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Unified Analytical Solution for Radial Flow to a Well in a Confined
Aquifer: Drawdowns generated by extracting water from a large diameter (e.g. water
supply) well are affected by wellbore storage. We present an analytical
solution in Laplace transformed space for drawdown in a uniform anisotropic
aquifer caused by withdrawing water at a constant rate from a partially
penetrating well with storage. The solution is back transformed into the time
domain numerically. When the pumping well is fully penetrating our solution
reduces to that of Papadopulos and Cooper [1967]; Hantush [1964] when the
pumping well has no wellbore storage; Theis [1935] when both conditions are
fulfilled and Yang et.al. [2006] when the pumping well is partially
penetrating, has finite radius but lacks storage. We use our solution to
explore graphically the effects of partial penetration, wellbore storage and
anisotropy on time evolutions of drawdown in the pumping well and in
observation wells. | math-ph |
Integrable quad equations derived from the quantum Yang-Baxter equation: This paper presents an explicit correspondence between two different types of
integrable equations; the quantum Yang-Baxter equation in its star-triangle
relation form, and the classical 3D-consistent quad equations in the
Adler-Bobenko-Suris (ABS) classification. Each of the 3D-consistent ABS quad
equations of $H$-type, are respectively derived from the quasi-classical
expansion of a counterpart star-triangle relation. Through these derivations it
is seen that the star-triangle relation provides a natural path integral
quantization of an ABS equation. The interpretation of the different
star-triangle relations is also given in terms of
(hyperbolic/rational/classical) hypergeometric integrals, revealing the
hypergeometric structure that links the two different types of integrable
systems. Many new limiting relations that exist between the star-triangle
relations/hypergeometric integrals are proven for each case. | math-ph |
Fourier law, phase transitions and the stationary Stefan problem: We study the one-dimensional stationary solutions of an integro-differential
equation derived by Giacomin and Lebowitz from Kawasaki dynamics in Ising
systems with Kac potentials, \cite{GiacominLebowitz}. We construct stationary
solutions with non zero current and prove the validity of the Fourier law in
the thermodynamic limit showing that below the critical temperature the limit
equilibrium profile has a discontinuity (which defines the position of the
interface) and satisfies a stationary free boundary Stefan problem.
Under-cooling and over-heating effects are also studied. We show that if
metastable values are imposed at the boundaries then the mesoscopic stationary
profile is no longer monotone and therefore the Fourier law is not satisfied.
It regains however its validity in the thermodynamic limit where the limit
profile is again monotone away from the interface. | math-ph |
Non-Central Potentials, Exact Solutions and Laplace Transform Approach: Exact bound state solutions and the corresponding wave functions of the
Schr\"odinger equation for some non-central potentials including Makarov
potential, modified-Kratzer plus a ring-shaped potential, double ring-shaped
Kratzer potential, modified non-central potential and ring-shaped non-spherical
oscillator potential are obtained by using the Laplace transform approach. The
energy spectrums of the Hartmann potential, modified-Kratzer potential and
ring-shaped oscillator potential are also briefly studied as special cases. It
is seen that our analytical results for all these potentials are consistent
with those obtained by other works. We also give some numerical results
obtained for the modified non-central potential for different values of the
related quantum numbers. | math-ph |
Structure of Noncommutative Solitons: Existence and Spectral Theory: We consider the Schr\"odinger equation with a Hamiltonian given by a second
order difference operator with nonconstant growing coefficients, on the half
one dimensional lattice. This operator appeared first naturally in the
construction and dynamics of noncommutative solitons in the context of
noncommutative field theory. We construct a ground state soliton for this
equation and analyze its properties. In particular we arrive at $\ell^{\infty}$
and $\ell^{1}$ estimates as well as a quasi-exponential spatial decay rate. | math-ph |
Quantum Energy Inequalities in Pre-Metric Electrodynamics: Pre-metric electrodynamics is a covariant framework for electromagnetism with
a general constitutive law. Its lightcone structure can be more complicated
than that of Maxwell theory as is shown by the phenomenon of birefringence. We
study the energy density of quantized pre-metric electrodynamics theories with
linear constitutive laws admitting a single hyperbolicity double-cone and show
that averages of the energy density along the worldlines of suitable observers
obey a Quantum Energy Inequality (QEI) in states that satisfy a microlocal
spectrum condition. The worldlines must meet two conditions: (a) the classical
weak energy condition must hold along them, and (b) their velocity vectors have
positive contractions with all positive frequency null covectors (we call such
trajectories `subluminal').
