];X_s)$ as
\begin{equation}
\mathcal{T}(U)\vcentcolon = U_h+S(\mathcal{F}(U)),
\end{equation}
which is well-defined by \eqref{prop: solution map} and the properties of $\mathcal{F}$. Next, we show that $\mathcal{T}$ has a unique fixed point $U=(U_1,U_2)$.
\medskip
\noindent\textit{Step 1.~Existence.} Let $U,V\in C([0,T];X_s)$, then by linearity of $S$, \eqref{eq: continuity estimate solution map} and \eqref{eq: Lipschitz estimate nonlinearity} we get
\[
\begin{split}
\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{X_s}&=\|S(\mathcal{F}(U))(t)-S(\mathcal{F}(V))(t)\|_{X_s}\\
&=\|S(\mathcal{F}(U)-\mathcal{F}(V))(t)\|_{X_s}\\
&\leq C\|\mathcal{F}(U)-\mathcal{F}(V)\|_{L^2(0,t;H^{-s}(\Omega))}\\
&\leq C\|U-V\|_{L^2(0,t;X_s)}.
\end{split}
\]
Next, let us define the following norm on $X_s$
\begin{equation}
\label{eq: new norm on Xs}
\|U\|_{\theta}\vcentcolon = \sup_{0\leq t\leq T}\left(e^{-\theta t}\|U(t)\|_{X_s}\right)
\end{equation}
for $\theta>0$, which will be fixed in a moment. Then we have the estimate
\[
\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{X_s}\leq C\left(\int_0^te^{2\theta \tau}\,d\tau\right)^{1/2}\|U-V\|_{\theta}\leq \frac{C}{(2\theta)^{1/2}}e^{\theta t}\|U-V\|_{\theta}
\]
and hence there holds
\[
\|\mathcal{T}(U)(t)-\mathcal{T}(V)(t)\|_{\theta}\leq \frac{C}{(2\theta)^{1/2}}\|U-V\|_{\theta}.
\]
Therefore, we deduce that $\mathcal{T}$ is a strict contraction from the complete metric space $(C([0,T];X_s),\|\cdot\|_{\theta})$ to itself, when $\theta>0$ is chosen such that $C/(2\theta)^{1/2}<1$. Now, we may invoke Banach's fixed point theorem to obtain a unique fixed point $U=(u,w)$ of $\mathcal{T}$. Next, observe that the definition of the solution map $S$ and $U=\mathcal{T}(U)=U_h+S(\mathcal{F}(U))$ imply
\[
u=u_h+u_n \text{ and } w=\partial_t u,
\]
where $u_n$ solves
\begin{equation}
\label{eq: nonlinear part}
\begin{cases}
(\partial_t^2+(-\Delta)^s) v = \mathcal{F}(U) & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega.
\end{cases}
\end{equation}
Going back to the definition of very weak solutions, we see this implies that $u$ solves \eqref{eq: well-posedness nonlinear very weak}.
\medskip
\noindent\textit{Step 2.~Uniqueness.} Suppose $\Tilde{u}\in C([0,T];\widetilde{L}^2(\Omega))\cap C^1([0,T];H^{-s}(\Omega))$ is any other solution to \eqref{eq: well-posedness nonlinear very weak}, then $\bar{u}\vcentcolon = u-\widetilde{u}$ solves
\begin{equation}
\label{eq: uniquenes very weak nonlinear}
\begin{cases}
(\partial_t^2+(-\Delta)^s) v = \mathcal{F}(u,\partial_t u)-\mathcal{F}(\widetilde{u},\partial_t \widetilde{u}) & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega.
\end{cases}
\end{equation}
Thus, applying the energy estimate \eqref{eq: energy estimate very weak without damping and potential} together with the Lipschitz assumption on $\mathcal{F}$, we see that
\[
\begin{split}
\|U(t)-\widetilde{U}(t)\|^2_{X_s}&\leq C\int_0^t\|\mathcal{F}(U)(\tau)-\mathcal{F}(\widetilde{U})(\tau)\|_{H^{-s}(\Omega)}^2\,d\tau\\
&\leq C\int_0^t\|U(t)-\widetilde{U}(t)\|^2_{X_s}\,d\tau,
\end{split}
\]
where $U=(u,\partial_tu)$ and $\widetilde{U}=(\widetilde{u},\partial_t\widetilde{u})$. So, Gronwall's inequality shows that $u=\widetilde{u}$. This establishes the uniqueness assertion and we can conclude the proof.
\end{proof}
As an application of Theorem \ref{thm: general well-posedness result of very weak solutions}, we can show the unique solvability of \eqref{eq: damped nonlocal wave equations well-posedness} for rough source and initial data.
\begin{theorem}[Very weak solutions to DNWEQ]
\label{thm: very weak sol DNWEQ}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Then for any $F\in L^2(0,T;H^{-s}(\Omega))$ and $(u_0,u_1)\in \widetilde{L}^2(\Omega)\times H^{-s}(\Omega)$, there exists a unique solution of
\begin{equation}
\label{eq: well-posedness very weak DNWEQ}
\begin{cases}
(L_\gamma+q) u = F & \text{ in } \Omega_T,\\
u =0 & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega.
\end{cases}
\end{equation}
\end{theorem}
\begin{proof}
Let us define the mapping $\mathcal{F}\colon C([0,T];X_s)\to L^2(0,T;H^{-s}(\Omega))$ by
\[
\mathcal{F}(U)(t)\vcentcolon =F-\gamma w(t)-qu(t),
\]
where $U=(u,w)\in C([0,T];X_s)$. On the one hand, using the estimate \eqref{eq: L2 estimate potential} we see that for any $u\in \widetilde{L}^2(\Omega)$ one has $qu\in H^{-s}(\Omega)$ and there holds
\begin{equation}
\label{eq: continuity estimate dual potential}
\begin{split}
\|qu\|_{H^{-s}(\Omega)}
&=\sup_{\|v\|_{\widetilde{H}^s(\Omega)}\leq 1}|\langle u,qv\rangle_{L^2(\Omega)}|\\
&\leq C\|q\|_{L^p(\Omega)}\|u\|_{L^2(\Omega)}.
\end{split}
\end{equation}
On the other hand, by applying \cite[Lemma 3.1]{Stability-fractional-conductivity} and $\partial\Omega\in C^0$ we deduce that for any $v\in \widetilde{H}^s(\Omega)$ one has $\gamma v\in \widetilde{H}^s(\Omega)$ and it obeys the estimate
\begin{equation}
\label{eq: multiplication by gamma}
\|\gamma v\|_{\widetilde{H}^s(\Omega)}\leq C\|\gamma\|_{C^{0,\alpha}(\R^n)}\|v\|_{\widetilde{H}^s(\Omega)}.
\end{equation}
Thus, we can again infer from a duality argument that $H^{-s}(\Omega)\ni w\mapsto \gamma w\in H^{-s}(\Omega)$ is a continuous map satisfying
\begin{equation}
\label{eq: continuity estimate dual damping}
\begin{split}
\|\gamma w\|_{H^{-s}(\Omega)}&=\sup_{\|v\|_{\widetilde{H}^s(\Omega)}\leq 1}|\langle \gamma w,v\rangle|\\
&=\sup_{\|v\|_{\widetilde{H}^s(\Omega)}\leq 1}|\langle w,\gamma v\rangle|\\
&\leq C\|\gamma\|_{C^{0,\alpha}(\R^n)}\|w\|_{H^{-s}(\Omega)}.
\end{split}
\end{equation}
From the estimates \eqref{eq: continuity estimate dual potential} and \eqref{eq: continuity estimate dual damping}, we easily deduce that $\mathcal{F}$ is well-defined and satisfies the Lipschitz estimate
\begin{equation}
\label{eq: Lipschitz estimate application}
\|\mathcal{F}(U)(t)-\mathcal{F}(V)(t)\|_{H^{-s}(\Omega)}\leq C(\|\gamma\|_{C^{0,\alpha}(\R^n)}+\|q\|_{L^p(\Omega)})\|U(t)-V(t)\|_{X_s}
\end{equation}
for all $U,V\in C([0,T];X_s)$. Thus, we can apply Theorem \ref{thm: general well-posedness result of very weak solutions} to get the existence of a unique solution to \eqref{eq: well-posedness very weak DNWEQ} in the sense that for any $G\in L^2(0,T;\widetilde{L}^2(\Omega))$ and corresponding solution $v$ of \eqref{eq: adjoint eq nonlinear}, there holds
\begin{equation}
\label{eq: prelim def of very weak sol}
\int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle (F-\gamma \partial_t u-qu),v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}.
\end{equation}
It remains to verify that $u$ is indeed a solution of \eqref{eq: well-posedness very weak DNWEQ} in the sense of Definition \ref{def: very weak solutions}. For this purpose let $G\in L^2(0,T;\widetilde{L}^2(\Omega))$ and suppose that $v$ is the unique solution to \eqref{eq: adjoint eq of def very weak}. Hence, $v$ solves
\[
\begin{cases}
(\partial_t^2+(-\Delta)^s) v = \widetilde{G} & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(T) = 0,\, \partial_{t}v(T) = 0 & \text{ on } \Omega
\end{cases}
\]
with $\widetilde{G}=G+\gamma \partial_t v-qv\in L^2(0,T;\widetilde{L}^2(\Omega))$ (see \eqref{eq: L2 estimate potential}). Next, we claim that there holds
\begin{equation}
\label{eq: integration by parts formula}
\int_0^T \langle \gamma \partial_t u,v\rangle\,dt=-\int_0^T \langle \gamma\partial_t v,u\rangle_{L^2(\Omega)}\,dt-\langle \gamma u_0,v(0)\rangle.
\end{equation}
For this purpose, let us consider for $\eps>0$ the unique solution $v_\eps\in H^1(0,T;\widetilde{H}^s(\Omega))$ with $\partial_t^2 v_\eps\in L^2(0,T;H^{-s}(\Omega))$ to the following parabolically regularized problem
\begin{equation}
\label{eq: regularization for equation of w}
\begin{cases}
(\partial_t^2 -\eps (-\Delta)^s \partial_t +(-\Delta)^s)v = \widetilde{G}&\text{ in }\Omega_T,\\
v=0 &\text{ in }(\Omega_e)_T,\\
v(T)= \partial_t v(T)=0 &\text{ in }\Omega
\end{cases}
\end{equation}
(see \cite[Chapter XVIII, Section 5.3.1]{DautrayLionsVol5}). By \cite[Chapter XVIII, Section 5.3.4]{DautrayLionsVol5} we know that there holds
\begin{equation}
\label{eq: convergence}
\begin{split}
v_{\eps}&\weakstar v \text{ in }L^{\infty}(0,T;\widetilde{H}^s(\Omega)),\\
\partial_t v_{\eps}&\weakstar \partial_t v \text{ in }L^{\infty}(0,T;\widetilde{L}^2(\Omega)),\\
v_\eps(t)&\to v(t) \text{ in } \widetilde{H}^s(\Omega)\text{ for all }0\leq t\leq T.
\end{split}
\end{equation}
First, note that the conditions $u\in C^1([0,T];H^{-s}(\Omega))$ and $v_\eps\in C^1([0,T];\widetilde{L}^2(\Omega))$, where the latter follows from the Sobolev embedding, guarantee that $\langle \gamma u,v_\eps\rangle\in C^1([0,T])$ with
\begin{equation}
\label{eq: product rule}
\partial_t \langle \gamma u,v_\eps\rangle=\langle \partial_t u,\gamma v_\eps\rangle+\langle u,\gamma \partial_t v_\eps\rangle_{L^2(\Omega)}.
\end{equation}
Thus, by the fundamental theorem of calculus we deduce that there holds
\[
\langle \gamma u(T),v_\eps(T)\rangle-\langle \gamma u_0,v_\eps(0)\rangle=\int_0^T\langle \partial_t u,\gamma v_\eps\rangle+\langle u,\gamma \partial_t v_\eps\rangle_{L^2(\Omega)}\,dt.
\]
By the convergence assertions \eqref{eq: convergence} and $v_\eps (T)=0$, we get
\[
-\langle \gamma u_0,v(0)\rangle=\int_0^T\langle \partial_t u,\gamma v\rangle+\langle u,\gamma \partial_t v\rangle_{L^2(\Omega)}\,dt.
\]
This proves \eqref{eq: integration by parts formula}. Hence, inserting this into \eqref{eq: prelim def of very weak sol}, we obtain
\[
\begin{split}
\int_0^T\langle\widetilde{G},u\rangle_{L^2(\Omega)}\,dt&=\int_0^T\langle (F-\gamma \partial_t u-qu),v\rangle\,dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}\\
&=\int_0^T\langle F,v\rangle\,dt+\int_0^T \langle u,\gamma \partial_t v\rangle_{L^2(\Omega)}\,dt-\int_0^T\langle u,qv\rangle\,dt\\
&\quad+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v(
0)\rangle.
\end{split}
\]
As $\widetilde{G}=G+\gamma \partial_t v-qv$ this gives
\[
\begin{split}
\int_0^T \langle G,u\rangle_{L^2(\Omega)}\,dt&=\int_0^T \langle F,v\rangle\, dt+\langle u_1,v(0)\rangle-\langle u_0,\partial_t v(0)\rangle_{L^2(\Omega)}+\langle \gamma u_0,v(0)\rangle.
