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Some improved nonperturbative bounds for Fermionic expansions: We reconsider the Gram-Hadamard bound as it is used in constructive quantum
field theory and many body physics to prove convergence of Fermionic
perturbative expansions. Our approach uses a recursion for the amplitudes of
the expansion, discovered originally by Djokic arXiv:1312.1185. It explains the
standard way to bound the expansion from a new point of view, and for some of
the amplitudes provides new bounds, which avoid the use of Fourier transform,
and are therefore superior to the standard bounds for models like the cold
interacting Fermi gas. | math-ph |
Weak Singularity for Two-Dimensional Nonlinear Equations of
Hydrodynamics and Propagation of Shock Waves: A system of two-dimensional nonlinear equations of hydrodynamics is
considered. It is shown that for the this system in the general case a solution
with weak discontinuity-type singularity behaves as a square root of S(x,y,t),
where S(x,y,t)>0 is a smooth function. The necessary conditions and series of
corresponding differential equations are obtained for the existence of a
solution. | math-ph |
Equations of hypergeometric type in the degenerate case: We consider the three most important equations of hypergeometric type,
${}_2F_1$, ${}_1F_1$ and ${}_1F_0$, in the so-called degenerate case. In this
case one of the parameters, usually denoted $c$, is an integer and the standard
basis of solutions consists of a hypergeometric-type function and a function
with a logarithmic singularity. This article is devoted to a thorough analysis
of the latter solution to all three equations. | math-ph |
Fusion procedure for the Yang-Baxter equation and Schur-Weyl duality: We use the fusion formulas of the symmetric group and of the Hecke algebra to
construct solutions of the Yang-Baxter equation on irreducible representations
of $\mathfrak{gl}_N$, $\mathfrak{gl}_{N|M}$, $U_q(\mathfrak{gl}_N)$ and
$U_q(\mathfrak{gl}_{N|M})$. The solutions are obtained via the fusion procedure
for the Yang--Baxter equation, which is reviewed in a general setting.
Distinguished invariant subspaces on which the fused solutions act are also
studied in the general setting, and expressed, in general, with the help of a
fusion function. Only then, the general construction is specialised to the four
situations mentioned above. In each of these four cases, we show how the
distinguished invariant subspaces are identified as irreducible
representations, using the relevant fusion formula combined with the relevant
Schur--Weyl duality. | math-ph |
On Generalized Monopole Spherical Harmonics and the Wave Equation of a
Charged Massive Kerr Black Hole: We find linearly independent solutions of the Goncharov-Firsova equation in
the case of a massive complex scalar field on a Kerr black hole. The solutions
generalize, in some sense, the classical monopole spherical harmonic solutions
previously studied in the massless cases. | math-ph |
Effective quantum gravity observables and locally covariant QFT: Perturbative algebraic quantum field theory (pAQFT) is a mathematically
rigorous framework that allows to construct models of quantum field theories on
a general class of Lorentzian manifolds. Recently this idea has been applied
also to perturbative quantum gravity, treated as an effective theory. The
difficulty was to find the right notion of observables that would in an
appropriate sense be diffeomorphism invariant. In this article I will outline a
general framework that allows to quantize theories with local symmetries (this
includes infinitesimal diffeomorphism transformations) with the use of the BV
(Batalin-Vilkovisky) formalism. This approach has been successfully applied to
effective quantum gravity in a recent paper by R. Brunetti, K. Fredenhagen and
myself. In the same paper we also proved perturbative background independence
of the quantized theory, which is going to be discussed in the present work as
well. | math-ph |
On certain new exact solutions of a diffusive predator-prey system: We construct exact solutions for a system of two nonlinear partial
differential equations describing the spatio-temporal dynamics of a
predator-prey system where the prey per capita growth rate is subject to the
Allee effect. Using the $\big(\frac{G'}{G}\big)$ expansion method, we derive
exact solutions to this model for two different wave speeds. For each wave
velocity we report three different forms of solutions. We also discuss the
biological relevance of the solutions obtained. | math-ph |
On the 3D steady flow of a second grade fluid past an obstacle: We study steady flow of a second grade fluid past an obstacle in three space
dimensions. We prove existence of solution in weighted Lebesgue spaces with
anisotropic weights and thus existence of the wake region behind the obstacle.
We use properties of the fundamental Oseen tensor together with results
achieved in \cite{Koch} and properties of solutions to steady transport
equation to get up to arbitrarily small $\ep$ the same decay as the Oseen
fundamental solution. | math-ph |
Noncommutative extensions of elliptic integrable Euler-Arnold tops and
Painleve VI equation: In this paper we suggest generalizations of elliptic integrable tops to
matrix-valued variables. Our consideration is based on $R$-matrix description
which provides Lax pairs in terms of quantum and classical $R$-matrices. First,
we prove that for relativistic (and non-relativistic) tops such Lax pairs with
spectral parameter follow from the associative Yang-Baxter equation and its
degenerations. Then we proceed to matrix extensions of the models and find out
that some additional constraints are required for their construction. We
describe a matrix version of ${\mathbb Z}_2$ reduced elliptic top and verify
that the latter constraints are fulfilled in this case. The construction of
matrix extensions is naturally generalized to the monodromy preserving
equation. In this way we get matrix extensions of the Painlev\'e VI equation
and its multidimensional analogues written in the form of non-autonomous
elliptic tops. Finally, it is mentioned that the matrix valued variables can be
replaced by elements of noncommutative associative algebra. In the end of the
paper we also describe special elliptic Gaudin models which can be considered
as matrix extensions of the (${\mathbb Z}_2$ reduced) elliptic top. | math-ph |
Lewis-Riesenfeld quantization and SU(1,1) coherent states for 2D damped
harmonic oscillator: In this paper we study a two-dimensional [2D] rotationally symmetric harmonic
oscillator with time-dependent frictional force. At the classical level, we
solve the equations of motion for a particular case of the time-dependent
coefficient of friction. At the quantum level, we use the Lewis-Riesenfeld
procedure of invariants to construct exact solutions for the corresponding
time-dependent Schr\"{o}dinger equations. The eigenfunctions obtained are in
terms of the generalized Laguerre polynomials. By mean of the solutions we
verify a generalization version of the Heisenberg's uncertainty relation and
derive the generators of the $su(1,1)$ Lie algebra. Based on these generators,
we construct the coherent states $\grave{\textrm{a}}$ la Barut-Girardello and
$\grave{\textrm{a}}$ la Perelomov and respectively study their properties. | math-ph |
Nonlinear dynamics of semiclassical coherent states in periodic
potentials: We consider nonlinear Schrodinger equations with either local or nonlocal
nonlinearities. In addition, we include periodic potentials as used, for
example, in matter wave experiments in optical lattices. By considering the
corresponding semiclassical scaling regime, we construct asymptotic solutions,
which are concentrated both in space and in frequency around the effective
semiclassical phase-space flow induced by Bloch's spectral problem. The
dynamics of these generalized coherent states is governed by a nonlinear
Schrodinger model with effective mass. In the case of nonlocal nonlinearities
we establish a novel averaging type result in the critical case. | math-ph |
Some Consequences of the Distributional Stress Equilibrium Condition: We derive two consequences of the distributional form of the stress
equilibrium condition while incorporating piecewise smooth stress and body
force fields with singular concentrations on an interface. First we obtain the
local equilibrium conditions in the bulk and at the interface, the latter
including conditions on the interfacial stress and stress dipole. Second we
obtain the necessary and the sufficient conditions on the divergence-free
non-smooth stress field for there to exist a stress function field such that
the equilibrium is trivially satisfied. In doing so we allow the domain to be
non-contractible with mutually disjoint connected boundary components. Both
derivations illustrate the utility of the theory of distributions in dealing
with singular stress fields. | math-ph |
Dirac Type Gauge Theories and the Mass of the Higgs Boson: We discuss the mass of the (physical component of the) Higgs boson in
one-loop and top-quark mass approximation. For this the minimal Standard Model
is regarded as a specific (parameterized) gauge theory of Dirac type. It is
shown that the latter formulation, in contrast to the usual description of the
Standard Model, gives a definite value for the Higgs mass. The predicted value
for the Higgs mass depends on the value addressed to the top mass m_T. We
obtain m_H= 186 \pm 8 GeV for m_T = 174 \pm 3 GeV (direct observation of top
events), resp. m_H = 184 \pm 22 GeV for m_T = 172 \pm 10 GeV (Standard Model
electroweak fit). Although the Higgs mass is predicted to be near the upper
bound, m_H is in full accordance with the range 114 \leq m_H < 193 GeV that is
allowed by the Standard Model.