After stating our general results, we explicitly quantize the electromagnetic
potential in a translationally invariant uniaxial birefringent crystal. Since
the propagation of light in such a crystal is governed by two nested
lightcones, the theory shows features absent in ordinary (quantized) Maxwell
electrodynamics. We then compute a QEI bound for worldlines of inertial
`subluminal' observers, which generalizes known results from the Maxwell
theory. Finally, it is shown that the QEIs fail along trajectories that have
velocity vectors which are timelike with respect to only one of the lightcones. | math-ph |
Finite range decomposition for a general class of elliptic operators: We consider a family of gradient Gaussian vector fields on $\Z^d$, where the
covariance operator is not translation invariant. A uniform finite range
decomposition of the corresponding covariance operators is proven, i.e., the
covariance operator can be written as a sum of covariance operators whose
kernels are supported within cubes of increasing diameter. An optimal
regularity bound for the subcovariance operators is proven. We also obtain
regularity bounds as we vary the coefficients defining the gradient Gaussian
measures. This extends a result of S. Adams, R. Koteck\'y and S. M\"uller
\cite{1202.1158}. | math-ph |
Non-standard matrix formats of Lie superalgebras: The standard format of matrices belonging to Lie superalgebras consists of
partitioning the matrices into even and odd blocks. In this paper, we study
other possible matrix formats and in particular the so-called diagonal format
which naturally occurs in various applications, e.g. in superconformal field
theory, superintegrable models, for super W-algebras and quantum supergroups. | math-ph |
Weakly resonant tunneling interactions for adiabatic quasi-periodic
Schrodinger operators: In this paper, we study spectral properties of the one dimensional periodic
Schrodinger operator with an adiabatic quasi-periodic perturbation. We show
that in certain energy regions the perturbation leads to resonance effects
related to the ones observed in the problem of two resonating quantum wells.
These effects affect both the geometry and the nature of the spectrum. In
particular, they can lead to the intertwining of sequences of intervals
containing absolutely continuous spectrum and intervals containing singular
spectrum. Moreover, in regions where all of the spectrum is expected to be
singular, these effects typically give rise to exponentially small "islands" of
absolutely continuous spectrum. | math-ph |
On the transport and concentration of enstrophy in 3D
magnetohydrodynamic turbulence: Working directly from the 3D magnetohydrodynamical equations and entirely in
physical scales we formulate a scenario wherein the enstrophy flux exhibits
cascade-like properties. In particular we show the inertially-driven transport
of current and vorticity enstrophy is from larger to smaller scale structures
and this inter-scale transfer is local and occurs at a nearly constant rate.
This process is reminiscent of the direct cascades exhibited by certain ideal
invariants in turbulent plasmas. Our results are consistent with the physically
and numerically supported picture that current and vorticity concentrate on
small-scale, coherent structures. | math-ph |
The free energy of the two-dimensional dilute Bose gas. II. Upper bound: We prove an upper bound on the free energy of a two-dimensional homogeneous
Bose gas in the thermodynamic limit. We show that for $a^2 \rho \ll 1$ and
$\beta \rho \gtrsim 1$ the free energy per unit volume differs from the one of
the non-interacting system by at most $4 \pi \rho^2 |\ln a^2 \rho|^{-1} (2 - [1
- \beta_{\mathrm{c}}/\beta]_+^2)$ to leading order, where $a$ is the scattering
length of the two-body interaction potential, $\rho$ is the density, $\beta$
the inverse temperature and $\beta_{\mathrm{c}}$ is the inverse
Berezinskii--Kosterlitz--Thouless critical temperature for superfluidity. In
combination with the corresponding matching lower bound proved in \cite{DMS19}
this shows equality in the asymptotic expansion. | math-ph |
Rota-Baxter operators on $sl(2,C)$ and solutions of the classical
Yang-Baxter equation: We explicitly determine all Rota-Baxter operators (of weight zero) on
$sl(2,C)$ under the Cartan-Weyl basis. For the skew-symmetric operators, we
give the corresponding skew-symmetric solutions of the classical Yang-Baxter
equation in $sl(2,C)$, confirming the related study by Semenov-Tian-Shansky. In
general, these Rota-Baxter operators give a family of solutions of the
classical Yang-Baxter equation in the 6-dimensional Lie algebra $sl(2,C)
\ltimes_{{\rm ad}^{\ast}} sl(2,C)^{\ast}$. They also give rise to 3-dimensional
pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter
equation in other 6-dimensional Lie algebras. | math-ph |
Heat conduction: a telegraph-type model with self-similar behavior of
solutions: For heat flux $q$ and temperature $T$ we introduce a modified
Fourier--Cattaneo law $q_t+ l \frac{q}{t}= - kT_x .$ The consequence of it is a
non-autonomous telegraph-type equation. % $\epsilon S_{tt} + \frac{a}{t} S_t =
S_{xx}$ . This model already has a typical self-similar solution which may be
written as product of two travelling waves modulo a time-dependent factor and
might play a role of intermediate asymptotics. | math-ph |
On a Random Matrix Models of Quantum Relaxation: Earlier two of us (J.L. and L.P.) considered a matrix model for a two-level
system interacting with a $n\times n$ reservoir and assuming that the
interaction is modelled by a random matrix. We presented there a formula for
the reduced density matrix in the limit $n\to \infty $ as well as several its
properties and asymptotic forms in various regimes. In this paper we give the
proofs of the assertions, and present also a new fact about the model. | math-ph |
Mechanics Systems on Para-Kaehlerian Manifolds of Constant J-Sectional
Curvature: The goal of this paper is to present Euler-Lagrange and Hamiltonian equations
on R2n which is a model of para-Kaehlerian manifolds of constant J-sectional
curvature. In conclusion, some differential geometrical and physical results on
the related mechanic systems have been given. | math-ph |
Poisson brackets after Jacobi and Plucker: We construct a symplectic realization and a bi-hamiltonian formulation of a
3-dimensional system whose solution are the Jacobi elliptic functions. We
generalize this system and the related Poisson brackets to higher dimensions.