\end{split}
\]
Hence, we observe that $u$ is indeed a solution of \eqref{eq: well-posedness very weak DNWEQ} in the sense of Definition \ref{def: very weak solutions}. By reversing the above arguments one can also observe that if $u$ is a solution in the sense of Definition \ref{def: very weak solutions}, then by \eqref{eq: integration by parts formula} it is a solution in the sense of \eqref{eq: prelim def of very weak sol} and thus the solution in the sense of Definition \ref{def: very weak solutions} is unique.
\end{proof}
\section{The inverse problem}
\label{sec: inverse problem}
After establishing the theory of very weak solutions to damped, nonlocal wave equations, we now turn our attention to the inverse problem part. First, in Section \ref{sec: Runge approx} we prove the optimal Runge approximation theorem (Theorem \ref{thm: Runge approx}) and in Section \ref{sec: integral identity} a suitable integral identity. Using these results, we then show in Section \ref{sec: linear uniqueness} our first main result dealing with linear perturbations (Theorem \ref{thm: uniqueness linear}). Finally, in Section \ref{sec: semilinear uniqueness} we prove Theorem \ref{thm: uniqueness semilinear} showing that the damping coefficient and the nonlinearity can be determined simultaneously.
\subsection{Runge approximation}
\label{sec: Runge approx}
With the material from Section \ref{sec: Very weak solutions to damped, nonlocal wave equations} at our disposal, we can now show the following Runge approximation theorem, whose proof is very similar to the one of \cite[Theorem 1.2]{Optimal-Runge-nonlocal-wave}.
\begin{theorem}[Runge approximation]
\label{thm: Runge approx}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Then for any measurement set $W\subset\Omega_e$ and initial conditions $(u_0,u_1)\in \widetilde{H}^s(\Omega)\times\widetilde{L}^2(\Omega)$, the \emph{Runge set}
\begin{equation}
\label{eq: Runge set}
\mathcal{R}^{u_0,u_1}_W\vcentcolon = \{u_\varphi-\varphi\,;\,\varphi\in C_c^{\infty}(W_T)\}
\end{equation}
is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$, where $u_\varphi$ is the unique solution to
\begin{equation}
\label{eq: PDE Runge}
\begin{cases}
(L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = u_0,\, \partial_{t}u(0) = u_1 & \text{ on } \Omega,
\end{cases}
\end{equation}
(see Proposition \ref{prop: Weak solutions to inhomogeneous DNWEQ}).
\end{theorem}
\begin{proof}
First of all note that it is enough to consider the case $(u_0,u_1)=0$. To see this assume that the density holds for $\mathcal{R}_W\vcentcolon =\mathcal{R}_W^{0,0}$ and let $f\in L^2(0,T;\widetilde{H}^s(\Omega))$. Let $v_0$ be the unique solution to
\begin{equation}
\begin{cases}
(L_{\gamma}+q)v = 0 & \text{ in } \Omega_T,\\
v =0 & \text{ on } (\Omega_e)_T,\\
v(0) = u_0,\, \partial_{t}v(0) = u_1 & \text{ on } \Omega
\end{cases}
\end{equation}
and define $\widetilde{f}\vcentcolon = f-v_0\in L^2(0,T;\widetilde{H}^s(\Omega))$. By assumption there exists $(\varphi_k)_{k\in\N}\subset C_c^{\infty}(W_T)$ such that $u_k-\varphi_k\to \widetilde{f}$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ as $k\to \infty$, where $u_k$ is the unique solution to
\begin{equation}
\label{eq: PDE Runge vanishing initial}
\begin{cases}
(L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
with $\varphi=\varphi_k$. Then $v_k\vcentcolon = u_k+v_0$ is the unique solution to \eqref{eq: PDE Runge} with $\varphi=\varphi_k$. The above convergence now implies $v_k-\varphi_k\to f$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ as $k\to\infty$ and we get that $\mathcal{R}_W^{u_0,u_1}$ is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$.
Therefore, it remains to show that $\mathcal{R}_W$ is dense in $L^2(0,T;\widetilde{H}^s(\Omega))$. As usual, we show this by a Hahn--Banach argument. Thus, suppose that $F\in L^2(0,T;H^{-s}(\Omega))$ vanishes on $\mathcal{R}_W$. Let us recall that if $\varphi\in C_c^{\infty}(W_T)$ and $u$ solves \eqref{eq: PDE Runge vanishing initial}, then by \eqref{eq: Usual cauchy homogeneous} and Lemma \ref{lemma: time reversal of solution} the function $v=(u-\varphi)^\star$ satisfies
\begin{equation}
\begin{cases}
(L_{-\gamma}+q)v = -(-\Delta)^s\varphi^\star &\text{ in }\Omega_T,\\
v=0 &\text{ in }(\Omega_e)_T,\\
v(T)=\partial_t v(T)=0 &\text{ in }\Omega.
\end{cases}
\end{equation}
Next, let $w$ be the unique solution to
\begin{equation}
\label{eq: solution to weak RHS}
\begin{cases}
(L_{\gamma}+q)w= F^\star &\text{ in }\Omega_T,\\
w=0 &\text{ in }(\Omega_e)_T,\\
w(0)= \partial_t w(0)=0 &\text{ in }\Omega
\end{cases}
\end{equation}
(see Theorem \ref{thm: very weak sol DNWEQ}). By testing the equation for $w$ by $v$, we get
\[
-\int_0^T \langle (-\Delta)^s\varphi^\star, w\rangle_{L^2(\Omega)}\,dt=\int_0^T\langle F^\star,v\rangle\,dt=\int_0^T\langle F,u_\varphi-\varphi\rangle\,dt=0
\]
for any $\varphi\in C_c^{\infty}(W_T)$. This ensures that there holds
\[
(-\Delta)^s w=0\quad \text{ in }W_T.
\]
Furthermore, by construction $w$ vanishes in $(\Omega_e)_T$ and hence the unique continuation principle for the fractional Laplacian guarantees $w=0$ in $\R^n_T$ (see \cite{GSU20}). As very weak solutions are distributional solutions, we get
\[
\int_0^T\langle F^\star,\Phi\rangle\,dt=\int_0^T \langle (L_{-\gamma}+q)\Phi,w\rangle_{L^2(\Omega)}\,dt=0
\]
for all $\Phi\in C_c^{\infty}(\Omega_T)$. To see that very weak solutions are distributional solutions, one can simply take $G=\chi_\Omega(L_\gamma+q)\Phi$ with $\Phi\in C_c^{\infty}(\Omega\times [0,T))$ in Definition \ref{def: very weak solutions}, where $\chi_\Omega$ denotes the characteristic function of $\Omega$ (see also \cite[Proposition 3.8]{Optimal-Runge-nonlocal-wave}). By density of $C_c^{\infty}(\Omega_T)$ in $L^2(0,T;\widetilde{H}^s(\Omega))$ we deduce that $F=0$.
This concludes the proof.
\end{proof}
As a consequence we have the following lemma.
\begin{lemma}[Convergence of time derivative]
\label{lemma: convergence of time derivatives}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma,q)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$. Let $\Phi,\Psi\in L^2(0,T;\widetilde{H}^s(\Omega))\cap H^1(0,T;H^{-s}(\Omega))$ and suppose $(\varphi_k)_{k\in\N}\subset C_c^{\infty}((\Omega_e)_T)$ is such that
\begin{equation}
\label{eq: convergence assertion for time derivative}
u_k-\varphi_k\to \Phi\text{ in }L^2(0,T;\widetilde{H}^s(\Omega))\text{ as }k\to\infty,
\end{equation}
where $u_k$ solves
\begin{equation}
\label{eq: approx time derivative}
\begin{cases}
(L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\
u =\varphi_k & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
for $k\in\N$. If $\Phi,\Psi$ satisfy one of the conditions
\begin{enumerate}[(a)]
\item \label{item cond 1 first} $\Psi(T)=\Phi(0)=0$
\item \label{item cond 2 first} or $\Psi(T)=\Psi(0)=0$,
\end{enumerate}
then we have
\begin{equation}
\label{eq: first order limit}
\lim_{k\to\infty}\int_0^T\langle \partial_t (u_k-\varphi_k),\Psi\rangle \,dt =\int_0^T\langle \partial_t\Phi,\Psi\rangle\,dt.
\end{equation}
\end{lemma}
\begin{remark}
Let us note that the same formula \eqref{eq: first order limit} holds for second order time derivatives under appropriate conditions.
\end{remark}
\begin{proof}
Using the integration by parts formula, we may compute
\[\small
\begin{split}
&\lim_{k\to\infty}\int_0^T\langle \partial_t (u_k-\varphi_k),\Psi\rangle \,dt \\
&= \lim_{k\to\infty}\left(\langle (u_k-\varphi_k)(T),\Psi(T)\rangle_{L^2(\Omega)}-\langle (u_k-\varphi_k)(0),\Psi(0)\rangle_{L^2(\Omega)}-\int_0^T\langle \partial_t\Psi,u_k-\varphi_k\rangle \,dt\right)\\
&=-\lim_{k\to\infty}\int_0^T\langle \partial_t\Psi,u_k-\varphi_k\rangle \,dt\\
&=-\int_0^T \langle \partial_t \Psi,\Phi\rangle \, dt\\
&=\langle \Phi(0),\Psi(0)\rangle_{L^2(\Omega)}-\langle \Phi(T),\Psi(T)\rangle_{L^2(\Omega)}+\int_0^T \langle \partial_t\Phi, \Psi\rangle \, dt\\
&=\int_0^T \langle \partial_t\Phi, \Psi\rangle \, dt.
\end{split}
\]
In the first equality sign we used an integration by parts, in the second equality we used \eqref{eq: approx time derivative}, $\Psi(T)=0$ and \eqref{eq: approx time derivative}, in the third equality the convergence \eqref{eq: convergence assertion for time derivative}, in the fourth equality again an integration by parts and finally in the last equality the conditions \ref{item cond 1 first} or \ref{item cond 2 first}.
\end{proof}
\subsection{DN map and integral identities}
\label{sec: integral identity}
Next, we define the \emph{Dirichlet to Neumann (DN) map} $\Lambda_{\gamma,q}$ related to
\begin{equation}
\label{eq: PDE for integral identity}
\begin{cases}
(L_{\gamma}+q)u = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega,
\end{cases}
\end{equation}
via
\begin{equation}
\label{eq: DN map}
\langle \Lambda_{\gamma,q}\varphi,\psi\rangle=\int_{\R^n_T}(-\Delta)^{s/2} u_\varphi (-\Delta)^{s/2}\psi\,dx
\end{equation}
for all $\varphi,\psi\in C_c^{\infty}((\Omega_e)_T)$, where $u_\varphi$ is the unique solution to \eqref{eq: PDE for integral identity} with exterior condition $\varphi$. Using the above preparation, we now establish the following integral identity.
\begin{proposition}[Integral identity for linear perturbations]
\label{prop: integral identity}
Let $\Omega \subset\R^n$ be a bounded Lipschitz domain, $T>0$, $0<s<\alpha\leq 1$ and suppose that $1\leq p\leq \infty$ satisfies \eqref{eq: restrictions on p}. Assume that we have given coefficients $(\gamma_j,q_j)\in C^{0,\alpha}(\R^n)\times L^{p}(\Omega)$ for $j=1,2$. Let $\varphi_j\in C_c^{\infty}((\Omega_e)_T)$ and denote by $u_j$ the corresponding solution of \eqref{eq: PDE for integral identity} with $(\gamma,q)=(\gamma_j,q_j)$. Then there holds
\begin{equation}
\label{eq: integral identity}
\begin{split}
&\langle (\Lambda_{\gamma_1,q_1}-\Lambda_{\gamma_2,q_2})\varphi_1,\varphi_2^\star\rangle\\
&\quad =\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
Let $(\Gamma_j,Q_j)\in C^{0,\alpha}(\R^n)\times L^p(\Omega)$, $j=1,2$, and suppose $U_j$ is the unique solutions of \eqref{eq: PDE for integral identity} with $(\gamma,q)=(\Gamma_j,Q_j)$ and exterior condition $\varphi=\psi_j$. Then we may compute
\begin{equation}
\label{eq: calculation for integral identity}
\begin{split}
&\int_{\Omega_T}\{[(\Gamma_1-\Gamma_2)\partial_t+Q_1-Q_2](U_1-\psi_1)\}(U_2-\psi_2)^\star\,dxdt\\
& =\int_{\Omega_T}\{[\Gamma_1\partial_t+Q_1](U_1-\psi_1)\}(U_2-\psi_2)^\star\,dxdt\\
&\quad -\int_{\Omega_T}(U_1-\psi_1)[- \Gamma_2\partial_t+Q_2](U_2-\psi_2)^\star\,dxdt\\
&=\int_{0}^T\langle L_{\Gamma_1,Q_1}(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt\\
&\quad -\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt\\
&\quad -\int_{0}^T\langle L_{-\Gamma_2,Q_2}(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\
&\quad +\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\
&=-\int_{0}^T\langle (-\Delta)^s\psi_1,(U_2-\psi_2)^\star\rangle_{L^2(\Omega)}\,dt +\int_{0}^T\langle (-\Delta)^s \psi_2^\star,(U_1-\psi_1)\rangle_{L^2(\Omega)}\,dt\\
&=\int_{\R^n_T} ((-\Delta)^s\psi_2^\star) U_1\,dxdt-\int_{\R^n_T} ((-\Delta)^s\psi_1) U_2^\star\,dxdt\\
&=\langle \Lambda_{\Gamma_1,Q_1}\psi_1,\psi_2^\star\rangle-\langle \Lambda_{\Gamma_2,Q_2}\psi_2,\psi_1^\star\rangle.