We show that the inclusion of (Dirac) massive neutrinos does not alter the
results presented. We also briefly discuss how the derived mass values are
related to those obtained within the frame of non-commutative geometry. | math-ph |
Sutherland-type Trigonometric Models, Trigonometric Invariants and
Multivariable Polynomials. II. $E_7$ case: It is shown that the $E_7$ trigonometric Olshanetsky-Perelomov Hamiltonian,
when written in terms of the Fundamental Trigonometric Invariants (FTI), is in
algebraic form, i.e., has polynomial coefficients, and preserves the infinite
flag of polynomial spaces with the characteristic vector $\vec \alpha =
(1,2,2,2,3,3,4)$. Its flag coincides with one of the minimal characteristic
vector for the $E_7$ rational model. | math-ph |
Mathematical models of topologically protected transport in twisted
bilayer graphene: Twisted bilayer graphene gives rise to large moir\'{e} patterns that form a
triangular network upon mechanical relaxation. If gating is included, each
triangular region has gapped electronic Dirac points that behave as bulk
topological insulators with topological indices depending on valley index and
the type of stacking. Since each triangle has two oppositely charged valleys,
they remain topologically trivial.
In this work, we address several questions related to the edge currents of
this system by analysis and computation of continuum PDE models. Firstly, we
derive the bulk invariants corresponding to a single valley, and then apply a
bulk-interface correspondence to quantify asymmetric transport along the
interface. Secondly, we introduce a valley-coupled continuum model to show how
valleys are approximately decoupled in the presence of small perturbations
using a multiscale expansion, and how valleys couple for larger defects.
Thirdly, we present a method to prove for a large class of continuum
(pseudo-)differential models that a quantized asymmetric current is preserved
through a junction such as a triangular network vertex. We support all of these
arguments with numerical simulations using spectral methods to compute relevant
currents and wavepacket propagation. | math-ph |
Abelian BF theory and Turaev-Viro invariant: The U(1) BF Quantum Field Theory is revisited in the light of
Deligne-Beilinson Cohomology. We show how the U(1) Chern-Simons partition
function is related to the BF one and how the latter on its turn coincides with
an abelian Turaev-Viro invariant. Significant differences compared to the
non-abelian case are highlighted. | math-ph |
Non-polynomial extensions of solvable potentials a la Abraham-Moses: Abraham-Moses transformations, besides Darboux transformations, are
well-known procedures to generate extensions of solvable potentials in
one-dimensional quantum mechanics. Here we present the explicit forms of
infinitely many seed solutions for adding eigenstates at arbitrary real energy
through the Abraham-Moses transformations for typical solvable potentials, e.g.
the radial oscillator, the Darboux-P\"oschl-Teller and some others. These seed
solutions are simple generalisations of the virtual state wavefunctions, which
are obtained from the eigenfunctions by discrete symmetries of the potentials.
The virtual state wavefunctions have been an essential ingredient for
constructing multi-indexed Laguerre and Jacobi polynomials through multiple
Darboux-Crum transformations. In contrast to the Darboux transformations, the
virtual state wavefunctions generate non-polynomial extensions of solvable
potentials through the Abraham-Moses transformations. | math-ph |
Regions of possible motion in mechanical systems: A method to study the topology of the integral manifolds basing on their
projections to some other manifold of lower dimension is proposed. These
projections are called the regions of possible motion and in mechanical systems
arise in a natural way as the regions on a space of configuration variables. To
classify such regions we introduce the notion of a generalized boundary of a
region of possible motion and give the equation to find the generalized
boundaries. The inertial motion of a gyrostat (the Euler--Zhukovsky case) is
considered as an example. Explicit parametric equations of generalized
boundaries are obtained. The investigation gives the main types of connected
components of the regions of possible motion (including the sets of the
admissible velocities over each point of the region). From this information,
the phase topology of the case is established. | math-ph |
Constant connections, quantum holonomies and the Goldman bracket: In the context of (2+1)--dimensional quantum gravity with negative
cosmological constant and topology R x T^2, constant matrix--valued connections
generate a q--deformed representation of the fundamental group, and signed area
phases relate the quantum matrices assigned to homotopic loops. Some features
of the resulting quantum geometry are explored, and as a consequence a quantum
version of the Goldman bracket is obtained | math-ph |
Conservation-dissipation formalism of irreversible thermodynamics: We propose a conservation-dissipation formalism (CDF) for coarse-grained
descriptions of irreversible processes. This formalism is based on a stability
criterion for non-equilibrium thermodynamics. The criterion ensures that
non-equilibrium states tend to equilibrium in long time. As a systematic
methodology, CDF provides a feasible procedure in choosing non-equilibrium
state variables and determining their evolution equations. The equations
derived in CDF have a unified elegant form. They are globally hyperbolic, allow
a convenient definition of weak solutions, and are amenable to existing
numerics. More importantly, CDF is a genuinely nonlinear formalism and works
for systems far away from equilibrium. With this formalism, we formulate novel
thermodynamics theories for heat conduction in rigid bodies and non-isothermal
compressible Maxwell fluid flows as two typical examples. In these examples,
the non-equilibrium variables are exactly the conjugate variables of the heat
fluxes or stress tensors. The new theory generalizes Cattaneo's law or
Maxwell's law in a regularized and nonlinear fashion. | math-ph |
Explicit representation of Green function for 3Dimensional exterior
Helmholtz equation: We have constructed a sequence of solutions of the Helmholtz equation forming
an orthogonal sequence on a given surface. Coefficients of these functions
depend on an explicit algebraic formulae from the coefficient of the surface.
Moreover, for exterior Helmholtz equation we have constructed an explicit
normal derivative of the Dirichlet Green function. In the same way the
Dirichlet-to-Neumann operator is constructed. We proved that normalized
coefficients are uniformly bounded from zero. | math-ph |
On asymptotic solvability of random graph's laplacians: We observe that the Laplacian of a random graph G on N vertices represents
and explicitly solvable model in the limit of infinitely increasing N. Namely,
we derive recurrent relations for the limiting averaged moments of the
adjacency matrix of G. These relations allow one to study the corresponding
eigenvalue distribution function; we show that its density has an infinite
support in contrast to the case of the ordinary discrete Laplacian. | math-ph |
Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical
Temperature Shift in a Weak Homogeneous Magnetic Field: Starting from the Bardeen-Cooper-Schrieffer (BCS) free energy functional, we
derive the Ginzburg-Landau functional in the presence of a weak homogeneous
magnetic field. We also provide an asymptotic formula for the BCS critical
temperature as a function of the magnetic field. This extends the previous
works arXiv:1102.4001 and arXiv:1410.2352 of Frank, Hainzl, Seiringer and
Solovej to the case of external magnetic fields with non-vanishing magnetic
flux through the unit cell. | math-ph |
Thomas rotation and Thomas precession: Exact and simple calculation of Thomas rotation and Thomas precessions along
a circular world line is presented in an absolute (coordinate-free) formulation
of special relativity. Besides the simplicity of calculations the absolute
treatment of spacetime allows us to gain a deeper insight into the phenomena of
Thomas rotation and Thomas precession. | math-ph |
On absence of bound states for weakly attractive
$δ^\prime$-interactions supported on non-closed curves in $\mathbb{R}^2$: Let $\Lambda\subset\mathbb{R}^2$ be a non-closed piecewise-$C^1$ curve, which
is either bounded with two free endpoints or unbounded with one free endpoint.
Let $u_\pm|_\Lambda \in L^2(\Lambda)$ be the traces of a function $u$ in the
Sobolev space $H^1({\mathbb R}^2\setminus \Lambda)$ onto two faces of
$\Lambda$. We prove that for a wide class of shapes of $\Lambda$ the
Schr\"odinger operator $\mathsf{H}_\omega^\Lambda$ with
$\delta^\prime$-interaction supported on $\Lambda$ of strength $\omega \in
L^\infty(\Lambda;\mathbb{R})$ associated with the quadratic form \[
H^1(\mathbb{R}^2\setminus\Lambda)\ni u \mapsto \int_{\mathbb{R}^2}\big|\nabla u
\big|^2 \mathsf{d} x
- \int_\Lambda \omega \big| u_+|_\Lambda - u_-|_\Lambda \big|^2 \mathsf{d} s
\] has no negative spectrum provided that $\omega$ is pointwise majorized by a
strictly positive function explicitly expressed in terms of $\Lambda$. If,
additionally, the domain $\mathbb{R}^2\setminus\Lambda$ is quasi-conical, we
show that $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$. For a bounded
curve $\Lambda$ in our class and non-varying interaction strength
$\omega\in\mathbb{R}$ we derive existence of a constant $\omega_* > 0$ such
that $\sigma(\mathsf{H}_\omega^\Lambda) = [0,+\infty)$ for all $\omega \in
(-\infty, \omega_*]$; informally speaking, bound states are absent in the weak
coupling regime. | math-ph |
A new approach for the strong unique continuation of electromagnetic
Schroedinger operator with complex-valued coefficient: This paper mainly addresses the strong unique continuation property for the
electromagnetic Schr\"{o}dinger operator with complex-valued coefficients.