These more general systems are parametrized by lines in projective space. For
these rank 2 Poisson brackets the Jacobi identity is satisfied only when the
Pl\" ucker relations hold. Two of these Poisson brackets are compatible if and
only if the corresponding lines in projective space intersect. We present
several examples of such systems. | math-ph |
Heat Determinant on Manifolds: We introduce and study new invariants associated with Laplace type elliptic
partial differential operators on manifolds. These invariants are constructed
by using the off-diagonal heat kernel; they are not pure spectral invariants,
that is, they depend not only on the eigenvalues but also on the corresponding
eigenfunctions in a non-trivial way. We compute the first three low-order
invariants explicitly. | math-ph |
Oscillator Algebra of Chiral Oscillator: For the chiral oscillator described by a second order and degenerate
Lagrangian with special Euclidean group of symmetries, we show, by cotangent
bundle Hamiltonian reduction, that reduced equations are Lie-Poisson on dual of
oscillator algebra, the central extension of special Euclidean algebra in two
dimensions. This extension, defined by symplectic two-cocycle of special
Euclidean algebra, seems to be an enforcement of reduction itself rooted to
Casimir function. | math-ph |
Singlets and reflection symmetric spin systems: We rigorously establish some exact properties of reflection symmetric spin
systems with antiferromagnetic crossing bonds: At least one ground state has
total spin zero and a positive semidefinite coefficient matrix. The crossing
bonds obey an ice rule. This augments some previous results which were limited
to bipartite spin systems and is of particular interest for frustrated spin
systems. | math-ph |
Decomposition of third-order constitutive tensors: Third-order tensors are widely used as a mathematical tool for modeling
physical properties of media in solid state physics. In most cases, they arise
as constitutive tensors of proportionality between basic physics quantities.
The constitutive tensor can be considered as a complete set of physical
parameters of a medium. The algebraic features of the constitutive tensor can
be seen as a tool for proper identification of natural material, as crystals,
and for design the artificial nano-materials with prescribed properties. In
this paper, we study the algebraic properties of a generic 3-rd order tensor
relative to its invariant decomposition. In a correspondence to different
groups acted on the basic vector space, we present the hierarchy of types of
tensor decomposition into invariant subtensors. In particular, we discuss the
problem of non-uniqueness and reducibility of high-order tensor decomposition.
For a generic 3-rd order tensor, these features are described explicitly.