\end{split}
\end{equation}
In the first equality we used that $U_1-\psi_1$ has vanishing initial conditions, $(U_2-\psi_2)^\star$ has vanishing terminal conditions and an integration by parts. In the third equality we used that the PDEs for $U_1-\psi_1$ and $(U_2-\psi_2)^\star$ hold in the sense of $L^2(0,T;H^{-s}(\Omega))=(L^2(0,T;\widetilde{H}^s(\Omega))'$ (see Lemma \ref{lemma: time reversal of solution}). In the fourth equality, we used the PDEs for $U_1$ and $U_2$, Lemma \ref{lemma: time reversal of solution} and that there holds
\[
\begin{split}
&\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_2-\psi_2)^\star,(U_1-\psi_1)\rangle\,dt\\
&=\int_{0}^T\langle (\partial_t^2+(-\Delta)^s)(U_1-\psi_1),(U_2-\psi_2)^\star\rangle\,dt,
\end{split}
\]
which can be established similarly as \cite[Claim 4.2]{Semilinear-nonlocal-wave-eq} (see also the proof of Theorem \ref{thm: very weak sol DNWEQ}). In the last equality, we have made the change of variables $\tau=T-t$ for the second integral. On the one hand, using \eqref{eq: calculation for integral identity} with
\[
\Gamma_1=\Gamma_2=\gamma_j\text{ and }Q_1=Q_2=q_j,
\]
we observe that
\begin{equation}
\label{eq: self-adjointness DN map}
\langle \Lambda_{\gamma_j,q_j}\psi_1,\psi_2^\star\rangle=\langle \Lambda_{\gamma_j,q_j}\psi_2,\psi_1^\star\rangle
\end{equation}
for all $\psi_j\in C_c^{\infty}((\Omega_e)_T)$, $j=1,2$. On the other hand, choosing
\[
\Gamma_j=\gamma_j,\,Q_j=q_j\text{ and }\psi_j=\varphi_j
\]
in \eqref{eq: calculation for integral identity} and taking into account the self-adjointness \eqref{eq: self-adjointness DN map}, we get \eqref{eq: integral identity}.
\end{proof}
\subsection{Simultaneous determination of damping coefficient and linear perturbations}
\label{sec: linear uniqueness}
\begin{proof}[Proof of Theorem \ref{thm: uniqueness linear}]
First note that by the integral identity in Proposition \ref{prop: integral identity}, we may deduce from the condition \eqref{eq: equality of DN maps} that there holds
\begin{equation}
\label{eq: uniqueness proof help identity}
\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt=0
\end{equation}
for all $\varphi_j\in C_c^{\infty}((W_j)_T)$, where $u_j$ denotes the unique solution to
\begin{equation}
\label{eq: PDE uniqueness proof}
\begin{cases}
(L_{\gamma_j}+q_j)u = 0 & \text{ in } \Omega_T,\\
u =\varphi_j & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega.
\end{cases}
\end{equation}
Let $\omega\Subset \Omega$ and choose a cutoff function $\Phi_1\in C_c^{\infty}(\Omega)$ satisfying $\Phi_1=1$ on $\omega$. Moreover, let $\Phi_2\in C_c^{\infty}(\omega_T)$. By the Runge approximation (Theorem \ref{thm: Runge approx}), there exist sequences $(\varphi_j^k)_{k\in\N}\subset C_c^{\infty}((W_j)_T)$ with corresponding solutions $u_j^k$ of \eqref{eq: PDE uniqueness proof} with $\varphi_j=\varphi_j^k$ such that $u_j^k-\varphi_j^k\to \Phi_j$ in $L^2(0,T;\widetilde{H}^s(\Omega))$.
Taking $\varphi_1=\varphi_1^k$ and $\varphi_2=\varphi_2^{\ell}$ in \eqref{eq: uniqueness proof help identity} gives
\[
\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u^k_1-\varphi^k_1)\}(u^{\ell}_2-\varphi^{\ell}_2)^\star\,dxdt=0
\]
for all $k,\ell\in\N$. First, we let $\ell\to\infty$ to deduce
\begin{equation}
\label{eq: uniqueness proof help identity 2}
\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t+q_1-q_2](u^k_1-\varphi^k_1)\}\Phi_2^\star\,dxdt=0
\end{equation}
for all $k\in\N$. As $\gamma_1-\gamma_2\in C^{0,\alpha}(\R^n)$ the estimate \eqref{eq: multiplication by gamma} ensures that we can apply Lemma \ref{lemma: convergence of time derivatives} under the condition \ref{item cond 2 first} and so $\partial_t\Phi_1=0$ shows that the first term in \eqref{eq: uniqueness proof help identity 2} goes to zero. So in the limit $k\to\infty$ what remains is
\[
\int_{\Omega_T}(q_1-q_2)\Phi_2^\star \,dxdt=0,
\]
where we used $\Phi_1=1$ on $\omega$. This ensures that $q_1=q_2$ on $\omega$. As the set $\omega$ is arbitrary, we get $q_1=q_2$ in $\Omega$. Now, the identity \eqref{eq: uniqueness proof help identity} reduces to
\[
\int_{\Omega_T}\{[(\gamma_1-\gamma_2)\partial_t](u_1-\varphi_1)\}(u_2-\varphi_2)^\star\,dxdt=0
\]
for all $\varphi_j\in C_c^{\infty}((W_j)_T)$. We choose $\eta\in C_c^{\infty}(\Omega_T)$, define
\[
\Phi_1(x,t)=\int_0^t\eta(x,\tau)\,d\tau\in C_c^{\infty}(\Omega\times (0,T])
\]
and take $\Phi_2\in C_c^{\infty}(\Omega_T)$. Then using $\partial_t \Phi_1=\eta$ and arguing as above via a Runge approximation and Lemma \ref{lemma: convergence of time derivatives}, we get from \eqref{eq: uniqueness proof help identity} the identity
\[
\int_{\Omega_T}(\gamma_1-\gamma_2)\eta\Phi_2^\star\,dxdt=0.
\]
This again implies $\gamma_1=\gamma_2$ in $\Omega$.
\end{proof}
\subsection{Simultaneous determination of damping coefficient and nonlinearity}
\label{sec: semilinear uniqueness}
Before turning to the proof of our second main result, let us recall that the \emph{DN map} related to the problem
\begin{equation}
\label{eq: PDE for semilinear problem}
\begin{cases}
L_{\gamma}u+f(u) = 0 & \text{ in } \Omega_T,\\
u =\varphi & \text{ on } (\Omega_e)_T,\\
u(0) = 0,\, \partial_{t}u(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
is defined by
\begin{equation}
\langle \Lambda_{\gamma,f}\varphi,\psi\rangle\vcentcolon = \int_{\R^n_T}(-\Delta)^{s/2}u_\varphi(-\Delta)^{s/2}\psi\,dxdt,
\end{equation}
where $\varphi,\psi\in C_c^{\infty}((\Omega_e)_T)$ and $u_\varphi$ is the unique solution to \eqref{eq: PDE for semilinear problem} (see \cite[Proposition 3.7]{Semilinear-nonlocal-wave-eq}).
\begin{proof}[Proof of Theorem \ref{thm: uniqueness semilinear}]
Let $\eps>0$ and denote by $u^{(j)}_\eps$ the unique solutions to \eqref{eq: PDE for semilinear problem} with $f=f_j$, $\gamma=\gamma_j$ and $\varphi=\eps \eta$ for some fixed $\eta\in C_c^{\infty}((W_1)_T)$. Let us observe that the UCP for the fractional Laplacian and the condition \eqref{eq: equality of DN maps semilinear} imply that $u_\eps\vcentcolon = u^{(1)}_\eps=u^{(2)}_\eps$. Next, let us note that we can write
\begin{equation}
\label{eq: decomposition of u eps}
u_\eps=\eps v_j+R^{(j)}_\eps
\end{equation}
for $j=1,2$, where $v_j$ and $R^{(j)}_\eps$ are the unique solutions of
\begin{equation}
\label{eq: linear part of u eps}
\begin{cases}
L_{\gamma_j}v = 0 & \text{ in } \Omega_T,\\
v =\eta & \text{ on } (\Omega_e)_T,\\
v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega
\end{cases}
\end{equation}
and
\begin{equation}
\label{eq: nonlinear part of u eps}
\begin{cases}
L_{\gamma_j}R = -f_j(u_\eps) & \text{ in } \Omega_T,\\
R =0 & \text{ on } (\Omega_e)_T,\\
R(0) = 0,\, \partial_{t}R(0) = 0 & \text{ on } \Omega,
\end{cases}
\end{equation}
respectively. This simply follows from the unique solvability of \eqref{eq: PDE for semilinear problem} and both functions $u_\eps$ and $\eps v_j+R^{(j)}_\eps$ are solutions. Furthermore, we notice that the energy estimate of \cite[Theorem 3.1]{Semilinear-nonlocal-wave-eq}, \cite[eq.~(3.18)]{Semilinear-nonlocal-wave-eq} and the $r+1$ homogeneity of $f_j$ ensure that $R^{(j)}_\eps$ satisfies
\begin{equation}
\label{eq: energy estimate remainder}
\begin{split}
\|\partial_t R^{(j)}_\eps\|_{L^{\infty}(0,T;L^2(\Omega))}+\|R^{(j)}_\eps (t)\|_{L^{\infty}(0,T;H^s(\R^n))}&\lesssim \|f_j(u_\eps)\|_{L^2(\Omega_T)}\\
&\lesssim \|u_\eps\|^{r+1}_{L^{\infty}(0,T;H^s(\R^n))}.
\end{split}
\end{equation}
Moreover, we may estimate
\begin{equation}
\label{eq: estimate u eps}
\begin{split}
&\|\partial_t u_\eps\|_{L^{\infty}(0,T;L^2(\R^n))}+\|u_\eps\|_{L^{\infty}(0,T;H^s(\R^n))}\\
&\lesssim \|\partial_t (u_\eps-\eps\eta)\|_{L^{\infty}(0,T;L^2(\Omega))}+\|u_\eps-\eps\eta\|_{L^{\infty}(0,T;H^s(\R^n))}+\eps\|\eta\|_{W^{1,\infty}(0,T;H^{2s}(\R^n))}\\
&\lesssim \eps\|\eta\|_{W^{1,\infty}(0,T;H^{2s}(\R^n))}.
\end{split}
\end{equation}
This follows from the following observations. If $u$ solves \eqref{eq: PDE for semilinear problem} for a damping coefficient $\gamma\in C^{0,\alpha}(\R^n)$, a weak nonlinearity $f$ and $\varphi\in C_c^{\infty}((\Omega_e)_T)$, then $v=u-\varphi$ solves
\begin{equation}
\begin{cases}
L_{\gamma}v+f(v) = -(-\Delta)^s\varphi & \text{ in } \Omega_T,\\
v =0& \text{ on } (\Omega_e)_T,\\
v(0) = 0,\, \partial_{t}v(0) = 0 & \text{ on } \Omega.
\end{cases}
\end{equation}
Now, we may invoke \cite[eq.~(3.15)]{Semilinear-nonlocal-wave-eq} to find that there holds
\[
\begin{split}
&\|\partial_t v(t)\|_{L^2(\Omega)}^2+\|v(t)\|_{H^s(\R^n)}^2\\
&\lesssim \int_0^t|\langle \gamma\partial_t v,\partial_t v\rangle_{L^2(\Omega)}|\,d\tau +\int_0^t|\langle (-\Delta)^{s}\varphi,\partial_t v\rangle_{L^2(\Omega)}|\,d\tau\\
&\lesssim \|(-\Delta)^{s}\varphi\|_{L^2(0,t;L^2(\Omega))}^2+\int_0^t\|\partial_t v\|_{L^2(\Omega)}^2\,d\tau.
\end{split}
\]
Thus, Gronwall's inequality gives
\[
\|\partial_t v(t)\|_{L^2(\Omega)}+\|v(t)\|_{H^s(\R^n)}\lesssim \|(-\Delta)^{s}\varphi\|_{L^2(0,t;L^2(\Omega))}.
\]
This ensures the validity of the second estimate in \eqref{eq: estimate u eps}.
Next, observe that by subtracting the PDEs for $u^{(1)}_\eps$ and $u^{(2)}_\eps$, we deduce that
\begin{equation}
\label{eq: condition for uniqueness semilinear}
(\gamma_1-\gamma_2)\partial_t u_\eps=f_2(u_\eps)-f_1(u_\eps)\text{ in }\Omega_T.
\end{equation}
By \eqref{eq: decomposition of u eps}, we may write
\begin{equation}
\label{eq: condition for uniqueness semilinear 2}
(\gamma_1-\gamma_2)(\eps\partial_tv_1+\partial_t R^{(1)}_\eps) =f_2(u_\eps)-f_1(u_\eps)\text{ in }\Omega_T.