Appropriate multipliers with physical backgrounds have been introduced to prove
a priori estimates. Moreover, its application in an exact controllability
problem has been shown, in which case, the boundary value determines the
interior value completely. | math-ph |
Convective Equations and a Generalized Cole-Hopf Transformation: Differential equations with convective terms such as the Burger's equation
appear in many applications and have been the subject of intense research. In
this paper we use a generalized form of Cole-Hopf transformation to relate the
solutions of some of these nonlinear equations to the solutions of linear
equations. In particular we consider generalized forms of Burger's equation and
second order nonlinear ordinary differential equations with convective terms
which can represent steady state one-dimensional convection. | math-ph |
The Symmetry Properties of a Non-Linear Relativistic Wave Equation:
Lorentz Covariance, Gauge Invariance and Poincare Transformation: The Lorentz covariance of a non-linear, time-dependent relativistic wave
equation is demonstrated; the equation has recently been shown to have highly
interesting and significant empirical consequences. It is established here that
an operator already exists which ensures the relativistic properties of the
equation. Furthermore, we show that the time-dependent equation is gauge
invariant. The equation however, breaks Poincare symmetry via time translation
in a way consistent with its physical interpretation. It is also shown herein
that the Casimir invariant PmuPmu of the Poincare group, which corresponds to
the square of the rest mass M-squared can be expressed in terms of quaternions
such that M is described by an operator Q which has a constant norm and a phase
phi which varies in hypercomplex space. | math-ph |
Quantum marginals from pure doubly excited states: The possible spectra of one-particle reduced density matrices that are
compatible with a pure multipartite quantum system of finite dimension form a
convex polytope. We introduce a new construction of inner- and outer-bounding
polytopes that constrain the polytope for the entire quantum system. The outer
bound is sharp. The inner polytope stems only from doubly excited states. We
find all quantum systems, where the bounds coincide giving the entire polytope.
We show, that those systems are: i) any system of two particles ii) $L$ qubits,
iii) three fermions on $N\leq 7$ levels, iv) any number of bosons on any number
of levels and v) fermionic Fock space on $N\leq 5$ levels. The methods we use
come from symplectic geometry and representation theory of compact Lie groups.
In particular, we study the images of proper momentum maps, where our method
describes momentum images for all representations that are spherical. | math-ph |
A note on $σ$-model with the target $S^n$: Naively the Hilbert space of a sigma model has to be defined as an L^2 space
of functions on the space of free loops of the target. This object is not well
defined. In this note we study a finite-dimensional approximations L_N(S^n) of
the free loops of the sphere S^n. Spaces L_N(S^n) are defined in terms of
finite Fourier series. L_N(S^n) finite-dimensional but singular. We compute
Riemann and Ricci curvatures of the smooth locus of this space and study
Schr\"odinger operator in the case of L_1(S^n) | math-ph |
Poisson Geometry of Monic Matrix Polynomials: We study the Poisson geometry of the first congruence subgroup
$G_1[[z^{-1}]]$ of the loop group $G[[z^{-1}]]$ endowed with the rational
r-matrix Poisson structure for $G=GL_m$ and $SL_m$. We classify all the
symplectic leaves on a certain ind-subvariety of $G_1[[z^{-1}]]$ in terms of
Smith Normal Forms. This classification extends known descriptions of
symplectic leaves on the (thin) affine Grassmannian and the space of
$SL_m$-monopoles. We show that a generic leaf is covered by open charts with
Poisson transition functions, the charts being birationally isomorphic to
products of coadjoint $GL_m$ orbits. Finally, we discuss our results in terms
of (thick) affine Grassmannians and Zastava spaces. | math-ph |
T-duality in rational homotopy theory via $L_\infty$-algebras: We combine Sullivan models from rational homotopy theory with Stasheff's
$L_\infty$-algebras to describe a duality in string theory. Namely, what in
string theory is known as topological T-duality between $K^0$-cocycles in type
IIA string theory and $K^1$-cocycles in type IIB string theory, or as Hori's
formula, can be recognized as a Fourier-Mukai transform between twisted
cohomologies when looked through the lenses of rational homotopy theory. We
show this as an example of topological T-duality in rational homotopy theory,
which in turn can be completely formulated in terms of morphisms of
$L_\infty$-algebras. | math-ph |
Perturbed Poeschl-Teller oscillators: Wave functions and energies are constructed in a short-range Poeschl-Teller
well (= negative quadratic secans hyperbolicus) with a quartic perturbation.
Within the framework of an innovated, Lanczos-inspired perturbation theory we
show that our choice of non-orthogonal basis makes all the corrections given by
closed formulae. The first few items are then generated using MAPLE. | math-ph |
Holomorphic Path Integrals in Tangent Space for Flat Manifolds: In this paper we study the quantum evolution in a flat Riemannian manifold.
The holomorphic functions are defined on the cotangent bundle of this manifold.
We construct Hilbert spaces of holomorphic functions in which the scalar
product is defined using the exponential map. The quantum evolution is proposed
by means of an infinitesimal propagator and the holomorphic Feynman integral is
developed via the exponential map. The integration corresponding to each step
of the Feynman integral is performed in the tangent space. Moreover, in the
case of $S^1$, the method proposed in this paper naturally takes into account
paths that must be included in the development of the corresponding Feynman
integral. | math-ph |
Homogenized description of defect modes in periodic structures with
localized defects: A spatially localized initial condition for an energy-conserving wave
equation with periodic coefficients disperses (spatially spreads) and decays in
amplitude as time advances. This dispersion is associated with the continuous
spectrum of the underlying differential operator and the absence of discrete
eigenvalues. The introduction of spatially localized perturbations in a
periodic medium leads to defect modes, states in which energy remains trapped
and spatially localized. In this paper we study weak, localized perturbations
of one-dimensional periodic Schr\"odinger operators. Such perturbations give
rise to such defect modes, and are associated with the emergence of discrete
eigenvalues from the continuous spectrum. Since these isolated eigenvalues are
located near a spectral band edge, there is strong scale-separation between the
medium period and the localization length of the defect mode. Bound states
therefore have a multi-scale structure: a "carrier Bloch wave" times a "wave
envelope", which is governed by a homogenized Schr\"odinger operator with
associated effective mass, depending on the spectral band edge which is the
site of the bifurcation. Our analysis is based on a reformulation of the
eigenvalue problem in Bloch quasi-momentum space, using the Gelfand-Bloch
transform and a Lyapunov-Schmidt reduction to a closed equation for the
near-band-edge frequency components of the bound state. A rescaling of the
latter equation yields the homogenized effective equation for the wave
envelope, and approximations to bifurcating eigenvalues and eigenfunctions. | math-ph |
Construction of Lie Superalgebras from Triple Product Systems: Any simple Lie superalgebras over the complex field can be constructed from
some triple systems. Examples of Lie superalgebras $D(2,1;\alpha)$, G(3) and
F(4) are given by utilizing a general construction method based upon $(-1,-1)$
balanced Freudenthal-Kantor triple system. | math-ph |
Classical and Quantum Systems: Alternative Hamiltonian Descriptions: In complete analogy with the classical situation (which is briefly reviewed)
it is possible to define bi-Hamiltonian descriptions for Quantum systems. We
also analyze compatible Hermitian structures in full analogy with compatible
Poisson structures. | math-ph |
Form Factors of the Heisenberg Spin Chain in the Thermodynamic Limit:
Dealing with Complex Bethe Roots: In this article we study the thermodynamic limit of the form factors of the
XXX Heisenberg spin chain using the algebraic Bethe ansatz approach. Our main
goal is to express the form factors for the low-lying excited states as
determinants of matrices that remain finite dimensional in the thermodynamic
limit. We show how to treat all types of the complex roots of the Bethe
equations within this framework. In particular we demonstrate that the Gaudin
determinant for the higher level Bethe equations arises naturally from the
algebraic Bethe ansatz. | math-ph |
A Multiparametric Quantum Superspace and Its Logarithmic Extension: We introduce a multiparametric quantum superspace with $m$ even generators
and $n$ odd generators whose commutation relations are in the sense of Manin
such that the corresponding algebra has a Hopf superalgebra. By using its Hopf
superalgebra structure, we give a bicovariant differential calculus and some
related structures such as Maurer-Cartan forms and the correspoinding vector
fields. It is also shown that there exists a quantum supergroup related with
these vector fields. Morever, we introduce the logarithmic extension of this
quantum superspace in the sense that we extend this space by the series
expansion of the logarithm of the grouplike generator, and we define new
elements with nonhomogeneous commutation relations. It is clearly seen that
this logarithmic extension is a generalization of the $\kappa-$Minkowski
superspace. We give the bicovariant differential calculus and the related
algebraic structures on this extension. All noncommutative results are found to
reduce to those of the standard superalgebra when the deformation parameters of
the quantum (m+n)-superspace are set to one. | math-ph |
An Attempt of Construction for the Grassmann Numbers: We will pursue a way of building up an algebraic structure that involves, in
a mathematical abstract way, the well known Grassmann variables. The problem
arises when we tried to understand the grassmannian polynomial expansion on the
scope of ring theory. The formalization of this kind of variables and its
properties will help us to have a better idea of some algebraic structures and
the way they are implemented in models concerning theoretical physics. | math-ph |
The density-density response function in time-dependent density
functional theory: mathematical foundations and pole shifting: We establish existence and uniqueness of the solution to the Dyson equation
for the density-density response function in time-dependent density functional
theory (TDDFT) in the random phase approximation (RPA). We show that the poles
of the RPA density-density response function are forward-shifted with respect
to those of the non-interacting response function, thereby explaining
mathematically the well known empirical fact that the non-interacting poles
(given by the spectral gaps of the time-independent Kohn-Sham equations)
underestimate the true transition frequencies. Moreover we show that the RPA
poles are solutions to an eigenvalue problem, justifying the approach commonly
used in the physics community to compute these poles. | math-ph |
On Complex Supermanifolds with Trivial Canonical Bundle: We give an algebraic characterisation for the triviality of the canonical
bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky
superalgebra structure. As an application, we study the Calabi-Yau case, in
which an explicit formula in terms of the Levi-Civita connection is achieved.