In the case of special tensors of a prescribed symmetry, the decomposition
turns out to be irreducible and unique. We present the explicit results for two
physically interesting models: the piezoelectric tensor as an example of a pair
symmetry and the Hall tensor as an example of a pair skew-symmetry. | math-ph |
The Elastic Theory of Shells using Geometric Algebra: We present a novel derivation of the elastic theory of shells. We use the
language of Geometric algebra, which allows us to express the fundamental laws
in component-free form, thus aiding physical interpretation. It also provides
the tools to express equations in an arbitrary coordinate system, which
enhances their usefulness. The role of moments and angular velocity, and the
apparent use by previous authors of an unphysical angular velocity, has been
clarified through the use of a bivector representation. In the linearised
theory, clarification of previous coordinate conventions which have been the
cause of confusion, is provided, and the introduction of prior strain into the
linearised theory of shells is made possible. | math-ph |
Relativistic Orbits and the Zeros of $\wp(Θ)$: A simple expression for the zeros of Weierstrass' function is given which
follows from a formula for relativistic orbits. | math-ph |
Mixed mode oscillations in the Bonhoeffer-van der Pol oscillator with
weak periodic perturbation: Following the paper of K. Shimizu et al. (2011) we consider the
Bonhoeffer-van der Pol oscillator with non-autonomous periodic perturbation. We
show that the presence of mixed mode oscillations reported in that paper can be
explained using the geometrical theory of singular perturbations. The
considered model can be re-written as a 4-dimensional (locally 3-dimensional)
autonomous system, which under certain conditions has a folded saddle-node
singularity and additionally can be treated as a three time scale one. | math-ph |
Catalan Solids Derived From 3D-Root Systems and Quaternions: Catalan Solids are the duals of the Archimedean solids, vertices of which can
be obtained from the Coxeter-Dynkin diagrams A3, B3 and H3 whose simple roots
can be represented by quaternions. The respective Weyl groups W(A3), W(B3) and
W(H3) acting on the highest weights generate the orbits corresponding to the
solids possessing these symmetries. Vertices of the Platonic and Archimedean
solids result as the orbits derived from fundamental weights. The Platonic
solids are dual to each others however duals of the Archimedean solids are the
Catalan solids whose vertices can be written as the union of the orbits, up to
some scale factors, obtained by applying the above Weyl groups on the
fundamental highest weights (100), (010), (001) for each diagram. The faces are
represented by the orbits derived from the weights (010), (110), (101), (011)
and (111) which correspond to the vertices of the Archimedean solids.
Representations of the Weyl groups W(A3), W(B3) and W(H3) by the quaternions
simplify the calculations with no reference to the computer calculations. | math-ph |
The Frobenius-Virasoro algebra and Euler equations: We introduce an $\mathfrak{F}$-valued generalization of the Virasoro algebra,
called the Frobenius-Virasoro algebra $\mathfrak{vir_F}$, where $\mathfrak{F}$
is a Frobenius algebra over $\mathbb{R}$. We also study Euler equations on the
regular dual of $\mathfrak{vir_F}$, including the $\mathfrak{F}$-$\mathrm{KdV}$
equation and the $\mathfrak{F}$-$\mathrm{CH}$ equation and the
$\mathfrak{F}$-$\mathrm{HS}$ equation, and discuss their Hamiltonian
properties. | math-ph |
Infinite-dimensional Hamilton-Jacobi theory and $L$-integrability: The classical Liouvile integrability means that there exist $n$ independent
first integrals in involution for $2n$-dimensional phase space. However, in the
infinite-dimensional case, an infinite number of independent first integrals in
involution don't indicate that the system is solvable. How many first integrals
do we need in order to make the system solvable? To answer the question, we
obtain an infinite dimensional Hamilton-Jacobi theory, and prove an infinite
dimensional Liouville theorem. Based on the theorem, we give a modified
definition of the Liouville integrability in infinite dimension. We call it the
$L$-integrability. As examples, we prove that the string vibration equation and
the KdV equation are $L$-integrable. In general, we show that an infinite
number of integrals is complete if all action variables of a Hamilton system
can reconstructed by the set of first integrals. | math-ph |
Universality at the edge of the spectrum in Wigner random matrices: We prove universality at the edge for rescaled correlation functions of
Wigner random matrices in the limit $n\to +\infty$. As a corollary, we show
that, after proper rescaling, the 1st, 2nd, 3rd, etc. eigenvalues of Wigner
random hermitian (resp. real symmetric) matrix weakly converge to the
distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases. | math-ph |
Conserved charges for rational electromagnetic knots: We revisit a newfound construction of rational electromagnetic knots based on
the conformal correspondence between Minkowski space and a finite
$S^3$-cylinder. We present here a more direct approach for this conformal
correspondence based on Carter-Penrose transformation that avoids a detour to
de Sitter space. The Maxwell equations can be analytically solved on the
cylinder in terms of $S^3$ harmonics $Y_{j;m,n}$, which can then be transformed
into Minkowski coordinates using the conformal map. The resultant "knot basis"
electromagnetic field configurations have non-trivial topology in that their
field lines form closed knots. We consider finite, complex linear combinations
of these knot-basis solutions for a fixed spin $j$ and compute all the $15$
conserved Noether charges associated with the conformal group. We find that the
scalar charges either vanish or are proportional to the energy. For the
non-vanishing vector charges, we find a nice geometric structure that
facilitates computation of their spherical components as well. We present
analytic results for all charges for up to $j{=}1$. We demonstrate possible
applications of our findings through some known previous results. | math-ph |
Structure of Matrix Elements in Quantum Toda Chain: We consider the quantum Toda chain using the method of separation of
variables. We show that the matrix elements of operators in the model are
written in terms of finite number of ``deformed Abelian integrals''. The
properties of these integrals are discussed. We explain that these properties
are necessary in order to provide the correct number of independent operators.