\end{equation}
Combining \eqref{eq: energy estimate remainder} and \eqref{eq: estimate u eps}, we see that
\begin{equation}
\label{eq: decay estimate R eps}
\|\partial_t R^{(j)}_\eps\|_{L^{\infty}(0,T;L^2(\Omega))}+\|R^{(j)}_\eps (t)\|_{L^{\infty}(0,T;H^s(\R^n))}\lesssim \eps^{r+1}.
\end{equation}
Multiplying \label{eq: condition for uniqueness semilinear 2} by $\eps^{-1}$ gives
\begin{equation}
\label{eq: condition for uniqueness semilinear 3}
\begin{split}
&(\gamma_1-\gamma_2)(\partial_tv_1+\eps^{-1}\partial_t R^{(1)}_\eps) =f_2(\eps^{-1/(r+1)}u_\eps)-f_1(\eps^{-1/(r+1)}u_\eps)\text{ in }\Omega_T.
\end{split}
\end{equation}
Next, let us focus one the case $2s<n$ as the other one can be treated similarly. As $r>0$ we deduce from \eqref{eq: estimate u eps} that $\eps^{-1/(1+r)}u_\eps\to 0$ in $L^{\infty}(0,T;H^s(\R^n))$ and so by Sobolev's embedding in $L^q(0,T;L^{2_s^*}(\Omega))$ for all $1\leq q\leq \infty$ and $2_s^*=\frac{2n}{n-2s}$. Hence, by our assumptions on $f_j$ and \cite[Lemma 3.6]{zimmermann2024calderon}, we get
\begin{equation}
\label{eq: zero convergence nonlinearity}
f_j(\eps^{-1/(r+1)}u_\eps)\to 0\text{ in }L^{q/(r+1)}(0,T;L^{2_s^*/(r+1)}(\Omega))
\end{equation}
for all $q\geq r+1$ as $\eps\to 0$. Additionally, using \eqref{eq: decay estimate R eps} we know that
\begin{equation}
\label{eq: zero convergence time derivative remainder}
\eps^{-1}\partial_t R^{(j)}_\eps\to 0\text{ in }L^{\infty}(0,T;L^2(\Omega)).
\end{equation}
Therefore, from \eqref{eq: condition for uniqueness semilinear 3}, \eqref{eq: zero convergence nonlinearity} and \eqref{eq: zero convergence time derivative remainder}, we infer
\[
(\gamma_1-\gamma_2)\partial_t v_1=0\text{ in }\Omega_T.
\]
In particular, this ensures that there holds
\[
\int_{\Omega_T}(\gamma_1-\gamma_2)\partial_t (v_1-\eta)(w_2-\psi)^\star\,dxdt=0
\]
for any $\psi \in C_c^{\infty}((W_2)_T)$, where $w_2$ is the unique solution of
\begin{equation}
\begin{cases}
L_{\gamma_2}w = 0 & \text{ in } \Omega_T,\\
w =\psi & \text{ on } (\Omega_e)_T,\\
w(0) = 0,\, \partial_{t}w(0) = 0 & \text{ on } \Omega.
\end{cases}
\end{equation}
Now, arguing as in the previous section, we get $\gamma_1=\gamma_2$ in $\Omega$. Hence, \eqref{eq: condition for uniqueness semilinear} reduces to
\begin{equation}
f_1(u_\eps)=f_2(u_\eps)\text{ in }\Omega_T.
\end{equation}
Multiplying this identity by $\eps^{-(r+1)}$ and arguing as before, we deduce that
\[
f_1(v)=f_2(v)\text{ in }\Omega_T,
\]
where $v\vcentcolon =v_1=v_2$ as $\gamma_1=\gamma_2$. One can now show $f_1(x,\tau)=f_2(x,\tau)$ for all $x\in\Omega$ and $\tau\in\R$ exactly as described in \cite[p.~29]{Optimal-Runge-nonlocal-wave}. Hence, we can conclude the proof.
\end{proof}
\medskip
\noindent\textbf{Acknowledgments.}
P.~Zimmermann was supported by the Swiss National Science Foundation (SNSF), under grant number 214500.
\section*{Statements and Declarations}
\subsection*{Data availability statement}
No datasets were generated or analyzed during the current study.
\subsection*{Conflict of Interests} Hereby we declare there are no conflict of interests.
\bibliography{refs}
\bibliographystyle{alpha}
\end{document}
|
2412.02042v1 | http://arxiv.org/abs/2412.02042v1 | Delta invariants of plumbed 3-manifolds | \documentclass[11pt,a4paper, reqno,dvipsnames]{amsart}
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\title{$\Delta$ invariants of plumbed manifolds}
\author{Shimal Harichurn, Andr\'as N\'emethi, Josef Svoboda}
\date{\today}
\begin{document}
\maketitle
\begin{abstract}
We study the minimal $q$-exponent $\Delta$ in the BPS $q$-series $\widehat{Z}$ of negative definite plumbed 3-manifolds equipped with a spin$^c$-structure. We express $\Delta$ of Seifert manifolds in terms of an invariant commonly used in singularity theory. We provide several examples illustrating the interesting behaviour of $\Delta$ for non-Seifert manifolds. Finally, we compare $\Delta$ invariants with correction terms in Heegaard--Floer homology.
\end{abstract}
\tableofcontents
\section{Introduction}
\(\Zhat_b(Y; q)\) is a $q$-series invariant of a negative definite plumbed 3-manifold \(Y\) equipped with a \(\spc\) structure \(b\) \cite{GPPV20}. It recovers Witten--Reshetikhin--Turaev $U_q(\mathfrak{sl}_2)$-invariants in radial limits to roots of unity as conjectured by \cite{GPPV20}, and recently proved in \cite{Mur23}.
In this paper, we focus on the behavior of \(\Zhat_b(Y; q)\) near \(q=0\). In other words, we study the smallest \(q\)-exponents \(\Delta_b\) in \(\Zhat_b(Y,q)\):
\[
\Zhat_b(Y,q) = q^{\Delta_b}(c_0+c_1 q + c_2 q^2 + \dots), \quad c_0 \neq 0.
\]
The rational numbers $\Delta_b$ were studied in \cite{GPP21} where their fractional part was related to various invariants of 3-manifolds. In this work, we focus on the actual value of $\Delta_b$.
For plumbed manifolds it is natural to consider the `canonical' $\spc$ structure $can$. Related to it, there is a numerical topological invariant \(\gamma(Y) := k^2+s\) (see \ref{ss:spinc}). It appears, e.g., in the study of Seiberg--Witten invariants \cite{NeNi02} of the associated plumbed 3-manifold, and its use in topology goes back to Gompf \cite{Gompf98}. It also plays an important role in singularity theory, e.g. in Laufer's formula \cite{Laufer1977} for Milnor number of a Gorenstein normal surface singularity.
As the main result of this paper, we prove that for Seifert manifolds, \(\Delta_{can}(Y)\) can be expressed using $\gamma(Y)$:
\begin{theorem}\label{thm:seifert_hom}
Let $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ be a Seifert manifold associated with negative definite plumbing graph. Let $\emph{can}$ be the canonical $\spc$ structure of $Y$. Then $\Delta_{can}$ satisfies
\begin{equation}
\Delta_{can} = -\frac{\gamma(Y)}{4} + \frac{1}{2}.
\end{equation}
If $Y$ is not a lens space, then $\Delta_{can}$ is minimal among all ${\Delta_b, b \in \spc(Y)}$.
\end{theorem}
In the special case of Seifert homology spheres $Y=\Sigma(a_1,a_2,\dots,a_n)$, we deduce that $\Delta_{can}(Y)$ grows polynomially with the leading term given by
\[
\Delta_{can}(Y) \approx (n-2)a_1 a_2 \cdots a_n.
\]
The idea of the proof is to express $\Delta_{can}$ as a minimum of a quadratic form, given by the linking form of the plumbing graph, over certain integral vectors. The key is identifying this set over which we minimize (\cref{lm:delta_minimizer}). We show that in this set, there exists a unique vector (up to sign) that minimizes the quadratic form.
Once we leave Seifert manifolds, the computation of \(\Delta_b\) invariants becomes much more complicated. We illustrate this on plumbing graphs with exactly two vertices of degree 3 and no vertices of degree $\geq 4$ in \cref{s:H_shaped}. In this case, \(\Delta_{can}\) is often smaller than $-\gamma(Y)/4 + 1/2$, because we have a larger freedom in finding minimizing vectors. To analyze $\Delta$ of these graphs, we use splice diagrams \cite{Sie80,NeumannWallBook86,NW05a} of plumbed manifolds, building on \cite{GKS23}.
Surprisingly, $\Delta_{can}$ can also be larger than $-\gamma(Y)/4 + 1/2$, as a result of interesting cancellations in the formula for $\Zhat_b(Y)$, see \cref{ex:cancel_H}. Namely, $\Zhat_b(Y)$ can be expressed as a weighted sum over certain lattice points and those can sometimes be organized in pairs with weights of opposite signs. This can be avoided by refining the weights, as provided by the two-variable series $\Zhathat_b(q,t)$ defined in \cite{AJK23}. However, \cref{ex:t_cancel_0} shows that cancellations may occur even in $\Zhathat_b(q,t)$.
An analogy between $\Delta_b(Y)$ invariants and the correction terms $d_b(Y)$ in Heegaard--Floer homology was proposed in \cite{GPP21}. In \cite{harichurn2023deltaa}, the first author demonstrated on Brieskorn spheres that unlike $d_b(Y)$, $\Delta_b(Y)$ are not cobordism invariants.
Using the explicit formula for $\Delta_{can}$ of Seifert manifolds given by \cref{thm:seifert_hom}, we compare $\Delta_{can}$ and $d_{can}$ for some classes of Brieskorn spheres, where $d_{can}$ is explicitly known \cite{BorNem2011}. We find that $\Delta_{can}$ is generically much larger than $d_{can}$. The reason for this discrepancy is that although $\Delta_{can}$ and $d_{can}$ are both minima of certain closely related quadratic forms, for $\Delta_{can}$, the quadratic form is being minimized over a much smaller set of vectors than the one for $d_{can}$.
\subsection*{Acknowledgement} We are grateful to Sergei Gukov, Mrunmay Jagadale and Sunghyuk Park for useful discussions. A. N\'emethi was partially supported by ``\'Elvonal (Frontier)'' Grant KKP 144148. J. Svoboda was supported by the Simons Foundation Grant {\it New structures in low-dimensional topology}. S. Harichurn was supported by the 2020 FirstRand FNB Fund Education Scholarship Award.
\section{Preliminaries}
In this section, we collect the necessary material on plumbed manifolds and \(\spc\) structures on them.
\subsection{Plumbed 3-manifolds}
Let $\Gamma = (V,E,m)$ be a finite tree with the set of vertices $V$ and edges $E$, and a vector \(m \in \ZZ^{\lvert V \rvert}\) consisting of integer labels \(m_v\) for each vertex \(v \in V\). Let $s = \lvert V\rvert$ and let $\delta = (\delta_v)_{v \in V} \in \ZZ^s$ be the vector of the degrees (valencies) of the vertices. We often implicitly order vertices of \(V\), so that \(V = {v_1,v_2,\dots,v_s}\) and write the quantities associated to \(v_i\) with subscript \(i\).
We can record \(\Gamma\) using $s\times s$ \emph{plumbing matrix} \(M=M(\Gamma)\), defined by
\[
M_{vw} =
\begin{cases}
m_v & \text{if} \quad v = w \\
1 & \text{if} \quad (v, w) \in E \\
0 & \text{otherwise.}
\end{cases}
\]
We always assume that \(M\) is a negative definite matrix. For a vector $l \in \Z^s$ we write
\[
l^2 = l^T M^{-1}l.
\]
Note that the quadratic form \(l \mapsto -l^2\) is a positive definite.
From \(\Gamma\), we can construct a closed oriented 3-manifold $Y := Y(\Gamma)$ by \emph{plumbing} \cite{Neu81}: For each vertex \(v\) we consider a circle bundle over \(S^2\) with Euler number \(m_v\). Then we glue the bundles together along tori corresponding to the edges in \(E\). Manifolds given by this construction, with \(M\) negative definite, are called \emph{negative definite plumbed manifolds}.
From the construction above, it follows that \(Y\) is a \emph{rational homology sphere}, that is, \(H_1(Y) = H_1(Y,\ZZ)\) is finite. We have \(H_1(Y) \cong \ZZ^s/M \ZZ^s\) and $\lvert H_1(Y)\rvert=\det M$. The quadratic form $l \mapsto -l^2$ takes values in $\lvert H\rvert^{-1}\ZZ$ as the adjugate matrix $\adj M = (\det M) M^{-1}$ of $M$ has integer entries.
\subsection{\texorpdfstring{\(\Spc\)}{Spinc} structures}\label{ss:spinc}
The set $\spc(Y)$ of \(\spc\) structures on $Y$ admits a natural free and transitive action of \(H_1(Y)\), hence it is finite. It can be identified with the set $(2\ZZ^s + m)/2M\ZZ^s$ of \emph{characteristic vectors}. For us, it is convenient to use another identification, with \(\spc(Y) \cong (2\ZZ^s + \delta)/2M\ZZ^s\), obtained from the usual characteristic vectors via the map \(l \to l - Mu\), where \(u = (1, 1, \dots, 1)\). This is justified by the identity \( \delta + m = Mu\).