Our methods include the use of complex integral forms and the recently
developed theory of superholonomy. | math-ph |
Galilei invariant theories. III. Wave equations for massless fields: Galilei invariant equations for massless fields are obtained via contractions
of relativistic wave equations. It is shown that the collection of
non-equivalent Galilei-invariant wave equations for massless fields with spin
equal 1 and 0 is very broad and describes many physically consistent systems.
In particular, there exist a huge number of such equations for massless fields
which correspond to various contractions of representations of the Lorentz
group to those of the Galilei one. | math-ph |
Saturation of uncertainty relations for twisted acceleration-enlarged
Newton-Hooke space-times: Using Fock representation we construct states saturating uncertainty
relations for twist-deformed acceleration-enlarged Newton-Hooke space-times. | math-ph |
The Existence and Uniqueness of Solutions to N-Body Problem of
Electrodynamics: Given $n$ charges interacting with each other according to Feynman's law. Let
$(r_j(t),v_j(t))$ denote the position and velocity of the charge $q_j.$ The
list $y(t)$ of all such vectors is called a trajectory. A Lipschitzian
trajectory $x(t), (t\le0),$ with continuous derivative, on which the velocities
do not exceed some limiting velocity $v<c,$ where $c$ denotes the speed of
light, is called an initial trajectory. A locally Lipschitzian trajectory
$y(t)$ is called relativistically admissible if the velocities on it stay below
the speed of light $c.$
The author constructs operators $\Phi_{j}$ of a trajectory whose values
$\Phi_j(y)(t)$ are linear transformations of $R^3$ into $R^3.$ A point $t=t_1$
on a trajectory $y$ is called singular if either some of the charges collide at
the time $t_1$ or the determinant is zero for at least one of the
transformations $\Phi_j(y)(t_1).$
The main result is the following: If $x(t) (t\le0)$ is an initial trajectory
with nonsingular point $t=0,$ then there exists a unique relativistically
admissible trajectory $y(t),$ defined for $t$ in an interval $I\subset <
0,\infty),$ extending the initial trajectory $x(t)$ and having the following
properties. (1) No point $t$ on the trajectory $y$ is singular. (2) The
trajectory represents a unique solution of the Newton-Einstein momentum-force
system of equations under Lorentz forces induced by electromagnetic field in
accord to Feynman's law for moving point charges. (3) The trajectory $y$
represents the maximal global solution of the system. | math-ph |
Yang-Baxter and reflection maps from vector solitons with a boundary: Based on recent results obtained by the authors on the inverse scattering
method of the vector nonlinear Schr\"odinger equation with integrable boundary
conditions, we discuss the factorization of the interactions of N-soliton
solutions on the half-line. Using dressing transformations combined with a
mirror image technique, factorization of soliton-soliton and soliton-boundary
interactions is proved. We discover a new object, which we call reflection map,
that satisfies a set-theoretical reflection equation which we also introduce.
Two classes of solutions for the reflection map are constructed. Finally, basic
aspects of the theory of set-theoretical reflection equations are introduced. | math-ph |
Distributing many points on spheres: minimal energy and designs: This survey discusses recent developments in the context of spherical designs
and minimal energy point configurations on spheres. The recent solution of the
long standing problem of the existence of spherical $t$-designs on
$\mathbb{S}^d$ with $\mathcal{O}(t^d)$ number of points by A. Bondarenko, D.
Radchenko, and M. Viazovska attracted new interest to this subject. Secondly,
D. P. Hardin and E. B. Saff proved that point sets minimising the discrete
Riesz energy on $\mathbb{S}^d$ in the hypersingular case are asymptotically
uniformly distributed. Both results are of great relevance to the problem of
describing the quality of point distributions on $\mathbb{S}^d$, as well as
finding point sets, which exhibit good distribution behaviour with respect to
various quality measures. | math-ph |
A path integral formalism for non-equilibrium Hamiltonian statistical
systems: A path integral formalism for non-equilibrium systems is proposed based on a
manifold of quasi-equilibrium densities. A generalized Boltzmann principle is
used to weight manifold paths with the exponential of minus the information
discrepancy of a particular manifold path with respect to full Liouvillean
evolution. The likelihood of a manifold member at a particular time is termed a
consistency distribution and is analogous to a quantum wavefunction. The
Lagrangian here is of modified generalized Onsager-Machlup form. For large
times and long slow timescales the thermodynamics is of Oettinger form. The
proposed path integral has connections with those occuring in the quantum
theory of a particle in an external electromagnetic field. It is however
entirely of a Wiener form and so practical to compute. Finally it is shown that
providing certain reasonable conditions are met then there exists a unique
steady-state consistency distribution. | math-ph |
Self-dual and anti-self-dual solutions of discrete Yang-Mills equations
on a double complex: We study a discrete model of the SU(2) Yang-Mills equations on a
combinatorial analog of $\Bbb{R}^4$. Self-dual and anti-self-dual solutions of
discrete Yang-Mills equations are constructed. To obtain these solutions we use
both techniques of a double complex and the quaternionic approach. Interesting
analogies between instanton, anti-instanton solutions of discrete and continual
self-dual, anti-self-dual equations are also discussed. | math-ph |
A Rigorous Real Time Feynman Path Integral: Using improper Riemann integrals, we will formulate a rigorous version of the
real-time, time-sliced Feynman path integral for the $L^2$ transition
probability amplitude. We will do this for nonvector potential Hamiltonians
with potential which has at most a finite number of discontinuities and
singularities. We will also provide a Nonstandard Analysis version of our
formulation. | math-ph |
A Review on Fish Swimming and Bird/Insect Flight: This expository review is devoted to fish swimming and bird/insect flight.
(i) The simple waving motion of an elongated flexible ribbon plate of constant
width, immersed in a fluid at rest, propagating a wave distally down the plate
to swim forward is first considered to provide a fundamental concept on energy
conservation. It is generalized to include variations in body width and
thickness, vortex shedding from appended dorsal, ventral and caudal fins to
closely simulate fish swimming for which a nonlinear theory is presented for
large-amplitude propulsion. (ii) For bird flight, the pioneering studies on
oscillating rigid wings are briefed, followed by presenting a nonlinear
unsteady theory for flexible wing with arbitrary variations in shape and
trajectory with a comparative study with experiments. (iii) For insect flight,
more recent advances are reviewed under aerodynamic theory and modeling,
computational methods, and experiments, on forward and hovering flights with
producing leading-edge vortex to give unsteady high lift. (iv) Prospects are
explored on extracting intrinsic flow energy by fish and bird to gain thrust
for propulsion. (v) The mechanical and biological principles are drawn together
for unified studies on the energetics in deriving metabolic power for animal
locomotion, leading to a surprising discovery that the hydrodynamic viscous
drag on swimming fish is largely associated with laminar boundary layers, thus
drawing valid and sound evidences for a resolution to the fish-swim paradox
proclaimed by Gray (1936, 1968). | math-ph |
A $C^*$-Algebraic Approach to Parametrized Quantum Spin Systems and
Their Phases in One Spatial Dimension: This thesis investigates parametrized quantum spin systems in the
thermodynamic limit from a $C^*$-algebraic point of view. Our main physical
result is the construction of a phase invariant for one-dimensional quantum
spin chains parametrized by a topological space $X$. This invariant is
constructed using $C^*$-algebraic techniques and takes values in degree one
\v{C}ech cohomology $H^1(X;\mathbb{P}\mathrm{U}(\mathscr{H}))$, where
$\mathbb{P}\mathrm{U}(\mathscr{H})$ is the projective unitary group of an
infinite-dimensional Hilbert space $\mathscr{H}$, endowed with the strong
operator topology. Using Dixmier-Douady theory [Bry93, DD63, HJJS08] one may
equivalently view this as an element of $H^3(X;\mathbb{Z})$. An exactly
solvable model of a one-dimensional spin system parametrized by the 3-sphere $X
= \mathbb{S}^3$ is presented and it is shown that its invariant is nontrivial
[WQB+22].