The comparison with the classical theory is done. | math-ph |
Some remarks on the visible points of a lattice: We comment on the set of visible points of a lattice and its Fourier
transform, thus continuing and generalizing previous work by Schroeder and
Mosseri. A closed formula in terms of Dirichlet series is obtained for the
Bragg part of the Fourier transform. We compare this calculation with the
outcome of an optical Fourier transform of the visible points of the 2D square
lattice. | math-ph |
Angular Gelfand--Tzetlin Coordinates for the Supergroup UOSp(k_1/2k_2): We construct Gelfand--Tzetlin coordinates for the unitary orthosymplectic
supergroup UOSp(k_1/2k_2). This extends a previous construction for the unitary
supergroup U(k_1/k_2). We focus on the angular Gelfand--Tzetlin coordinates,
i.e. our coordinates stay in the space of the supergroup. We also present a
generalized Gelfand pattern for the supergroup UOSp(k_1/2k_2) and discuss
various implications for representation theory. | math-ph |
Manifolds obtained by soldering together points, lines, etc: This text is the extended version of a talk given at the conference Geometry,
Topology, QFT and Cosmology hold from May 28 to May 30, 2008 at the
Observatoire de Paris. We explore the notion of solder (or soldering form) in
differential geometry and propose an alternative interpretation of it,
motivated by the search of an accurate mathematical description of the General
Relativity. This new interpretation leads naturally to imagine a rich family of
new geometries which has not yet a satisfactory definition in general. We try
however to communicate to the reader an intuition of such geometries through
some examples and review quickly some possible applications in physics. The
basic objects in this geometry are not points (i.e. 0-dimensional), but
(p-1)-dimensional. | math-ph |
Extended Z-invariance for integrable vector and face models and
multi-component integrable quad equations: In a previous paper, the author has established an extension of the
Z-invariance property for integrable edge-interaction models of statistical
mechanics, that satisfy the star-triangle relation (STR) form of the
Yang-Baxter equation (YBE). In the present paper, an analogous extended
Z-invariance property is shown to also hold for integrable vector models and
interaction-round-a-face (IRF) models of statistical mechanics respectively. As
for the previous case of the STR, the Z-invariance property is shown through
the use of local cubic-type deformations of a 2-dimensional surface associated
to the models, which allow an extension of the models onto a subset of next
nearest neighbour vertices of $\mathbb{Z}^3$, while leaving the partition
functions invariant. These deformations are permitted as a consequence of the
respective YBE's satisfied by the models. The quasi-classical limit is also
considered, and it is shown that an analogous Z-invariance property holds for
the variational formulation of classical discrete Laplace equations which arise
in this limit. From this limit, new integrable 3D-consistent multi-component
quad equations are proposed, which are constructed from a degeneration of the
equations of motion for IRF Boltzmann weights. | math-ph |
Mass Dependence of Quantum Energy Inequality Bounds: In a recent paper [J. Math. Phys. 47 082303 (2006)], Quantum Energy
Inequalities were used to place simple geometrical bounds on the energy
densities of quantum fields in Minkowskian spacetime regions. Here, we refine
this analysis for massive fields, obtaining more stringent bounds which decay
exponentially in the mass. At the technical level this involves the
determination of the asymptotic behaviour of the lowest eigenvalue of a family
of polyharmonic differential equations, a result which may be of independent
interest. We compare our resulting bounds with the known energy density of the
ground state on a cylinder spacetime. In addition, we generalise some of our
technical results to general $L^p$-spaces and draw comparisons with a similar
result in the literature. | math-ph |
Solutions for the Klein-Gordon and Dirac equations on the lattice based
on Chebyshev polynomials: The main goal of this paper is to adopt a multivector calculus scheme to
study finite difference discretizations of Klein-Gordon and Dirac equations for
which Chebyshev polynomials of the first kind may be used to represent a set of
solutions. The development of a well-adapted discrete Clifford calculus
framework based on spinor fields allows us to represent, using solely
projection based arguments, the solutions for the discretized Dirac equations
from the knowledge of the solutions of the discretized Klein-Gordon equation.