For a vector $b \in 2\ZZ^s + \delta,$ we denote the corresponding $\spc$ structure as $[b] \in (2\ZZ^s + \delta)/2M\ZZ^s$. We often omit the brackets when it is clear from the context, e.g. we write $\Zhat_b$ for $\Zhat_{[b]}$.
We consider the vector $2u - \delta$ and the corresponding `canonical' $\spc$ structure $can = [2u-\delta]$. Its characteristic vector is \(k = 2u-\delta + Mu = m + 2u\). The rational number \(\gamma(Y) := k^2+s = (2u-\delta +Mu)^2+s\) does not depend on the plumbing representation of $Y$, so it is a topological invariant of $Y$.
Denote by $\Tr(M) = \sum_{v \in V}m_v$ the trace of the plumbing matrix $M$. Then $\gamma(Y)$ can be expressed as follows:
\begin{proposition}[\cite{NeNi02}]\label{base-k2-s}
Let $Y:=Y(\Gamma)$ be a negative definite plumbed manifold, which is a rational homology sphere. Then
\begin{equation}
\begin{split}
\gamma(Y) &= 3s + \Tr(M) + 2 + \sum_{v,w \in V} (2-\delta_v)(2-\delta_w)M_{vw}^{-1}\\
&=3s + \Tr(M) + 2 + (2u-\delta)^2.
\end{split}
\end{equation}
\end{proposition}
\subsection{The \texorpdfstring{\(\Zhat_b\)}{Zhat} invariants}
For the rest of the paper, let \(Y = Y(\Gamma)\) be a negative definite plumbed rational homology sphere with a chosen negative definite plumbing matrix \(M\) of $\Gamma$.
\begin{definition}
Let \(b \in 2\ZZ^s + \delta\) be a vector representing a $\spc$ structure $[b]$. For $\lvert q \rvert<1$, the GPPV $q$-series $\Zhat_b(Y;q)$ is defined by
\begin{equation}\label{def:zhat}
\Zhat_b(Y;q) = q^{\frac{-3s - \Tr(M)}{4}} \vp\oint\limits_{\lvert z_i \rvert = 1}\prod_{v_i\in V} \frac{dz_i}{2\pi i z_i} \left(z_i - \frac{1}{z_i}\right)^{2-\delta_i} \Theta_{b}(z),
\end{equation}
where
\begin{equation}\label{eq:theta}
\Theta_{b}(z) = \sum_{l \in 2M\ZZ^s + b} z^l q^{-\frac{l^2}{4}},
\end{equation}
and $\vp$ denotes the principal part of the integral.
\end{definition}
We often omit the manifold $Y$ when it is clear from the context, e.g. we write $\Zhat_b(q)$ for $\Zhat_b(Y,q)$. Clearly, $\Zhat_b(q)$ does not depend on the choice of $b \in [b]$. The convergence of $\Zhat_b(Y)$ for $\lvert q \rvert<1$ follows from the negative definiteness of \(M\).
\section{\texorpdfstring{$\Delta_b$}{Delta} invariants}
\begin{definition}
The \emph{Delta invariant} \(\Delta_b\) is the smallest $q$-exponent in \(\Zhat_b(q)\). If $\Zhat_b(q) = 0$, we set \(\Delta_b = \infty\).
\end{definition}
Using $\Delta_b$, we can write the $\Zhat_b(q)$ in the following form:
\[
\Zhat_b(q) = 2^{-s} q^{\Delta_b}\sum_{i=0}^\infty r_iq^i = 2^{-s} q^{\Delta_b}(r_0 + r_1q^1 + r_2q^2 + \cdots)
\]
for some integers $r_i$ with $r_0 \neq 0$. This is justified by part (1) of the following Lemma (recall that $\lvert H\rvert$ is the order of $H_1(Y,\ZZ)$):
\begin{lemma}\label{delta-h-mod-1}
\begin{enumerate}
\item The differences between the exponents of \(\Zhat_b(q)\) are integers.
\item The fractional part of the exponents (and in particular of \(\Delta_b\)) is given by
\[
\frac{-3s - \Tr(M) -b^2}{4} \Mod{1}.
\]
Consequently $4 \lvert H \rvert \Delta_b \in \ZZ \cup \{\infty\}$.
\item $\lvert H\rvert\Delta_a \equiv \lvert H\rvert\Delta_{b} \Mod{1}$ for any two $\spc$ structures $[a],[b]$ on $Y$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Consider two vectors \(l, l+2Mn \in [b]\), where $l,n \in \ZZ^s$. The difference of the corresponding exponents is
\[
\frac{-l^2}{4}+\frac{(l+2Mn)^2}{4} = l^Tn + n^T Mn \in \ZZ.
\]
\item The exponent of a representative $b \in [b]$ reads
\[
\frac{-3s - \Tr(M) - b^2}{4} \in \frac{1}{4\lvert H\rvert}\ZZ
\]
By (1), all the other exponents have the same fractional part.
\item
We have
\[
\lvert H\rvert\Delta_b(Y) - \lvert H\rvert\Delta_{a}(Y) \equiv \frac{\lvert H\rvert(b^2 -a^2)}{4} \Mod{1}.
\]
By writing $b = 2l + \delta$ and $a = 2n+\delta$ with $l,n \in \ZZ$, we see that
\[
\frac{1}{4}\lvert H\rvert (b^2 - a^2) = \lvert H\rvert (l^2 + l^TM^{-1}\delta - n^2 + n^TM^{-1}\delta) \in \ZZ.
\]
\end{enumerate}
\end{proof}
\begin{remark}
The set of $\spc$ structures admits a natural involution, called conjugation, denoted by $b \mapsto \bar{b}$. It is known that \(\Zhat_b(q) = \Zhat_{\bar{b}}(q)\), hence we have \(\Delta_b =\Delta_{\bar{b}}\).
\end{remark}
\subsection{The exponents of \texorpdfstring{$\Zhat_b$}{Zhat}}\label{computation-via-set-c}
The \(\Delta_b\) invariant is the minimal $q$-exponent in the series \(\Zhat_b(q)\). The exponents are, up to an overall shift, given by the quadratic form \(l \mapsto -l^2\). Here $l$ are lattice vectors that run over a certain subset \(\LC_b \subset \ZZ^s\). We need to identify this subset.
We order the set of vertices \(V\) of the plumbing tree $\Gamma$ by their degree:
\[
V = \{v_1, \dots, v_{s_1}, v_{s_{1}+1}, \dots, v_{s_{1}+s_{2}}, v_{s_{1}+s_{2}+1}, \dots, v_{s_{1}+s_{2} + s_{3} = s}\}.
\]
We have \(s_1\) leaves, \(s_2\) vertices of degree 2 and \(s_3\) vertices of degree \(\geq 3\).
The integrand of \(\Zhat\) contains the rational function
\begin{equation}\label{eq:rat_function}
\prod_{i=1}^s(z_i- z_i^{-1})^{2-\delta_i} = \prod_{i=1}^{s_1} (z_i-z_i^{-1}) \prod_{i=s_1+s_2+1}^{s} \frac{1}{(z_i-z_i^{-1})^{\delta_i-2}}.
\end{equation}
The integration in \eqref{def:zhat} is equivalent to the following procedure: First, we expand each term of the product above using the \emph{symmetric expansion}---the average of Laurent expansions as $z_i \rightarrow 0$ and $z_i \rightarrow \infty$, and multiply these together, giving an element of $\Z[z_1^{\pm 1},\dots, z_s^{\pm 1}]$:
\begin{equation}\label{eq:rat_function_expn}
\prod_{i=1}^s \se (z_i- z_i^{-1})^{2-\delta_i} = \sum_{l \in \ZZ^s} \ct_l z^l = \sum_{l \in \ZZ^s} \ct_l z_1^{l_1}z_2^{l_2}\cdots z_s^{l_s}.
\end{equation}
Then we multiply the result with the theta function $\Theta_b(z)$ in \eqref{eq:theta}, and we extract the constant coefficient in variables $z_i$, giving the $q$-series $\Zhat_b(q)$.
Let \(\LCt\) denote the set of all the vectors \(l \in \ZZ^s\) with nonvanishing coefficient $\ct_l \neq 0$ in \eqref{eq:rat_function_expn}. Similarly, define $\LCt_b = \LCt \cap -(2\ZZ^s+b)$. The reason for the sign is that we are pairing $l \in \LCt$ with $-l \in 2\ZZ^s+b$ when extracting the constant coefficient. Note that while \(\LCt\) is symmetric about the origin, $2\ZZ^s+b$ in general is not.
\begin{lemma}\label{description-c}
The set $\LCt$ consists of vectors whose components satisfy the following conditions:
\[\LCt =
\left\{(l_1,\dots,l_{s_1}, 0, \dots 0, m_1, \dots, m_{s_3}) \in \ZZ^s
\;\middle\vert\;
\begin{aligned}
&l_i = \pm 1 \\
&m_i \equiv \delta_i \Mod{2} \\
&\lvert m_i \rvert \geq \delta_i-2
\end{aligned}
\right\}.
\]
\end{lemma}
\begin{proof}
The expansion for 1-vertices is simply \(z_i-z_i^{-1}\), giving the entries \(l_i = \pm 1\). The variables corresponding to 2-vertices are absent in \ref{eq:rat_function_expn}. For a vertex \(v_i\) of degree \(\delta_{i} \geq 3\), put \(d := \delta_{i}-2 \geq 1\). We have the following symmetric expansion:
\begin{align*}
2 \cdot \se (z-z^{-1})^{-d} &=
\underset{z \to \infty}{\expn} \frac{z^{-d}}{(1-z^{-2})^{d}}
+ \underset{z \to 0}{\expn} \frac{z^{d}}{(z^2-1)^{d}}\\
&=
z^{-d} \sum_{k=0}^\infty \binom{k-1+d}{k}z^{-2k} + (-1)^d z^{d} \sum_{k=0}^\infty \binom{k-1+d}{k} z^{2k}\\
&=
(z^{-d} + d z^{-d-2} + \dots) + (-1)^{d} (z^{d} + d z^{d+2} +\dots)
\end{align*}
It follows that the corresponding entry \(m_i\) must have the same parity as \(\delta_i\) and \(\lvert m_i \rvert \geq d = \delta_i-2\).
\end{proof}
\subsection{Cancellations}\label{ss:cancel}
By the previous section, the $q$-series \(\Zhat_b(Y,q)\) can then be expressed as a sum over $\LCt_b$:
\begin{equation}\label{eq:before_cancel}
\Zhat_b(Y,q) = q^{\frac{-3s - \Tr(M)}{4}} \sum_{l \in \LCt_b} \ct_l q^{\frac{-l^2}{4}}.
\end{equation}
The $q$-exponents of $\Zhat_b(q)$ are therefore given by
$( -3s - \Tr(M) -l^2)/4$ for those $l \in \LCt_b$ whose contribution does not cancel out in the sum above, in other words:
\begin{equation}\label{eq:cancel}
c_l:= \sum_{\substack{l' \in \LCt_b\\l'^2 = l^2}} \ct_{l'} \neq 0.
\end{equation}
However, $c_l$ can vanish even if there are more nonzero terms $\tilde{c}_l$ is \eqref{eq:cancel}. We refer to this phenomenon as `cancellations' -- see Examples \ref{ex:cancel_H} and \ref{ex:cancel_seifert}.
Motivated by this, we define
\[
\LC_b := \{l \mid l \in \LCt_b; l \text{ satisfies \eqref{eq:cancel}} \} \subseteq \LCt_b.
\]
It follows that $\Delta_b$ is determined by minimizing $-l^2$ over the set $\LC_b$. We refer to the elements $l$ in $C_b$ for which $-l^2$ is minimal, as \emph{minimizing vectors}.
\begin{lemma}\label{lm:delta_minimizer}
\begin{equation}
\Delta_b(Y) = \frac14 \left(-3s - \Tr(M) + \min_{l \in \LC_b} \{ -l^2 \}\right).
\end{equation}
\end{lemma}
\section{$\Delta$ for Seifert Manifolds}
In this section, we will prove \cref{thm:seifert} which gives an explicit formula for $\Delta_{can}(Y)$ of a Seifert manifold $Y$ equipped with the canonical $\spc$ structure $can$.
\subsection{Seifert manifolds}
Seifert manifold $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$, fibered over $S^2$, is given by an integer $b_0$ and tuples of integers $0<\omega_i<a_i$ for $1 \leq i \leq n$. It can be represented by a star-shaped plumbing as shown in \cref{fig:seifert}. The graph consists of a central vertex with label $-b_0$ and $n$ `strings'. The labels $-b_{j_1},-b_{j_2},\dots, -b_{j_{s_j}}$ on the $j$-th string are determined by Hirzebruch--Jung continued fraction \[\frac{a_j}{\omega_j} = [b_{j_1}, \dots, b_{j_{s_j}}] =b_{j_1} - \cfrac{1}{b_{j_2} - \cfrac{1}{\cdots - \cfrac{1}{b_{j_{s_j}}}}}. \]
\begin{figure}
\centering
\includestandalone{graphics/seifert/new_seifert_plumbing}
\caption{Plumbing graph of a Seifert manifold.}
\label{fig:seifert}
\end{figure}
The intersection form $M$ is negative definite if and only if the orbifold Euler number $e = -b_0 + \sum_i \omega_i/a_i$ is negative.