We also prove several mathematical results on topological aspects of the pure
state space $\mathscr{P}(\mathfrak{A})$ of a $C^*$-algebra $\mathfrak{A}$. We
prove that $\mathscr{P}(\mathfrak{A})$, endowed with the weak* topology, has
trivial fundamental group for every UHF algebra $\mathfrak{A}$ [BHM+23]. We
review the results of [SMQ+22] that show how the outputs of the GNS
representation and Kadison transitivity theorem can be understood to depend
continuously on their inputs. These results are expanded upon with the
construction of a distinguished \v{C}ech class in
$H^1(\mathscr{P}(\mathfrak{A});\mathrm{U}(1))$ that generalizes the principal
$\mathrm{U}(1)$-bundle $\mathbb{S} \mathscr{H} \rightarrow \mathbb{P}
\mathscr{H}$ in a representation independent way, where $\mathbb{S}
\mathscr{H}$ and $\mathbb{P} \mathscr{H}$ are the unit sphere and projective
Hilbert space of $\mathscr{H}$. Finally, we prove a selection theorem in the
vein of [SMQ+22] for the weak* topology on $\mathscr{P}(\mathfrak{A})$. | math-ph |
A variational principle and its application to estimating the electrical
capacitance of a perfect conductor: Assume that A is a bounded selfadjoint operator in a Hilbert space H. Then,
the variational principle is obtained for some functional. As an application of
this principle, a variational principle for the electrical capacitance of a
conductor of an arbitrary shape is derived. | math-ph |
Conformal Mappings and Dispersionless Toda hierarchy II: General String
Equations: In this article, we classify the solutions of the dispersionless Toda
hierarchy into degenerate and non-degenerate cases. We show that every
non-degenerate solution is determined by a function $\mathcal{H}(z_1,z_2)$ of
two variables. We interpret these non-degenerate solutions as defining
evolutions on the space $\mathfrak{D}$ of pairs of conformal mappings $(g,f)$,
where $g$ is a univalent function on the exterior of the unit disc, $f$ is a
univalent function on the unit disc, normalized such that $g(\infty)=\infty$,
$f(0)=0$ and $f'(0)g'(\infty)=1$. For each solution, we show how to define the
natural time variables $t_n, n\in\Z$, as complex coordinates on the space
$\mathfrak{D}$. We also find explicit formulas for the tau function of the
dispersionless Toda hierarchy in terms of $\mathcal{H}(z_1, z_2)$. Imposing
some conditions on the function $\mathcal{H}(z_1, z_2)$, we show that the
dispersionless Toda flows can be naturally restricted to the subspace $\Sigma$
of $\mathfrak{D}$ defined by $f(w)=1/\overline{g(1/\bar{w})}$. This recovers
the result of Zabrodin. | math-ph |
The system of three three-dimensional charged quantum particles:
asymptotic behavior of the eigenfunctions of the continuous spectrum at
infinity: The asymptotic behavior in the leading order of the continuous spectrum
eigenfunctions $\Psi(\bz,\bq)$ as $|\bz|\rightarrow\infty$ for the system of
three three-dimensional charged quantum particles has been obtained on the
heuristic level. The equality of the masses and the equality of the absolute
values of charges of particles are not crucial for the method. | math-ph |
Regularization of central forces with damping in two and
three-dimensions: Regularization of damped motion under central forces in two and
three-dimensions are investigated and equivalent, undamped systems are
obtained. The dynamics of a particle moving in $\frac{1}{r}$ potential and
subjected to a damping force is shown to be regularized a la Levi-Civita. We
then generalize this regularization mapping to the case of damped motion in the
potential $r^{-\frac{2N}{N+1}}$. Further equation of motion of a damped Kepler
motion in 3-dimensions is mapped to an oscillator with inverted sextic
potential and couplings, in 4-dimensions using Kustaanheimo-Stiefel
regularization method. It is shown that the strength of the sextic potential is
given by the damping co-efficient of the Kepler motion. Using homogeneous
Hamiltonian formalism, we establish the mapping between the Hamiltonian of
these two models. Both in 2 and 3-dimensions, we show that the regularized
equation is non-linear, in contrast to undamped cases. Mapping of a particle
moving in a harmonic potential subjected to damping to an undamped system with
shifted frequency is then derived using Bohlin-Sudman transformation. | math-ph |
Riemann-Hilbert Approach to the Helmholtz Equation in a quarter-plane.
Revisited: We revisit the Helmholts equation in a quarter-plane in the framework of the
Riemann-Hilbert approach to linear boundary value problems suggested in late
90s by A. Fokas. We show the role of the Sommerfeld radiation condition in
Fokas's scheme. | math-ph |
A comprehensive analysis of the geometry of TDOA maps in localisation
problems: In this manuscript we consider the well-established problem of TDOA-based
source localization and propose a comprehensive analysis of its solutions for
arbitrary sensor measurements and placements. More specifically, we define the
TDOA map from the physical space of source locations to the space of range
measurements (TDOAs), in the specific case of three receivers in 2D space. We
then study the identifiability of the model, giving a complete analytical
characterization of the image of this map and its invertibility. This analysis
has been conducted in a completely mathematical fashion, using many different
tools which make it valid for every sensor configuration. These results are the
first step towards the solution of more general problems involving, for
example, a larger number of sensors, uncertainty in their placement, or lack of
synchronization. | math-ph |
Smallest eigenvalue distribution of the fixed trace Laguerre
beta-ensemble: In this paper we study entanglement of the reduced density matrix of a
bipartite quantum system in a random pure state.
It transpires that this involves the computation of the smallest eigenvalue
distribution of the fixed trace Laguerre ensemble of $N\times N$ random
matrices. We showed that for finite $N$ the smallest eigenvalue distribution
may be expressed in terms of Jack polynomials.
Furthermore, based on the exact results, we found, a limiting distribution,
when the smallest eigenvalue is suitably scaled with $N$ followed by a large
$N$ limit. Our results turn out to be the same as the smallest eigenvalue
distribution of the classical Laguerre ensembles without the fixed trace
constraint. This suggests in a broad sense, the global constraint does not
influence local correlations, at least, in the large $N$ limit.
Consequently, we have solved an open problem: The determination of the
smallest eigenvalue distribution of the reduced density matrix---obtained by
tracing out the environmental degrees of freedom---for a bipartite quantum
system of unequal dimensions. | math-ph |
Low-Pass Filters, Fourier Series and Partial Differential Equations: When Fourier series are used for applications in physics, involving partial
differential equations, sometimes the process of resolution results in
divergent series for some quantities. In this paper we argue that the use of
linear low-pass filters is a valid way to regularize such divergent series. In
particular, we show that these divergences are always the result of
oversimplification in the proposition of the problems, and do not have any
fundamental physical significance. We define the first-order linear low-pass
filter in precise mathematical terms, establish some of its properties, and
then use it to construct higher-order filters. We also show that the
first-order linear low-pass filter, understood as a linear integral operator in
the space of real functions, commutes with the second-derivative operator. This
can greatly simplify the use of these filters in physics applications, and we
give a few simple examples to illustrate this fact. | math-ph |
Quantum unique ergodicity: This short note proves that a Laplacian cannot be quantum uniquely ergodic if
it possesses a quasimode of order zero which (i) has a singular limit, and (ii)
is a linear combination of a uniformly bounded number of eigenfunctions (modulo
an o(1) error). Bouncing ball quasimodes of the stadium are believed to have
this property (E.J. Heller et al) and so are analogous quasimodes recently
constructed by H. Donnelly on certain non-positively curved surfaces. The main
ingredient is the proof that all sequences of off-diagonal matrix elements of
QUE systems with vanishing spectral gaps tend to zero. | math-ph |
The electromagnetic energy-momentum tensor: We clarify the relation between canonical and metric energy-momentum tensors.
In particular, we show that a natural definition arises from Noether's Theorem
which directly leads to a symmetric and gauge invariant tensor for
electromagnetic field theories on an arbitrary space-time of any dimension. | math-ph |
Algebraic formulas for the structure constants in symmetric functions: Littlewood-Richardson rule gives the decomposition formula for the
multiplication of two Schur functions, while the decomposition formula for the
multiplication of two Hall-Littlewood functions or two universal characters is
also given by the combinatorial method. In this paper, using the vertex
operator realizations of these symmetric functions, we construct the algebraic
forms of these decomposition formulas. | math-ph |
Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface
Superconductivity Regime: We study the 2D Ginzburg-Landau theory for a type-II superconductor in an
applied magnetic field varying between the second and third critical value. In
this regime the order parameter minimizing the GL energy is concentrated along
the boundary of the sample and is well approximated to leading order by a
simplified 1D profile in the direction perpendicular to the boundary. Motivated
by a conjecture of Xing-Bin Pan, we address the question of whether this
approximation can hold uniformly in the boundary region. We prove that this is
indeed the case as a corollary of a refined, second order energy expansion
including contributions due to the curvature of the sample. Local variations of
the GL order parameter are controlled by the second order term of this energy
expansion, which allows us to prove the desired uniformity of the surface
superconductivity layer. | math-ph |
Simplicity of eigenvalues in Anderson-type models: We show almost sure simplicity of eigenvalues for several models of
Anderson-type random Schr\"odinger operators, extending methods introduced by
Simon for the discrete Anderson model. These methods work throughout the
spectrum and are not restricted to the localization regime. We establish
general criteria for the simplicity of eigenvalues which can be interpreted as
separately excluding the absence of local and global symmetries, respectively.