Implications of those findings on the interpretation of the lattice fermion
doubling problem is briefly discussed. | math-ph |
A 2-adic approach of the human respiratory tree: We propose here a general framework to address the question of trace
operators on a dyadic tree. This work is motivated by the modeling of the human
bronchial tree which, thanks to its regularity, can be extrapolated in a
natural way to an infinite resistive tree. The space of pressure fields at
bifurcation nodes of this infinite tree can be endowed with a Sobolev space
structure, with a semi-norm which measures the instantaneous rate of dissipated
energy. We aim at describing the behaviour of finite energy pressure fields
near the end. The core of the present approach is an identification of the set
of ends with the ring Z_2 of 2-adic integers. Sobolev spaces over Z_2 can be
defined in a very natural way by means of Fourier transform, which allows us to
establish precised trace theorems which are formally quite similar to those in
standard Sobolev spaces, with a Sobolev regularity which depends on the growth
rate of resistances, i.e. on geometrical properties of the tree. Furthermore,
we exhibit an explicit expression of the "ventilation operator", which maps
pressure fields at the end of the tree onto fluxes, in the form of a
convolution by a Riesz kernel based on the 2-adic distance. | math-ph |
A $\mathbb{Z}_{2}$-Topological Index for Quasi-Free Fermions: We use infinite dimensional self-dual $\mathrm{CAR}$ $C^{*}$-algebras to
study a $\mathbb{Z}_{2}$-index, which classifies free-fermion systems embedded
on $\mathbb{Z}^{d}$ disordered lattices. Combes-Thomas estimates are pivotal to
show that the $\mathbb{Z}_{2}$-index is uniform with respect to the size of the
system. We additionally deal with the set of ground states to completely
describe the mathematical structure of the underlying system. Furthermore, the
weak$^{*}$-topology of the set of linear functionals is used to analyze paths
connecting different sets of ground states. | math-ph |
Review of a Simplified Approach to study the Bose gas at all densities: In this paper, we will review the results obtained thus far by Eric A.
Carlen, Elliott H. Lieb and me on a Simplified Approach to the Bose gas. The
Simplified Approach yields a family of effective one-particle equations, which
capture some non-trivial physical properties of the Bose gas at both low and
high densities, and even some of the behavior at intermediate densities. In
particular, the Simplified Approach reproduces Bogolyubov's estimates for the
ground state energy and condensate fraction at low density, as well as the
mean-field estimate for the energy at high densities. We will also discuss a
phase that appears at intermediate densities with liquid-like properties. The
simplest of the effective equations in the Simplified Approach can be studied
analytically, and we will review several results about it; the others are so
far only amenable to numerical analysis, and we will discuss several numerical
results. We will start by reviewing some results and conjectures on the Bose
gas, and then introduce the Simplified Approach and its derivation from the
Bose gas. We will then discuss the predictions of the Simplified Approach and
compare these to results and conjectures about the Bose gas. Finally, we will
discuss a few open problems about the Simplified Approach. | math-ph |
KdV waves in atomic chains with nonlocal interactions: We consider atomic chains with nonlocal particle interactions and prove the
existence of near-sonic solitary waves. Both our result and the general proof
strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV
limit of chains with nearest neighbor interactions but differ in the following
two aspects: First, we allow for a wider class of atomic systems and must hence
replace the distance profile by the velocity profile. Second, in the asymptotic
analysis we avoid a detailed Fourier pole characterization of the nonlocal
integral operators and employ the contraction mapping principle to solve the
final fixed point problem. | math-ph |
Modeling error in Approximate Deconvolution Models: We investigate the assymptotic behaviour of the modeling error in approximate
deconvolution model in the 3D periodic case, when the order $N$ of
deconvolution goes to $\infty$. We consider successively the generalised
Helmholz filters of order $p$ and the Gaussian filter. For Helmholz filters, we
estimate the rate of convergence to zero thanks to energy budgets, Gronwall's
Lemma and sharp inequalities about Fouriers coefficients of the residual
stress. We next show why the same analysis does not allow to conclude
convergence to zero of the error modeling in the case of Gaussian filter,
leaving open issues. | math-ph |
Quasilocal conservation laws in XXZ spin-1/2 chains: open, periodic and
twisted boundary conditions: A continuous family of quasilocal exact conservation laws is constructed in
the anisotropic Heisenberg (XXZ) spin-1/2 chain for periodic (or twisted)
boundary conditions and for a set of commensurate anisotropies densely covering
the entire easy plane interaction regime. All local conserved operators follow
from the standard (Hermitian) transfer operator in fundamental representation
(with auxiliary spin s=1/2), and are all even with respect to a spin flip
operation. However, the quasilocal family is generated by differentiation of a
non-Hermitian highest weight transfer operator with respect to a complex
auxiliary spin representation parameter s and includes also operators of odd
parity. For a finite chain with open boundaries the time derivatives of
quasilocal operators are not strictly vanishing but result in operators
localized near the boundaries of the chain. We show that a simple modification
of the non-Hermitian transfer operator results in exactly conserved, but still