We have the following formula for $\gamma(Y)$ of a Seifert manifold $Y$ \cite[p. 296]{NeNi02}, a consequence of \cref{base-k2-s}:
\begin{equation}\label{eq:formula_gamma}
\gamma(Y) = \frac{1}{e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 + e + 5 + 12 \sum_{i=1}^n \ds(\omega_i, a_i).
\end{equation}
Here $\ds$ denotes the Dedekind sum.
\subsection{\texorpdfstring{$\Delta_{can}$}{Delta} of Seifert manifolds}
\begin{theorem}\label{thm:seifert}
Let $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ be a negative definite Seifert manifold. Then $\Delta_{can}$ of the canonical $\spc$ structure satisfies
\begin{equation}\label{eq:seifert}
\Delta_{can} = -\frac{\gamma(Y)}{4} + \frac{1}{2}.
\end{equation}
If $Y$ is not a lens space, then $\Delta_{can}$ is minimal among all ${\Delta_b, b \in \spc(Y)}$.
\end{theorem}
\begin{proof}
We will treat the lens spaces separately in \cref{ssec:lens}, so we may assume that $n \geq 3$ and \(a_i \geq 2\) for each \(i\). Denote $A = \prod_{i=1}^n a_i$.
We will first show that over the set $\LCt$ from \cref{description-c}, \(-l^2\) is minimized by exactly be vectors $\pm(2u-\delta)$. From that, it will follow that $2u-\delta \in \LC_{can}$ is a true minimizing vector in the sense of \cref{ss:cancel}. The formula \eqref{eq:seifert} for $\Delta_{can}$ then follows from \cref{base-k2-s}.
The set $\LCt$ consists of the vectors
\[
l = (l_1, l_2, \dots, l_n, 0, \dots, 0, m)
\]
where $l_i = \pm 1$ and $m \equiv n \Mod{2}$ and $\lvert m \rvert \geq n-2$ by \cref{description-c}. The quadratic form \(l^2\) can be expressed using Seifert data as follows:
\begin{equation}\label{eq:quadratic}
-l^2 = m^2 A + \sum_{i=1}^n l_i m \frac{A}{a_i} +
\sum_{i \neq j}^n l_i l_j \frac{A}{a_ia_j} - \sum_{i=1}^n M^{-1}_{ii}.
\end{equation}
By the symmetry \(l^2 = (-l)^2\), we may assume that $m \geq n-2$. Taking the derivative with respect to \(m\), we obtain
\[
\frac{1}{A} \frac{\partial}{\partial m}\left(-l^2\right) = 2m + \sum_{i=1}^n \frac{l_i}{a_i} \geq 2(n-2) - \frac{n}{2} > 0,
\]
so the minimum is attained when \(\lvert m \rvert = n-2.\)
Similarly, pick \(j \in 1,\dots, s_1.\) Let \(l^+, l^-\) be two vectors with \(m = n-2\) that only differ by having \(l_j^+=1\), \(l^-_j=-1\), respectively. We have
\begin{align*}
l_+^2 - l_-^2 & = \frac{2A}{a_j} (n-2 + \sum_{i \neq j} \frac{l_i}{a_i}) \\
& \geq \frac{2A}{a_j} (n-2 - \frac{n-1}{2}) \geq 0.
\end{align*}
It follows that
\[
l = (-1,\dots, -1, 0, \dots, 0, n-2) = \delta - 2u.
\]
minimizes \eqref{eq:quadratic}. The argument also implies that the only other vector giving the same value of $l^2$ is $-l=2u-\delta$ which represents the canonical $\spc$ structure $can$. These two vectors have the same coefficients in the expansion of \eqref{eq:rat_function}:
\[
c_l = c_{-l} = \frac12\operatorname{sgn}(n-2)^n \binom{\frac{n+\lvert n-2 \rvert}{2}-2}{n-3}.
\]
Therefore even in the case that $\pm l$ belong to the same $\spc$ structure, i.e. $can$ is $\spin$, their contributions do not cancel in $\Zhat_{can}(q)$ and consequently $-l \in \LC_{can}$.
Thus the minimum of \(-l^2\) over \(\LC_{can}\) is given by $-(2u-\delta)^2$ and we have
\begin{equation}\label{delta-cleaning-up}
\Delta = \frac{-3s - \Tr(M)}{4} - \frac{(2u-\delta)^2}{4} = - \frac{\gamma(Y)}{4} + \frac{1}{2}
\end{equation}
The last equality follows the formula for \(\gamma(Y)\) in \cref{base-k2-s}.
\end{proof}
\subsection{Lens spaces}\label{ssec:lens}
In this section, we compute $\Delta_b$ of lens spaces. Let \(Y= L(p,r) \) be a lens space with \(p>r>0\). Denote by \(g\) a generator of \(H_1(Y,\ZZ) \cong \ZZ/p\ZZ\) and $can$ the canonical \(\spc\) structure.
We have the following formula for \(\Zhat_b(q)\) of lens spaces:
\[
\sum_{i=0}^{p-1} \Zhat_{g^i can}(q)g^i
= q^{3\ds(r,p)} \left((g^{-r-1}+1) q^{1/2p} - (g^{-r}+g^{-1}) q^{-1/2p}\right)
\]
Here \(\ds(r,p)\) denotes the Dedekind sum. From the formula, we read off the four finite \(\Delta_b\) invariants.
\begin{equation}\label{eq:delta_lens}
\Delta_{g^{-r-1}can} = \Delta_{can} = 3\ds(r,p) + \frac{1}{2p}, \quad
\Delta_{g^{-r}can} = \Delta_{g^{-1}can} = 3\ds(r,p) - \frac{1}{2p}.
\end{equation}
\noindent \(\gamma(Y)\) can be described using Dedekind sums in the following manner \cite[p. 304]{NeNi02}:
\[
\gamma(Y) = 2 - \frac{2}{p} - 12\ds(p,r).
\]
Comparing with \eqref{eq:delta_lens}, we obtain that \(\Delta_{can}\) satisfies the same formula as for the other Seifert manifolds:
\[
\Delta_{can} = -\frac{\gamma(Y)}{4} + \frac12.
\]
Note that \eqref{eq:delta_lens} shows that for lens spaces, $\Delta_{can}$ is not minimal among all $\Delta_b$. See also \cref{ex:cancel_H}.
\begin{remark}
We originally proved \cref{thm:seifert} using the reduction theorem \cite[Thm. 4.2]{GKS23} which may be used to compute all $q$-exponents of $\Zhat_b(q)$ of Seifert manifolds. Later, we found a simpler argument presented here, which focuses on the smallest exponent. It also emphasizes the role of the vector $2u-\delta$, making the presence of the invariant $\gamma(Y)$ more transparent.
\end{remark}
\begin{remark}
As we have seen above, \(\Delta_b\) invariants are often infinite. In \cite{GPPV20}, the authors conjectured, based on Physics considerations, the existence of a categorification of \(\Zhat_b(q)\), i.e. a doubly-graded cohomology theory \(\mathcal{H}_b^{i,j}(Y)\) whose graded Euler characteristic is \(\Zhat_b(q)\):
\[
\Zhat_b(q) = 2^{-s} q^{\Delta_b} \sum_{i, j \in \ZZ} q^i (-1)^j \dim \mathcal{H}_b^{i,j}(Y).
\]
Smaller \(q\)-exponents than \(\Delta_b\) could appear in the corresponding two-variable generating series \(q^{\Delta_b}\sum_{i,j} q^i t^j \dim \mathcal{H}_b^{i,j}(Y)\). In \cite{GPV16}, a Poincar\'e series of this sort was defined for lens spaces \(L(p,1)\). Its minimal \(q\)-power is finite for all \(\spc\) structures, in contrast with \(\Delta_b\).
\end{remark}
\section{Beyond Seifert manifolds}\label{s:H_shaped}
In the proof of \cref{thm:seifert} giving the formula for $\Delta_{can}$ of Seifert manifolds, we used a special form \eqref{eq:quadratic} of the quadratic from $l \mapsto -l^2$ following from the fact that Seifert manifolds admit a star-shaped graph. For more general graphs, we need a generalization of \eqref{eq:quadratic}. This is realized by some properties of \emph{splice diagrams}, which are certain weighted graphs built from plumbing graphs. We will illustrate this method on plumbing graphs with exactly two vertices of degree 3 and no vertices of degree 4 or more. The corresponding splice diagrams are ``H-shaped'' graphs with 6 vertices, as in \cref{fig:H_splice}.
In general, there is no uniform choice of minimizing vector as was the vector $2u-\delta$ for $\Delta_{can}$ of Seifert manifolds. Therefore we cannot hope for a simple universal formula for $\Delta_{can}$ as in \cref{thm:seifert}. Nevertheless, the minimizing vectors keep a specific form in certain regions given by the relative size of the weights in splice diagram. On the boundaries of these regions, we may see multiple minimizing vectors.
In principle, one can divide the study into those particular cases. Some of these can be effectively reduced to the case of Seifert manifolds, as we will illustrate in \cref{ss:computation_h_shaped}. The techniques described here can be used for more general plumbings, but the number of cases grows significantly with the complexity of the splice diagram.
\subsection{Splice diagrams}
Following \cite{NW05a} (see also \cite{SavelievBook}), given a plumbed manifold $Y$ with plumbing graph $\Gamma$, we construct a splice diagram $\Omega$ of $Y$ as follows: $\Omega$ is a tree obtained by replacing
each maximal string in $\Gamma$ (a simple path in $\Gamma$ whose interior is open in $\Gamma$) by a single edge. Thus $\Omega$ is homeomorphic to $\Gamma$ but has no vertices of degree two. We identify the vertices of $\Omega$ with the corresponding vertices of $\Gamma$.
At each vertex \(v\) of $\Omega$ of degree \(\geq 3\), we assign a weight $w_{v\varepsilon}$ on an incident edge $\varepsilon$ as follows. The edge $\varepsilon$ in $\Omega$ corresponds to a string in $\Gamma$ starting in $v$ with some edge $e$. Let $\Gamma_{ve}$ be the subgraph of $\Gamma$ cut off by the edge of $\Gamma$ at $v$ in the direction of $e$, as in the following picture.
\begin{center}
\includestandalone{graphics/H_shaped/explanation_subgraph}
\end{center}
The corresponding weight \(w_{v\varepsilon}\) is given by the determinant of \(-M(\Gamma_{ve})\), where \(M(\Gamma_{ve})\) denotes the intersection matrix of $\Gamma_{ve}$. We draw the weight \(w_{v \varepsilon}\) on the edge \(\varepsilon\) near \(v\). For example, the following is a plumbing graph and its splice diagram:
\begin{center}
\includestandalone{graphics/H_shaped/explanation_splice}
\end{center}
If $Y(\Gamma)$ is a homology sphere, the splice diagram uniquely determines the plumbing graph, see \cite{NW02}. Although $\Gamma$ and $\Omega$ are essentially equivalent in this case, $\Omega$ is much smaller. More importantly, it is the relative size of the weights of $\Omega$, which directly influences the minimizing vectors. For example, if one weights is significantly larger than the others, the minimizing vectors must be of some specific form, see \cref{ss:computation_h_shaped}.
The values of the quadratic form $l \mapsto -l^2=-l^T M^{-1}l$ can be computed from the splice diagram as follows: For two vertices \(v\), \(v'\) of the splice diagram, consider the shortest path $P$ connecting them. Let \(N_{vv'}\) be the product of all weights adjacent to vertices of $P$, but not lying on $P$.
\begin{center}
\includestandalone{graphics/H_shaped/explanation_path}
\end{center}
\begin{theorem}[{\cite[Thm. 12.2]{NW05a}}]\label{theorem:neumann} With the notation above, we have
\[
M^{-1}_{vv'} = -\frac{N_{vv'}}{\det M}.
\]
\end{theorem}
Recall that the vectors $l \in \Z^s$ that contribute to $\Zhat_b(q)$ lie in the set $\LCt$ from \cref{description-c}. If $l \in \LCt$, we have $l_v=0$ if $v$ is of degree 2 and $l_v^2 = (\pm 1)^2=1$ if $v$ is of degree 1. We obtain the following expression for $-l^2$ generalizing \eqref{eq:quadratic}:
\begin{equation}\label{eq:quadr_form_restricted}
-l^2 = \sum_{\substack{v \neq v'\\ \delta_v,\delta_{v'} \neq 2}} M^{-1}_{vv'} l_v l_{v'} + \sum_{\delta_v \geq 3} M^{-1}_{vv} l_v^2 + \sum_{\delta_v =1} M^{-1}_{vv}
\end{equation}
Note that the coefficients in the first and second sum can be expressed using the weights of the splice diagram $\Omega$, up to the multiplication by $\det M$. On the other hand, the third sum is not expressed in terms of weights of $\Omega$ in a simple way, but it is independent of $l$ (whenever $l \in \LCt$), so it does not influence which vectors $l \in \LCt$ have the minimal value of $-l^2$.