The criteria are applied to Anderson models with matrix-valued potential as
well as with single-site potentials supported on a finite box. | math-ph |
Extensions of Noether's Second Theorem: from continuous to discrete
systems: A simple local proof of Noether's Second Theorem is given. This proof
immediately leads to a generalization of the theorem, yielding conservation
laws and/or explicit relationships between the Euler--Lagrange equations of any
variational problem whose symmetries depend upon a set of free or
partly-constrained functions. Our approach extends further to deal with finite
difference systems. The results are easy to apply; several well-known
continuous and discrete systems are used as illustrations. | math-ph |
Self-adjoint extensions and spectral analysis in the generalized Kratzer
problem: We present a mathematically rigorous quantum-mechanical treatment of a
one-dimensional nonrelativistic motion of a particle in the potential field
$V(x)=g_{1}x^{-1}+g_{2}x^{-2}$. For $g_{2}>0$ and $g_{1}<0$, the potential is
known as the Kratzer potential and is usually used to describe molecular energy
and structure, interactions between different molecules, and interactions
between non-bonded atoms. We construct all self-adjoint Schrodinger operators
with the potential $V(x)$ and represent rigorous solutions of the corresponding
spectral problems. Solving the first part of the problem, we use a method of
specifying s.a. extensions by (asymptotic) s.a. boundary conditions. Solving
spectral problems, we follow the Krein's method of guiding functionals. This
work is a continuation of our previous works devoted to Coulomb, Calogero, and
Aharonov-Bohm potentials. | math-ph |
Critical two-point functions for long-range statistical-mechanical
models in high dimensions: We consider long-range self-avoiding walk, percolation and the Ising model on
$\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the
form $D(x)\asymp|x|^{-d-\alpha}$ with $\alpha>0$. The upper-critical dimension
$d_{\mathrm{c}}$ is $2(\alpha\wedge2)$ for self-avoiding walk and the Ising
model, and $3(\alpha\wedge2)$ for percolation. Let $\alpha\ne2$ and assume
certain heat-kernel bounds on the $n$-step distribution of the underlying
random walk. We prove that, for $d>d_{\mathrm{c}}$ (and the spread-out
parameter sufficiently large), the critical two-point function
$G_{p_{\mathrm{c}}}(x)$ for each model is asymptotically
$C|x|^{\alpha\wedge2-d}$, where the constant $C\in(0,\infty)$ is expressed in
terms of the model-dependent lace-expansion coefficients and exhibits crossover
between $\alpha<2$ and $\alpha>2$. We also provide a class of random walks that
satisfy those heat-kernel bounds. | math-ph |
Singularities of the scattering kernel related to trapping rays: An obstacle $K \subset \R^n,\: n \geq 3,$ $n$ odd, is called trapping if
there exists at least one generalized bicharacteristic $\gamma(t)$ of the wave
equation staying in a neighborhood of $K$ for all $t \geq 0.$ We examine the
singularities of the scattering kernel $s(t, \theta, \omega)$ defined as the
Fourier transform of the scattering amplitude $a(\lambda, \theta, \omega)$
related to the Dirichlet problem for the wave equation in $\Omega = \R^n
\setminus K.$ We prove that if $K$ is trapping and $\gamma(t)$ is
non-degenerate, then there exist reflecting $(\omega_m, \theta_m)$-rays
$\delta_m,\: m \in \N,$ with sojourn times $T_m \to +\infty$ as $m \to \infty$,
so that $-T_m \in {\rm sing}\:{\rm supp}\: s(t, \theta_m, \omega_m),\: \forall
m \in \N$. We apply this property to study the behavior of the scattering
amplitude in $\C$. | math-ph |
Asymptotics of spacing distributions 50 years later: In 1962 Dyson used a physically based, macroscopic argument to deduce the
first two terms of the large spacing asymptotic expansion of the gap
probability for the bulk state of random matrix ensembles with symmetry
parameter \beta. In the ensuing years, the question of asymptotic expansions of
spacing distributions in random matrix theory has shown itself to have a rich
mathematical content. As well as presenting the main known formulas, we give an
account of the mathematical methods used for their proofs, and provide some new
formulas. We also provide a high precision numerical computation of one of the
spacing probabilities to illustrate the accuracy of the corresponding
asymptotics. | math-ph |
Poisson Hypothesis for Information Networks II. Cases of Violations and
Phase Transitions: We present examples of queuing networks that never come to equilibrium. That
is achieved by constructing Non-linear Markov Processes, which are non-ergodic,
and possess eternal transience property. | math-ph |
Hamilton-Jacobi Formalism on Locally Conformally Symplectic Manifolds: In this article we provide a Hamilton-Jacobi formalism in locally conformally
symplectic manifolds. Our interest in the Hamilton-Jacobi theory comes from the
suitability of this theory as an integration method for dynamical systems,
whilst our interest in the locally conformal character will account for
physical theories described by Hamiltonians defined on well-behaved line
bundles, whose dynamic takes place in open subsets of the general manifold. We
present a local l.c.s. Hamilton-Jacobi in subsets of the general manifold, and
then provide a global view by using the Lichnerowicz-deRham differential. We
show a comparison between the global and local description of a l.c.s.
Hamilton--Jacobi theory, and how actually the local behavior can be glued to
retrieve the global behavior of the Hamilton-Jacobi theory. | math-ph |
Bounds on the Lyapunov exponent via crude estimates on the density of
states: We study the Chirikov (standard) map at large coupling $\lambda \gg 1$, and
prove that the Lyapounov exponent of the associated Schroedinger operator is of
order $\log \lambda$ except for a set of energies of measure $\exp(-c
\lambda^\beta)$ for some $1 < \beta < 2$. We also prove a similar (sharp) lower
bound on the Lyapunov exponent (outside a small exceptional set of energies)
for a large family of ergodic Schroedinger operators, the prime example being
the $d$-dimensional skew shift. | math-ph |
Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations: In this article, we study the self-similar solutions of the 2-component
Camassa-Holm equations% \begin{equation} \left\{ \begin{array} [c]{c}%
\rho_{t}+u\rho_{x}+\rho u_{x}=0
m_{t}+2u_{x}m+um_{x}+\sigma\rho\rho_{x}=0 \end{array} \right. \end{equation}
with \begin{equation} m=u-\alpha^{2}u_{xx}. \end{equation} By the separation
method, we can obtain a class of blowup or global solutions for $\sigma=1$ or
$-1$. In particular, for the integrable system with $\sigma=1$, we have the
global solutions:% \begin{equation} \left\{ \begin{array} [c]{c}%
\rho(t,x)=\left\{ \begin{array} [c]{c}% \frac{f\left( \eta\right)
}{a(3t)^{1/3}},\text{ for }\eta^{2}<\frac {\alpha^{2}}{\xi}
0,\text{ for }\eta^{2}\geq\frac{\alpha^{2}}{\xi}% \end{array} \right.
,u(t,x)=\frac{\overset{\cdot}{a}(3t)}{a(3t)}x
\overset{\cdot\cdot}{a}(s)-\frac{\xi}{3a(s)^{1/3}}=0,\text{ }a(0)=a_{0}%
>0,\text{ }\overset{\cdot}{a}(0)=a_{1}
f(\eta)=\xi\sqrt{-\frac{1}{\xi}\eta^{2}+\left( \frac{\alpha}{\xi}\right)
^{2}}% \end{array} \right. \end{equation}
where $\eta=\frac{x}{a(s)^{1/3}}$ with $s=3t;$ $\xi>0$ and $\alpha\geq0$ are
arbitrary constants.\newline Our analytical solutions could provide concrete
examples for testing the validation and stabilities of numerical methods for
the systems. | math-ph |
Higher Symplectic Geometry: We consider generalizations of symplectic manifolds called n-plectic
manifolds. A manifold is n-plectic if it is equipped with a closed,
nondegenerate form of degree n+1. We show that higher structures arise on these
manifolds which can be understood as the categorified or homotopy analogues of
important structures studied in symplectic geometry and geometric quantization.
Just as a symplectic manifold gives a Poisson algebra of functions, we show
that any n-plectic manifold gives a Lie n-algebra containing certain
differential forms which we call Hamiltonian. Lie n-algebras are examples of
strongly homotopy Lie algebras. They consist of an n-term chain complex
equipped with a collection of skew-symmetric multi-brackets that satisfy a
generalized Jacobi identity. We then develop the machinery necessary to
geometrically quantize n-plectic manifolds. In particular, just as a
prequantized symplectic manifold is equipped with a principal U(1)-bundle with
connection, a prequantized 2-plectic manifold is equipped with a U(1)-gerbe
with 2-connection. A gerbe is a categorified sheaf, or stack, which generalizes
the notion of a principal bundle. Furthermore, over any 2-plectic manifold
there is a vector bundle equipped with extra structure called a Courant
algebroid. This bundle is the 2-plectic analogue of the Atiyah algebroid over a
prequantized symplectic manifold. Its space of global sections also forms a Lie
2-algebra, which we use to prequantize the Lie 2-algebra of Hamiltonian forms.