quasilocal operators for periodic or generally twisted boundary conditions. As
an application, we demonstrate that implementing the new exactly conserved
operator family for estimating the high-temperature spin Drude weight results,
in the thermodynamic limit, in exactly the same lower bound as for almost
conserved family and open boundaries. Under the assumption that the bound is
saturating (suggested by agreement with previous thermodynamic Bethe ansatz
calculations) we propose a simple explicit construction of infinite time
averages of local operators such as the spin current. | math-ph |
Pade approximants of random Stieltjes series: We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 +
>...))) where the s_n are independent random variables with the same gamma
distribution. For every realisation of the sequence, S(t) defines a Stieltjes
function. We study the convergence of the finite truncations of the continued
fraction or, equivalently, of the diagonal Pade approximants of the function
S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional
disordered system, and the results of Marklof et al. (2005), we obtain explicit
formulae (in terms of modified Bessel functions) for the almost-sure rate of
convergence of these approximants, and for the almost-sure distribution of
their poles. | math-ph |
Universal microscopic correlation functions for products of truncated
unitary matrices: We investigate the spectral properties of the product of $M$ complex
non-Hermitian random matrices that are obtained by removing $L$ rows and
columns of larger unitary random matrices uniformly distributed on the group
${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or random
contractions. We first derive the joint probability distribution for the
eigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by a
standard determinantal point process in the complex plane. The weight however
is non-standard and can be expressed in terms of the Meijer G-function. The
explicit knowledge of all eigenvalue correlation functions and the
corresponding kernel allows us to take various large $N$ (and $L$) limits at
fixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condense
on a domain inside the unit circle. At the edge and in the bulk we find the
same universal microscopic kernel as for a single complex non-Hermitian matrix
from the Ginibre ensemble. At the origin we find the same new universality
classes labelled by $M$ as for the product of $M$ matrices from the Ginibre
ensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinity
leads to weak non-unitarity, with most eigenvalues on the unit circle as for
unitary matrices. Here we find a new microscopic edge kernel that generalizes
the known results for M=1. We briefly comment on the case when each product
matrix results from a truncation of different size $L_j$. | math-ph |
Localization for One Dimensional, Continuum, Bernoulli-Anderson Models: We use scattering theoretic methods to prove strong dynamical and exponential
localization for one dimensional, continuum, Anderson-type models with singular
distributions; in particular the case of a Bernoulli distribution is covered.
The operators we consider model alloys composed of at least two distinct types
of randomly dispersed atoms. Our main tools are the reflection and transmission
coefficients for compactly supported single site perturbations of a periodic
background which we use to verify the necessary hypotheses of multi-scale
analysis. We show that non-reflectionless single sites lead to a discrete set
of exceptional energies away from which localization occurs. | math-ph |
Generalisation of the Eyring-Kramers transition rate formula to
irreversible diffusion processes: In the small noise regime, the average transition time between metastable
states of a reversible diffusion process is described at the logarithmic scale
by Arrhenius' law. The Eyring-Kramers formula classically provides a
subexponential prefactor to this large deviation estimate. For irreversible
diffusion processes, the equivalent of Arrhenius' law is given by the
Freidlin-Wentzell theory. In this paper, we compute the associated prefactor
and thereby generalise the Eyring-Kramers formula to irreversible diffusion
processes. In our formula, the role of the potential is played by
Freidlin-Wentzell's quasipotential, and a correction depending on the
non-Gibbsianness of the system along the instanton is highlighted. Our analysis
relies on a WKB analysis of the quasistationary distribution of the process in
metastable regions, and on a probabilistic study of the process in the
neighbourhood of saddle-points of the quasipotential. | math-ph |
A Lieb-Thirring inequality for extended anyons: We derive a Pauli exclusion principle for extended fermion-based anyons of
any positive radius and any non-trivial statistics parameter. That is, we
consider 2D fermionic particles coupled to magnetic flux tubes of non-zero
radius, and prove a Lieb-Thirring inequality for the corresponding many-body
kinetic energy operator. The implied constant is independent of the radius of
the flux tubes, and proportional to the statistics parameter. | math-ph |
A perturbation theory approach to the stability of the Pais-Uhlenbeck
oscillator: We present a detailed analysis of the orbital stability of the Pais-Uhlenbeck
oscillator, using Lie-Deprit series and Hamiltonian normal form theories. In
particular, we explicitly describe the reduced phase space for this Hamiltonian
system and give a proof for the existence of stable orbits for a certain class
of self-interaction, found numerically in previous works, by using singular
symplectic reduction. | math-ph |
Solutions of Painlevé II on real intervals: novel approximating
sequences: Novel sequences of approximants to solutions of Painlev\'e II on finite
intervals of the real line, with Neumann boundary conditions, are constructed.