\subsection{\texorpdfstring{$\Delta$}{Delta} for homology spheres with $H$-shaped splice diagrams} \label{ss:computation_h_shaped}
We now consider a plumbing graph $\Gamma$ with exactly two vertices of degree 3 and no vertices of higher degree, e.g. the graph in \cref{fig:plumbing_from_splice}. For simplicity, we assume that the associated plumbed manifold $Y$ is a homology sphere. The associated splice diagram $\Omega$ is an `H-shaped' graph with six vertices, see \cref{fig:H_splice}. Its six weights are denoted \(a_1,a_2,a_3\) and \(a'_1,a'_2,a'_3\). They are pairwise coprime integers, which we further assume to be $\geq 2$. In this case, $Y$ can be realized as splicing of Brieskorn spheres $\Sigma(a_1,a_2,a_3)$ and $\Sigma(a'_1,a'_2,a'_3)$ along their third singular fibers.
We consider the projection $\ZZ^s \to \ZZ^6$, denoted by $l \mapsto \bar{l}$, which removes the components corresponding to the vertices of degree 2. We order the components of $\bar{l} = (x_1,x_2,x_3,x'_1,x'_2,x'_3)$ as in \cref{fig:H_splice} and keep the ordering throughout this section.
\begin{figure}[ht]
\centering
\includestandalone{graphics/H_shaped/splice_H_shaped}
\caption{H-shaped splice diagram $\Omega$}
\label{fig:H_splice}
\end{figure}
The quadratic form $l \mapsto -l^2$, restricted to the set $\LCt$ from \cref{description-c}, can be expressed in terms of the weights of $\Omega$ as:
\begin{equation}\label{eq:form_H_shaped}
\begin{split}
-l^2 &= x_3^2 a_1 a_2 a_3 + x_1 x_3 a_2 a_3 + x_2 x_3 a_1 a_3 + x_1 x_2 a_3 + (\dots)' + \\
& + x_1 x'_2 a_2 a'_1 + x_1' x_3 a_1 a_2 a'_2 + x'_2 x_3 a_1 a_2 a'_1 + (\dots)' + \\
& + x_1 x'_1 a_2 a'_2 + x_2 x'_2 a_1 a'_1 + x_3x'_3a_1a_2a'_1a'_2 + C.
\end{split}
\end{equation}
Here $(\dots)'$ means that we repeat the terms on each line with the usual and dashed variables reversed. \(C\) is constant on the set $\LCt$ so it does not influence the minimizing vectors for $\Delta_{can}$.
The general strategy to identify $\Delta_{can}$ can be described as follows: We know that $x_1,x_2,x'_1,x'_2 \in \{\pm1\}$ because $l \in \LCt$. For each of the $2^4$ possibilities of the signs, the form above reduces to a quadratic form in variables $x_3$, and $x_3'$. We can then minimize these forms over odd integers, using standard optimization methods. Finally, we must check that the resulting vectors do not cancel out, as in \cref{ss:cancel}, so they are true minimizing vectors.
Clearly, the form of minimizing vectors depends on the relative size of the coefficients \(a_i, a'_i\). We describe in greater detail the case when \(a_3\) is very large compared to other \(a_i\) and \(a'_i\). This allows us to effectively reduce the minimizing problem to the Seifert case. If $a_3 \gg a_1,a_2,a'_1,a'_2,a'_3$, the substantial terms are those containing \(a_3\):
\begin{equation}\label{eq:half_graph}
x^2_3 a_1 a_2 a_3 + x_1 x_3 a_2 a_3 + x_2 x_3 a_1 a_3 + x_1 x_2 a_3
\end{equation}
Any vector $l \in \LCt$ with the minimal value of $-l^2$ also minimizes the expression \eqref{eq:half_graph}. As this is (almost) a quadratic form of a star-shaped graph, we can repeat the first part of the argument in the proof of \cref{thm:seifert}. Namely, assuming that $a_i \geq 2$ and taking the $x_3$-derivative, we obtain that \(x_3=\pm 1\) in any minimizing vector, say $x_3=1$. The signs of $x_1$ and $x_2$ are easily determined by the relative size of $a_1$ and $a_2$.
This consideration `freezes' the left-hand side of the graph and for each choice of $x'_1 = \pm 1$ and $x'_2 = \pm 1$, we are left with a quadratic function of a single variable $x'_3$. Explicitly, in the case $x_1=x_2=-1$, the minimum in the variable $x'_3$ (over $\R$) is given by
\begin{equation}
\begin{split}
x_3' &= \frac{x'_1 a'_2 a'_3 + x'_2 a'_1 a'_3 - a_1 a_2 a'_2 + a_1 a_2 a'_1 a'_2 - a_1 a_2 a'_1}{2a'_1 a'_2 a'_3}\\
&= \frac12 \left(\frac{x_1'}{a'_1}+\frac{x_2'}{a'_2}-\frac{a_1 a_2}{a'_1 a'_3}+\frac{a_1 a_2}{a'_2 a'_3}-\frac{a_1 a_2}{a'_3}\right).
\end{split}
\end{equation}
This computation illustrates that the value of $x'_3$ can be arbitrarily large. This shows that the minimizing vector is very different from the vector $2u-\delta$ which was minimizing in the case of Seifert manifolds.
In a similar way, we could analyze other cases of the relative size of the weights, giving (at least approximate) `explicit formulas' for $\Delta_{can}$. However, in practice, it is easier to use a computer to search for the minimizing vectors. We illustrate the variability of possible minimizing vectors and possible values of $\Delta_{can}$ on several examples in this and the following section.
\begin{example}\label{ex:original} Consider the integral homology sphere $Y$ associated with the splice diagram
\begin{center}
\includestandalone{graphics/H_shaped/example_splice_big}
\end{center}
Then $\Delta(Y) = \frac{3045}{1000}$, with minimizing vectors
$\pm(-1,-1,3,1,1,-1).$
$\Delta(Y)$ is strictly smaller than $-\frac{\gamma(Y)}{4} + \frac{1}{2} = \frac{3885}{1000}$ given by the vector $2u-\delta=(-1,-1,1,-1,-1,1)$. The corresponding plumbing graph on $28$ vertices is shown in Figure \ref{fig:plumbing_from_splice}.
\begin{figure}[ht]
\centering
\resizebox{\columnwidth}{!}{
\includestandalone{graphics/H_shaped/example_plumbing_from_splice}
}
\caption{The plumbing graph associated to the splice diagram in Example \ref{ex:original}}
\label{fig:plumbing_from_splice}
\end{figure}
\end{example}
\begin{example} Consider the manifold $Y$ given by the following plumbing:
\begin{center}
\includestandalone{graphics/H_shaped/example_H_shaped/plumbing_H}
\end{center}
Then $Y$ is an integral homology sphere with $\Delta(Y) = \frac{1}{2}$ and the minimizing vectors are $\{\pm v, \pm w \}$, where $v=(1, 1, -1, -1, 1, 1)$, $w = (1, 1, -1, 1, -1, 1)$. Again, $\Delta(Y) < - \frac{\gamma(Y)}{4} + \frac{1}{2} = \frac{5}{2}$. Note that this plumbing corresponds to the following splice diagram:
\begin{center}
\includestandalone{graphics/H_shaped/example_H_shaped/splice_H}
\end{center}
\end{example}
\section{Upper Bounds and Cancellations}
In this section, we focus on upper bounds for $\Delta_b$. In principle, bounding $\Delta_b$ from above should be easy---the (shifted) norm $(-3s - \Tr(M) -l^2)/4$ of any element $l \in \mathcal{C}_b$ gives an upper bound.
However, when finding elements of $\LC_b$, we are facing two issues. First of all, $\LCt_b$ can be empty, e.g. for some $\spc$ structures on lens spaces, giving $\Zhat_b(q)=0$ and $\Delta_b = \infty$. Secondly, even if $\LCt_b$ is non-empty, there may be drastic cancellations of the coefficients, as we will illustrate on the Seifert manifold in \cref{ex:cancel_seifert}.
This prevents us from establishing general results in this direction. In particular, the vector $2u-\delta$ does not always give an upper bound for $\Delta_{can}$, unlike for Seifert manifolds (where it was optimal), see \cref{ex:cancel_H}.
We believe that those cancellations are rather special, being related to some additional symmetry of the plumbing graph. In particular, it would be rather surprising if they occurred for all $\spc$ structures at once. Therefore, we expect the following:
\begin{conjecture}\label{conj:ineq} For a negative-definite plumbed manifold $Y$ we have
\[\min_{b \in \spc{Y}} \Delta_b \leq -\frac{\gamma(Y)}{4}+\frac12.\]
\end{conjecture}
We now proceed with several examples of manifolds for which the cancellations occur. We also compare $\Zhat(q)$ with the two-variable extension $\Zhathat$ defined in \cite{AJK23}. The new variable \(t\) can distinguish vectors $l$ with the same value of $l^2$, removing some cancellations, but not all of them, see \cref{ex:t_cancel_0}.
\begin{example}\label{ex:cancel_H}
Consider the following plumbing: \begin{center}
\includestandalone{graphics/H_shaped/example_cancel/plumbing_cancel}
\end{center}
The resulting plumbed manifold $Y$ admits $20$ $\spc$ structures. For the canonical $\spc$ structure $can$ we have $\Delta_{can} = \frac{33}{20}$ with the (sole) minimizing vector $(1, -1, 1, -1, -1, 1)$. This is strictly larger than $-\frac{\gamma(Y)}{4} + \frac12 = \frac{13}{20}$.
The only vectors within $\LCt_{can}$ that satisfy $\frac{1}{4}(-3s - \operatorname{Tr}(M) - l^2) = \frac{13}{20}$ are $l_1 = (1, 1, -1, 1, 1, -1)$ and $l_2 = (-1, -1, 1, 1, 1, -3)$. However their coefficients are $c_{l_1} = \frac{1}{4} = -c_{l_2}$ and so they cancel out in $\Zhat_{can}(q)$.
The above cancellation does not happen for $\Zhathat_{can}(q,t)$:
\[
\Zhathat_{can}(q,t) = -\frac{1}{4}\left((t^{-1} -t)q^\frac{13}{20} +q^\frac{33}{20} -t^{-1}q^\frac{53}{20} + (t^{-2} + t^2)q^\frac{73}{20} - t^{-1}q^\frac{93}{20} +\cdots\right).
\]
\end{example}
\begin{example}\label{ex:cancel_seifert}
Consider Seifert manifold \(M(2;(3,1),(3,2),(3,2))\). It is described by the following negative definite plumbing:
\begin{center}
\includestandalone{graphics/seifert/plumbingM2_13_23_23}
\end{center}
Let \(b\) be the \(\spc\) structure with the associated vector \(\delta = (1,2,3,2,1,1) \in 2\ZZ^s+\delta\). The $q$-series $\Zhat_b(q)$ is a single monomial:
\[
\Zhat_b(q) = \frac12 q^{-5/6}
\]
This is due to the following cancellation: All vectors but one in the set $\mathcal{C}$ can be split into pairs \(l_1,l_2\) satisfying \(l_1^2=l_2^2\) and \(c_{l_1} = -c_{l_2}\).
Similarly to the previous example, $\Zhathat_b(q,t)$ removes the cancellations and it gives an infinite series:
\[
\Zhathat_b(q,t) = \frac12 ( q^{-5/6} t^{-1} + q^{13/6} (1-t^{-2}) + q^{67/6} (t-t^{-3}) + q^{157/6} (t^2 - t^{-4}) \dots).
\]
\end{example}
\begin{example}\label{ex:t_cancel_0}
Consider the following plumbing: \begin{center}
\includestandalone{graphics/H_shaped/example_cancel/plumbing_t_cancel}
\end{center}
The resulting plumbed manifold $Y$ admits $21$ $\spc$ structures. For the canonical $\spc$ structure $can$ represented by $2u - \delta = ( 1, 1, -1, 1, 1, -1)$ we have that $\Delta_{can}' = \frac{5}{2}$ and the minimizing vectors for the quadratic form that produces $\Delta'_{can}$ are given by $l_1 = (-1, -1, -3, 1, 1, 1)$, $l_2 = (-1, -1, 7, -1, -1, -1)$, $l_3 = (1, 1, -7, 1, 1, 1)$ and $l_4 = (1, 1, 3, -1, -1, -1)$ with coefficients $c_{l_1} = c_{l_3}= -\frac{1}{4}t^{-1}$ and $c_{l_2} = c_{l_4 }= -\frac{1}{4}t$. This is strictly larger than $-\frac{\gamma(Y)}{4} + \frac12 = \frac{1}{2}$.