Finally, we introduce the 2-plectic analogue of the Bohr-Sommerfeld variety
associated to a real polarization, and use this to geometrically quantize
2-plectic manifolds. The output of this procedure is a category of quantum
states. We consider a particular example in which the objects of this category
can be identified with representations of the Lie group SU(2). | math-ph |
Quantum quasi-Lie systems: properties and applications: A Lie system is a non-autonomous system of ordinary differential equations
describing the integral curves of a $t$-dependent vector field taking values in
a finite-dimensional Lie algebra of vector fields. Lie systems have been
generalised in the literature to deal with $t$-dependent Schr\"odinger
equations determined by a particular class of $t$-dependent Hamiltonian
operators, the quantum Lie systems, and other differential equations through
the so-called quasi-Lie schemes. This work extends quasi-Lie schemes and
quantum Lie systems to cope with $t$-dependent Schr\"odinger equations
associated with the here called quantum quasi-Lie systems. To illustrate our
methods, we propose and study a quantum analogue of the classical nonlinear
oscillator searched by Perelomov and we analyse a quantum one-dimensional fluid
in a trapping potential along with quantum $t$-dependent
Smorodinsky--Winternitz oscillators. | math-ph |
On the norm of the $q$-circular operator: The $q$-commutation relations, formulated in the setting of the $q$-Fock
space of Bo\.zjeko and Speicher, interpolate between the classical commutation
relations (CCR) and the classical anti-commutation relations (CAR) defined on
the classical bosonic and fermionic Fock spaces, respectively. Interpreting the
$q$-Fock space as an algebra of "random variables" exhibiting a specific
commutativity structure, one can construct the so-called $q$-semicircular and
$q$-circular operators acting as $q$-deformations of the classical Gaussian and
complex Gaussian random variables, respectively. While the $q$-semicircular
operator is generally well understood, many basic properties of the
$q$-circular operator (in particular, a tractable expression for its norm)
remain elusive. Inspired by the combinatorial approach to free probability, we
revist the combinatorial formulations of these operators. We point out that a
finite alternating-sum expression for $2n$-norm of the $q$-semicircular is
available via generating functions of chord-crossing diagrams developed by
Touchard in the 1950s and distilled by Riordan in 1974. Extending these norms
as a function in $q$ onto the complex unit ball and taking the $n\to\infty$
limit, we recover the familiar expression for the norm of the $q$-semicircular
and show that the convergence is uniform on the compact subsets of the unit
ball. In contrast, the $2n$-norms of the $q$-circular are encoded by
chord-crossing diagrams that are parity-reversing, which have not yet been
characterized in the combinatorial literature. We derive certain combinatorial
properties of these objects, including closed-form expressions for the number
of such diagrams of any size with up to eleven crossings. These properties
enable us to conclude that the $2n$-norms of the $q$-circular operator are
significantly less well behaved than those of the $q$-semicircular operator. | math-ph |
The Structure of the Ladder Insertion-Elimination Lie algebra: We continue our investigation into the insertion-elimination Lie algebra of
Feynman graphs in the ladder case, emphasizing the structure of this Lie
algebra relevant for future applications in the study of Dyson-Schwinger
equations. We work out the relation of this Lie algebra to some classical
infinite dimensional Lie algebra and we determine its cohomology. | math-ph |
A unifying perspective on linear continuum equations prevalent in
science. Part I: Canonical forms for static, steady, and quasistatic
equations: Following some past advances, we reformulate a large class of linear
continuum science equations in the format of the extended abstract theory of
composites so that we can apply this theory to better understand and
efficiently solve those equations. Here in part I we elucidate the form for
many static, steady, and quasistatic equations. | math-ph |
Quantum integrable systems and special functions: The wave functions of quantum Calogero-Sutherland systems for trigonometric
case are related to polynomials in l variables (l is a rank of root system) and
they are the generalization of Gegenbauer polynomials and Jack polynomials.
Using the technique of \kappa-deformation of Clebsch-Gordan series developed in
previous authors papers we investigate some new properties of generalized
Gegenbauer polynomials.Note that similar results are also valid in A_2 case for
more general two-parameter deformation ((q,t)-deformation) introduced by
Macdonald. | math-ph |
The Quasi-Reversibility Method for the Thermoacoustic Tomography and a
Coefficient Inverse Problem: An inverse problem of the determination of an initial condition in a
hyperbolic equation from the lateral Cauchy data is considered. This problem
has applications to the thermoacoustic tomography, as well as to linearized
coefficient inverse problems of acoustics and electromagnetics. A new version
of the quasi-reversibility method is described. This version requires a new
Lipschitz stability estimate, which is obtained via the Carleman estimate.
Numerical results are presented. | math-ph |
Asymptotic properties of MUSIC-type imaging in two-dimensional inverse
scattering from thin electromagnetic inclusions: The main purpose of this paper is to study the structure of the well-known
non-iterative MUltiple SIgnal Classification (MUSIC) algorithm for identifying
the shape of extended electromagnetic inclusions of small thickness located in
a two-dimensional homogeneous space. We construct a relationship between the
MUSIC-type imaging functional for thin inclusions and the Bessel function of
integer order of the first kind. Our construction is based on the structure of
the left singular vectors of the collected multistatic response matrix whose
elements are the measured far-field pattern and the asymptotic expansion
formula in the presence of thin inclusions. Some numerical examples are shown
to support the constructed MUSIC structure. | math-ph |
Symmetric Function Theory and Unitary Invariant Ensembles: Representation theory and the theory of symmetric functions have played a
central role in Random Matrix Theory in the computation of quantities such as
joint moments of traces and joint moments of characteristic polynomials of
matrices drawn from the Circular Unitary Ensemble and other Circular Ensembles
related to the classical compact groups. The reason is that they enable the
derivation of exact formulae, which then provide a route to calculating the
large-matrix asymptotics of these quantities. We develop a parallel theory for
the Gaussian Unitary Ensemble of random matrices, and other related unitary
invariant matrix ensembles. This allows us to write down exact formulae in
these cases for the joint moments of the traces and the joint moments of the
characteristic polynomials in terms of appropriately defined symmetric
functions. As an example of an application, for the joint moments of the traces
we derive explicit asymptotic formulae for the rate of convergence of the
moments of polynomial functions of GUE matrices to those of a standard normal
distribution when the matrix size tends to infinity. | math-ph |
The Identification of Thresholds and Time Delay in Self-Exciting
Threshold AR Model by Wavelet: In this paper we studied about the wavelet identification of the thresholds
and time delay for more general case without the constraint that the time delay
is smaller than the order of the model. Here we composed an empirical wavelet
from the SETAR (Self-Exciting Threshold Autoregressive) model and identified
the thresholds and time delay in the model using it. | math-ph |
The 2-category of species of dynamical patterns: A new category $\mathfrak{dp}$, called of dynamical patterns addressing a
primitive, nongeometrical concept of dynamics, is defined and employed to
construct a $2-$category $2-\mathfrak{dp}$, where the irreducible plurality of
species of context-depending dynamical patterns is organized. We propose a
framework characterized by the following additional features. A collection of
experimental settings is associated with any species, such that each one of
them induces a collection of experimentally detectable trajectories. For any
connector $T$, a morphism between species, any experimental setting $E$ of its
target species there exists a set such that with each of its elements $s$
remains associated an experimental setting $T[E,s]$ of its source species,
$T[\cdot,s]$ is called charge associated with $T$ and $s$. The vertical
composition of connectors is contravariantly represented in terms of charge
composition. The horizontal composition of connectors and $2-$cells of
$2-\mathfrak{dp}$ is represented in terms of charge transfer. A collection of
trajectories induced by $T[E,s]$ corresponds to a collection of trajectories
induced by $E$ (equiformity principle). Context categories, species and
connectors are organized respectively as $0,1$ and $2$ cells of
$2-\mathfrak{dp}$ with factorizable functors via $\mathfrak{dp}$ as $1-$cells
and as $2-$cells, arranged themself to form objects of categories, natural
transformations between $1-$cells obtained as horizontal composition of natural
transformations between the corresponding factors. We operate a
nonreductionistic interpretation positing that the physical reality holds the
structure of $2-\mathfrak{dp}$, where the fibered category $\mathfrak{Cnt}$ of
connectors is the only empirically knowable part..... | math-ph |
Density of Complex Critical Points of a Real Random SO(m+1) Polynomials: We study the density of complex critical points of a real random SO(m+1)
polynomial in m variables. In a previous paper [Mac09], the author used the
Poincare- Lelong formula to show that the density of complex zeros of a system
of these real random polynomials rapidly approaches the density of complex
zeros of a system of the corresponding complex random polynomials, the SU(m+1)
polynomials. In this paper, we use the Kac- Rice formula to prove an analogous
result: the density of complex critical points of one of these real random
polynomials rapidly approaches the density of complex critical points of the
corresponding complex random polynomial. In one variable, we give an exact
formula and a scaling limit formula for the density of critical points of the
real random SO(2) polynomial as well as for the density of critical points of
the corresponding complex random SU(2) polynomial. | math-ph |
OPEs of rank two W-algebras: In this short note, we provide OPEs for several affine W-algebras associated
with Lie algebras of rank two and give some direct applications. | math-ph |
Ultraviolet Renormalisation of a Quantum Field Toy Model II: We consider a class of toy models describing a fermion field coupled with a
boson field. The model can be viewed as a Yukawa model but with scalar
fermions. As in our first paper, the interaction kernels are assumed bounded in
the fermionic momentum variable and decaying like $|q|^{-p}$ for large boson
momenta $q$. With no restrictions on the coupling strength, we prove norm
resolvent convergence to an ultraviolet renormalized Hamiltonian, when the
ultraviolet cutoff is removed. We do this by subtracting a sufficiently large,
but finite, number of recursively defined self-energy counter-terms, which may
be interpreted as arising from a perturbation expansion of the ground state
energy. The renormalization procedure requires a spatial cutoff and works in
three dimensions provided $p>\frac12$, which is as close as one may expect to
the physically natural exponent $p = \frac12$. | math-ph |
Relativistic Corrections to the Moyal-Weyl Spacetime: We define a coordinate operator in a QFT-fashion to obtain by a deformation
procedure a relativistic Moyal-Weyl spacetime. The idea is extracted from
recent progress in deformation theory concerning the emergence of the quantum
plane of the Landau-quantization. The obtained spacetime is not equal to the
standard Moyal-Weyl plane but relativistic corrections occur. | math-ph |
Laplace-Runge-Lenz symmetry in general rotationally symmetric systems: The universality of the Laplace-Runge-Lenz symmetry in all rotationally
symmetric systems is discussed. The independence of the symmetry on the type of
interaction is proven using only the most generic properties of the Poisson
brackets. Generalized Laplace-Runge-Lenz vectors are definable to be constant
(not only piece-wise conserved) for all cases, including systems with open
orbits. Applications are included for relativistic Coulomb systems and
electromagnetic/gravitational systems in the post-Newtonian approximation. The
evidence for the relativistic origin of the symmetry are extended to all
centrally symmetric systems. | math-ph |
On the rational invariants of quantum systems of $n$-qubits: For an $n$-qubit system, a rational function on the space of mixed states
which is invariant with respect to the action of the group of local symmetries
may be viewed as a detailed measure of entanglement. We show that the field of
all such invariant rational functions is purely transcendental over the complex
numbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence
basis is also exhibited. | math-ph |
Invariant Classification and Limits of Maximally Superintegrable Systems
in 3D: The invariant classification of superintegrable systems is reviewed and
utilized to construct singular limits between the systems. It is shown, by
construction, that all superintegrable systems on conformally flat, 3D complex
Riemannian manifolds can be obtained from singular limits of a generic system
on the sphere. By using the invariant classification, the limits are
geometrically motivated in terms of transformations of roots of the classifying
polynomials. | math-ph |
Entropy Anomaly in Langevin-Kramers Dynamics with a Temperature
Gradient, Matrix Drag, and Magnetic Field: We investigate entropy production in the small-mass (or overdamped) limit of
Langevin-Kramers dynamics. The results generalize previous works to provide a
rigorous derivation that covers systems with magnetic field as well as
anisotropic (i.e. matrix-valued) drag and diffusion coefficients that satisfy a
fluctuation-dissipation relation with state-dependent temperature. In
particular, we derive an explicit formula for the anomalous entropy production
which can be estimated from simulated paths of the overdamped system.
As a part of this work, we develop a theory for homogenizing a class of
integral processes involving the position and scaled-velocity variables. This
allows us to rigorously identify the limit of the entropy produced in the
environment, including a bound on the convergence rate. | math-ph |
Complex Structures for Klein-Gordon Theory on Globally Hyperbolic
Spacetimes: We develop a rigorous method to parametrize complex structures for
Klein-Gordon theory in globally hyperbolic spacetimes that satisfy a
completeness condition. The complex structures are conserved under
time-evolution and implement unitary quantizations. They can be interpreted as
corresponding to global choices of vacuum. The main ingredient in our
construction is a system of operator differential equations. We provide a
number of theorems ensuring that all ingredients and steps in the construction
are well-defined. We apply the method to exhibit natural quantizations for
certain classes of globally hyperbolic spacetimes. In particular, we consider
static, expanding and Friedmann-Robertson-Walker spacetimes. Moreover, for a
huge class of spacetimes we prove that the differential equation for the
complex structure is given by the Gelfand-Dikki equation. | math-ph |
Soliton equations in N-dimensions as exact reductions of the Self-Dual
Yang-Mills equation V. Simplest (2+1)-dimensional soliton equations: Some aspects of the multidimensional soliton geometry are considered. It is
shown that some simples (2+1)-dimensional equations are exact reductions of the
Self-Dual Yang-Mills equation or its higher hierarchy. | math-ph |
Exact solution of the six-vertex model with domain wall boundary
conditions. Critical line between disordered and antiferroelectric phases: In the present article we obtain the large $N$ asymptotics of the partition
function $Z_N$ of the six-vertex model with domain wall boundary conditions on
the critical line between the disordered and antiferroelectric phases. Using
the weights $a=1-x,b=1+x,c=2,|x|<1$, we prove that, as $N\rightarrow\infty$,
$Z_N=CF^{N^2}N^{1/12}(1+O(N^{-1}))$, where $F$ is given by an explicit
expression in $x$ and the $x$-dependency in $C$ is determined. This result
reproduces and improves the one given in the physics literature by Bogoliubov,
Kitaev and Zvonarev. Furthermore, we prove that the free energy exhibits an
infinite order phase transition between the disordered and antiferroelectric
phases. Our proofs are based on the large $N$ asymptotics for the underlying
orthogonal polynomials which involve a non-analytical weight function, the
Deift-Zhou nonlinear steepest descent method to the corresponding
Riemann-Hilbert problem, and the Toda equation for the tau-function. | math-ph |
Introduction to Sporadic Groups for physicists: We describe the collection of finite simple groups, with a view on physical
applications. We recall first the prime cyclic groups $Z_p$, and the
alternating groups $Alt_{n>4}$. After a quick revision of finite fields
$\mathbb{F}_q$, $q = p^f$, with $p$ prime, we consider the 16 families of
finite simple groups of Lie type. There are also 26 \emph{extra} "sporadic"
groups, which gather in three interconnected "generations" (with 5+7+8 groups)
plus the Pariah groups (6). We point out a couple of physical applications,
including constructing the biggest sporadic group, the "Monster" group, with
close to $10^{54}$ elements from arguments of physics, and also the relation of
some Mathieu groups with compactification in string and M-theory. | math-ph |
Dual Lindstedt series and KAM theorem: We prove that exists a Lindstedt series that holds when a Hamiltonian is
driven by a perturbation going to infinity. This series appears to be dual to a
standard Lindstedt series as it can be obtained by interchanging the role of
the perturbation and the unperturbed system. The existence of this dual series
implies that a dual KAM theorem holds and, when a leading order Hamiltonian
exists that is non degenerate, the effect of tori reforming can be observed
with a system passing from regular motion to fully developed chaos and back to
regular motion with the reappearance of invariant tori. We apply these results
to a perturbed harmonic oscillator proving numerically the appearance of tori
reforming. Tori reforming appears as an effect limiting chaotic behavior to a
finite range of parameter space of some Hamiltonian systems. Dual KAM theorem,
as proved here, applies when the perturbation, combined with a kinetic term,
provides again an integrable system. | math-ph |
Magnon-phonon coupling from a crossing symmetric screened interaction: The magnon-phonon coupling has received growing attention in recent years due
to its central role in spin caloritronics and the emerging field of acoustic
spintronics. At resonance, this magnetoelastic interaction drives the formation
of magnon polarons, which underpin exotic phenomena such as magnonic heat
currents and phononic spin, but has with a few recent exceptions only been
investigated using mesoscopic spin-lattice models. Motivated to integrate the
magnon-phonon coupling into first-principle many-body electronic structure
theory, we set up to derive the non-relativistic exchange-contribution, which
is more subtle than the spin-orbit contribution, using Schwinger's method of
functional derivatives. To avoid having to solve the famous Hedin-Baym
equations self-consistently, the phonons are treated as a perturbation to the
electronic structure. A formalism is developed around the idea of imposing
crossing symmetry on the interaction, in order to treat charge and spin on
equal footing. By an iterative scheme, we find that the spin-flip component of
the ${\mathit collective}$ four-point interaction, $\mathcal{V}$, which is used
to calculate the magnon spectrum, contains a first-order "screened T matrix"
part and an arguably more important second-order part, which in the limit of
local spins describes the same processes of phonon emission and absorption as
obtained from phenomenological magnetoelastic models. Here, the "order" refers
to the ${\mathit screened}$ ${\mathit collective}$ four-point interaction,
$\mathcal{W}$ - the crossing-symmetric analog of Hedin's $W$.
Proof-of-principle model calculations are performed at varying temperatures for
the isotropic magnon spectrum in three dimensions in the presence of a flat
optical phonon branch. | math-ph |
Closed form Solutions to Some Nonlinear equations by a Generalized
Cole-Hopf Transformation: In the first part of this paper we linearize and solve the Van der Pol and
Lienard equations with some additional nonlinear terms by the application of a
generalized form of Cole-Hopf transformation. We then show that the same
transformation can be used to linearize Painleve III equation for certain
combinations of its parameters. Finally we linearize new forms of Burger's and
related convective equations with higher order nonlinearities. | math-ph |
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