For this manifold, the only vectors within $\tilde{\mathcal{C}}_{can}$ which satisfy $\frac{1}{4}(-3s - \operatorname{Tr}(M) - l^2) = \frac{1}{2}$ are $l_1 = (-1, -1, 1, -1, -1, 1)$, $l_2 = (-1, -1, 3, 1, 1, -1)$, $l_3 = (1, 1, -1, 1, 1, -1)$ and $l_4 = (1, 1, -3, -1, -1, 1)$. However their coefficients are $c_{l_1} = \frac{1}{4}t^{-1} = -c_{l_4}$ and $c_{l_2} = -\frac{1}{4}t = -c_{l_3}$ and so they cancel out in $\Zhathat_{can}(q,t)$. The entire $(t, q)$ series is given by:
\[
\Zhathat_{can}(q,t) = -\frac{1}{4}\left((2t^{-1} + 2t)q^\frac{5}{2} + (2t^{-3} +2t^3)q^\frac{9}{2} + (-t^{-4} + t^{-2} +t^2 -t^4)q^\frac{15}{2} + \cdots\right)
\]
This example shows that cancellations do occur even for $\Zhathat_{can}(q,t)$ and as a result $\Delta_{can}' > -\frac{\gamma(Y)}{4} + \frac{1}{2}$ where $\Delta_{can}'$ denotes the smallest $q$-exponent of $\Zhathat_{can}(q,t)$.
\end{example}
\section{Comparison with correction terms}
In the last section, we compare $\Delta_b(Y)$ with correction terms $d_b(Y)=d(Y,[b])$ in Heegaard--Floer homology\footnote{Again, we omit the brackets for $\spc$ structures.}. We include this discussion because there were some expectations that $\Delta_b(Y)$ and $d_b(Y)$ might be related \cite{GPP21, AJK23}. In the Seifert case, we have an explicit formula for $\Delta_{can}(Y)$ in terms of the $\gamma(Y)$ invariant by \cref{thm:seifert}. We can then use elementary bounds for Dedekind sums to obtain estimates on $\Delta_{can}(Y)$. Finally, we compare $\Delta_{can}(Y)$ to $d_{can}(Y)$ for some classes of Brieskorn spheres, where $d_{can}(Y)$ is known, finding that they are very different.
\subsection{Quadratic forms}
Correction terms can be expressed as minimizers of a quadratic form over the characteristic vectors:
\begin{theorem}[{\cite{Nem05}}]\label{thm:cor_terms} For an almost rational graph $\Gamma$, the correction terms are given by
\[
d_k(Y) = \max_{k' \in [k]} \frac{(k')^2+s}{4}
\]
\end{theorem}
Note that this formula holds in many other cases and was conjectured by the second author to be true for any negative definite rational homology sphere \cite{Nemethi_lattice08}.
To compare with $\Delta$, we need to shift to our conventions on $\spc$ structures, so we set $l = k' - Mu$. Then we have
\begin{align*}
\frac{(k')^2+s}{4} &= \frac{(l + Mu)^T M^{-1}(l + Mu) + s}{4}\\
&= \left(\frac{\Tr(M) +3s}{4} + \frac{l^2}{4} \right) + \frac{l^T u}{2} - \frac12
\end{align*}
We see that the minimized quadratic forms for $d$ and $\Delta$ differ in the linear term $l^T u/2= (\sum_{i=1}^s l_i)/2$.
\begin{remark}
In \cite{GPP21}, the authors observed that in the setting of \cref{thm:cor_terms}, the difference between $d_b(Y)$ and $\Delta_b(Y)-\frac12$ is an integer. For Seifert manifold $Y$ with the canonical $\spc$ structure, this can be explained as follows:
By \cite[Thm. 8.3.]{Nem08}, $d_{can}(Y)$ and $\gamma(Y)$ (and therefore $\Delta_{can}(Y)$) satisfy the following relation:
\begin{equation}
d_{can}(Y)=\frac{\gamma(Y)}{4} - 2 \chi_{can} \left( = -\Delta_{can} + \frac12 - 2 \chi_{can} \right).
\end{equation}
Here $\chi_{can}$ is the holomorphic Euler characteristic of a certain holomorphic line bundle on the corresponding quasihomogeneous singularity. In particular, it is an integer.
\end{remark}
\subsubsection{Lower Bound for Seifert Manifolds}
We end this section with some elementary estimates considering the $\gamma(Y)$ invariant of Seifert manifolds, and hence of $\Delta_{can}(Y)$. The formula \eqref{eq:formula_gamma} for $\gamma(Y)$ contains Dedekind sums. We have the following well-known inequality for $p>0$, $a \in \ZZ$:
\begin{equation}
-\mathbf{s}(1, p) \leq \mathbf{s}(a, p) \leq \mathbf{s}(1, p) = \frac{p}{12} + \frac{1}{6p} - \frac{1}{4}.
\end{equation}
Combining this with \eqref{eq:formula_gamma}, we obtain the following result.
\begin{proposition}\label{lower-bound-delta-general-seifert}
For the Seifert manifold $Y = M(b_0;(a_1, \omega_1), \dots, (a_n, \omega_n))$ and the canonical $\spc$ structure, we have
\[
\Delta_{can} \geq - \frac{1}{4e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 - \frac{e+3(n+1)}{4} + \sum_{i=1}^n \frac{a_i}{4} + \frac{1}{2a_i}
\]
and
\[
\Delta_{can} \leq -\frac{1}{4e}\left(2-n + \sum_{i=1}^n \frac{1}{a_i}\right)^2 -\frac{e+3(1-n)}{4} - \sum_{i=1}^n \frac{a_i}{4} + \frac{1}{2a_i}
\]
where $e$ is the orbifold Euler number of $Y$.
\end{proposition}
If $Y$ is a Brieskorn sphere $\Sigma(a_1, \dots, a_n)$, then $e = -\prod_{i=1}^n a_i^{-1}=-A^{-1}$. In particular, if $a_i \gg 1$, the leading term of $\Delta$ is given by $(n-2)^2A$. We obtain that the $\Delta$ invariant grows polynomially with $a_i$ for large values of $a_i$.
\subsubsection{Brieskorn spheres}
We illustrate the difference between $\Delta = \Delta_{can}$ and $d = d_{can}$ for some families of Brieskorn spheres, for which correction terms are explicitly known \cite{BorNem2011}. For $p, q > 0$ set $\rho = (p-1)(q-1)/2$. Then for $\Sigma(p, q, p q+1)= S^3_{-1}(T_{p, q})$ we have
$\gamma(Y) = -4\rho(\rho-1)$. From \cref{thm:seifert} we immediately obtain the next result:
\begin{corollary}
For $Y = \Sigma(p, q, pq+1)$, \begin{equation}\label{eq:Delta-p-p+1}
\Delta(Y) = \rho(\rho-1) + \frac{1}{2}.
\end{equation}
\end{corollary}
In contrast to this, the correction term vanishes, so that $\Delta(Y) \gg d(Y) = 0$.
For $Y = \Sigma(p, p+1, p(p+1)-1)$, we have \begin{equation}\label{eq:d}
d(Y) = \bigg\lfloor \frac{p}{2} \bigg\rfloor\left(\bigg\lfloor \frac{p}{2} \bigg\rfloor+1\right).\end{equation}
From \cref{lower-bound-delta-general-seifert} we obtain, after some manipulations, that for $p>2$
\begin{equation}\label{inequality-delta-d-comparison}
\Delta(Y) \geq \frac{1}{4}\left(p^4 + 2p^3-5p \right)-3 > d(Y).
\end{equation}
The case of $p = 2$ correspond to Poincar\'e sphere $Y = \Sigma(2, 3, 5)$, where we have $\Delta(Y) = -\frac{3}{2} < d(Y) = 2$. This seems to be a boundary case and in general, we expect the following conjecture:
\begin{conjecture}
For all but finitely many Seifert manifolds $Y$ we have
\[
d_{can}(Y) < \Delta_{can}(Y).
\]
\end{conjecture}
\printbibliography
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\begin{tikzpicture}
\begin{scope}[every node/.style={circle,fill,draw,inner sep=1.5pt}]
\node (1) at (-2,0.5) {};
\node (2) at (-2,-0.5) {};
\node (3) at (-1,0) {};
\node (4) at (0,0) {};
\node (5) at (1,0.5) {};
\node (6) at (1,-0.5) {};
\end{scope}
\node[xshift=-.3cm] at (1) {-2};
\node[xshift=-.3cm] at (2) {-3};
\node[yshift=.3cm] at (3) {-1};
\node[yshift=.3cm] at (4) {-7};
\node[xshift=.3cm] at (5) {-2};
\node[xshift=.3cm] at (6) {-3};
\begin{scope}[every edge/.style ={draw}]
\draw (1) -- (3);
\draw (2) -- (3);
\draw (3) -- (4);
\draw (4) -- (5);
\draw (4) -- (6);
\end{scope}
\end{tikzpicture}
\end{document}
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\begin{scope}[every node/.style={circle,fill,draw,inner sep=1.5pt}]
\node (1) at (-2,0) {};
\node (2) at (-1,0) {};
\node (3) at (-0,0) {};
\node (4) at (1,0) {};
\node (5) at (2,0) {};
\node (6) at (0,-1) {};
\end{scope}
\node[yshift=.3cm] at (1) {-2};
\node[yshift=.3cm] at (2) {-2};
\node[yshift=.3cm] at (3) {-2};
\node[yshift=.3cm] at (4) {-2};
\node[yshift=.3cm] at (5) {-2};
\node[xshift=-.3cm] at (6) {-3};
\begin{scope}[every edge/.style ={draw}]
\draw (1) -- (2);
\draw (2) -- (3);
\draw (3) -- (6);
\draw (3) -- (4);
\draw (4) -- (5);
\end{scope}
\end{tikzpicture}
\end{document}
\documentclass[tikz]{standalone}
\begin{document}
\begin{tikzpicture}
\begin{scope}[every node/.style={circle,fill,draw,inner sep=1pt}]
\node (1) at (0,0) {};
\node (2) at (1,1) {};
\node (3) at (2,1) {};
\node (4) at (4,1) {};
\node (5) at (1,-1) {};
\node (6) at (2,-1) {};
\node (7) at (4,-1) {};
\end{scope}
\node (14) at (2,0) {$\dots$};
\node (15) at (3,1) {$\dots$};
\node (16) at (3,-1) {$\dots$};
\node[xshift=-.4cm] at (1) {$-b_0$};
\node[yshift=.4cm] at (2) {$-b_{1_1}$};
\node[yshift=.4cm] at (3) {$-b_{1_2}$};
\node[yshift=.4cm] at (4) {$-b_{1_{s_1}}$};
\node[yshift=-.4cm] at (5) {$-b_{n_1}$};
\node[yshift=-.4cm] at (6) {$-b_{n_2}$};
\node[yshift=-.4cm] at (7) {$-b_{n_{s_n}}$};
\begin{scope}[every edge/.style ={draw}]
\draw (1) -- (2);
\draw (2) -- (3);
\draw (3) -- (2.5,1);
\draw (3.5,1) -- (4);
\draw (1) -- (5);
\draw (5) -- (6);
\draw (6) -- (2.5,-1);
\draw (3.5,-1) -- (7);
\draw (1) -- (1,0.25);
\draw (1) -- (1,-0.25);
\end{scope}
\end{tikzpicture}
\end{document}
|
2412.02123v3 | http://arxiv.org/abs/2412.02123v3 | Self-embedding similitudes of Bedford-McMullen carpets with dependent ratios | "\\documentclass[11pt]{amsart}\n\n\\usepackage{amsfonts,amsthm,amsmath,enumerate,amssymb,latexsym,co(...TRUNCATED) |
2412.02118v2 | http://arxiv.org/abs/2412.02118v2 | Algebraic properties of Indigenous semirings | "\\documentclass[psamsfonts]{amsart}\n\n\\usepackage{amssymb,amsfonts}\n\\usepackage[all,arc]{xy}\n\(...TRUNCATED) |
2412.02093v1 | http://arxiv.org/abs/2412.02093v1 | Twist Coefficients of Periodic Orbits of Minkowski Billiards | "\\documentclass[reqno]{amsart}\n\\usepackage[left=1in, top=1in, right=1in, bottom=1in]{geometry}\n\(...TRUNCATED) |
2412.03601v3 | http://arxiv.org/abs/2412.03601v3 | Relations between average shortest path length and another centralities in graphs | "\\documentclass[12pt, a4paper]{extarticle}\n\\usepackage[margin=2cm]{geometry}\n\\usepackage[affil-(...TRUNCATED) |
2412.02420v1 | http://arxiv.org/abs/2412.02420v1 | Fitting parameters of a Fokker-Planck-like equation with constraint | "\\documentclass[12pt]{article}\n\\usepackage{amsmath,amsfonts,amssymb}\n\\usepackage{mathrsfs}\n\\u(...TRUNCATED) |
2412.02523v1 | http://arxiv.org/abs/2412.02523v1 | "Density formulas for $p$-adically bounded primes for hypergeometric series with rational and quadra(...TRUNCATED) | "\n\\documentclass[12pt]{amsart}\n\\usepackage{amssymb,latexsym,amsmath,amsthm,amscd}\n\\usepackage{(...TRUNCATED) |
2412.02494v2 | http://arxiv.org/abs/2412.02494v2 | On the hit problem for the polynomial algebra and the algebraic transfer | "\n\\PassOptionsToPackage{top=1cm, bottom=1cm, left=2cm, right=2cm}{geometry}\n\\documentclass[final(...TRUNCATED) |
2412.02620v1 | http://arxiv.org/abs/2412.02620v1 | The Dimension of the Disguised Toric Locus of a Reaction Network | "\n\\documentclass[11pt]{article}\n\\usepackage{amsmath,amsfonts,amssymb,amsthm}\n\\usepackage{enume(...TRUNCATED) |
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