Numerical experiments strongly suggest convergence of these sequences in a
surprisingly wide range of cases, even ones where ordinary perturbation series
fail to converge. These sequences are here labeled extraordinary because of
their unusual properties. Each element of such a sequence is defined on its own
interval. As the sequence (apparently) converges to a solution of the
corresponding boundary value problem for Painlev\'e II, these intervals
themselves (apparently) converge to the defining interval for that problem, and
an associated sequence of constants (apparently) converges to the constant term
in the Painlev\'e II equation itself. Each extraordinary sequence is
constructed in a nonlinear fashion from a perturbation series approximation to
the solution of a supplementary boundary value problem, involving a
generalization of Painlev\'e II that arises in studies of electrodiffusion. | math-ph |
A Classical Limit of Noumi's $q$-Integral Operator: We demonstrate how a known Whittaker function integral identity arises from
the $t=0$ and $q\to 1$ limit of the Macdonald polynomial eigenrelation
satisfied by Noumi's $q$-integral operator. | math-ph |
A Gaussian Beam Construction of de Haas-vanAlfven Resonances: The de Haas-van Alfven Effect arises when a metallic crystal is placed in a
constant magnetic field. One observes equally spaced peaks in its physical
properties as the strength of the magnetic field is varied. Onsager explained
that the spacing of these peaks depended on the areas in pseudo momentum space
of the regions bounded by the intersections of a Fermi surface with planes
perpendicular to the magnetic field. Hence, the dHvA effect has been quite
useful in mapping Fermi surfaces. The purpose of this note is to explain
Onsager's observation using gaussian beams. | math-ph |
Comparison of methods to determine point-to-point resistance in nearly
rectangular networks with application to a hammock network: Considerable progress has recently been made in the development of techniques
to exactly determine two-point resistances in networks of various topologies.
In particular, two types of method have emerged. One is based on potentials and
the evaluation of eigenvalues and eigenvectors of the Laplacian matrix
associated with the network or its minors. The second method is based on a
recurrence relation associated with the distribution of currents in the
network. Here, these methods are compared and used to determine the resistance
distances between any two nodes of a network with topology of a hammock. | math-ph |
On the Thermodynamics of Particles Obeying Monotone Statistics: The aim of the present paper is to provide a preliminary investigation of the
thermodynamics of particles obeying monotone statistics. To render the
potential physical applications realistic, we propose a modified scheme called
block-monotone, based on a partial order arising from the natural one on the
spectrum of a positive Hamiltonian with compact resolvent. The block-monotone
scheme is never comparable with the weak monotone one and is reduced to the
usual monotone scheme whenever all the eigenvalues of the involved Hamiltonian
are non-degenerate. Through a detailed analysis of a model based on the quantum
harmonic oscillator, we can see that: (a) the computation of the
grand-partition function does not require the Gibbs correction factor $n!$
(connected with the indistinguishability of particles) in the various terms of
its expansion with respect to the activity; and (b) the decimation of terms
contributing to the grand-partition function leads to a kind of "exclusion
principle" analogous to the Pauli exclusion principle enjoined by Fermi
particles, which is more relevant in the high-density regime and becomes
negligible in the low-density regime, as expected. | math-ph |
The Simplified approach to the Bose gas without translation invariance: The Simplified approach to the Bose gas was introduced by Lieb in 1963 to
study the ground state of systems of interacting Bosons. In a series of recent
papers, it has been shown that the Simplified approach exceeds earlier
expectations, and gives asymptotically accurate predictions at both low and
high density. In the intermediate density regime, the qualitative predictions
of the Simplified approach have also been found to agree very well with Quantum
Monte Carlo computations. Until now, the Simplified approach had only been
formulated for translation invariant systems, thus excluding external
potentials, and non-periodic boundary conditions. In this paper, we extend the
formulation of the Simplified approach to a wide class of systems without
translation invariance. This also allows us to study observables in translation
invariant systems whose computation requires the symmetry to be broken. Such an
observable is the momentum distribution, which counts the number of particles
in excited states of the Laplacian. In this paper, we show how to compute the
momentum distribution in the Simplified approach, and show that, for the Simple
Equation, our prediction matches up with Bogolyubov's prediction at low
densities, for momenta extending up to the inverse healing length. | math-ph |
Algebraic area enumeration of random walks on the honeycomb lattice: We study the enumeration of closed walks of given length and algebraic area
on the honeycomb lattice. Using an irreducible operator realization of
honeycomb lattice moves, we map the problem to a Hofstadter-like Hamiltonian
and show that the generating function of closed walks maps to the grand
partition function of a system of particles with exclusion statistics of order
$g=2$ and an appropriate spectrum, along the lines of a connection previously
established by two of the authors. Reinterpreting the results in terms of the
standard Hofstadter spectrum calls for a mixture of $g=1$ (fermion) and $g=2$
exclusion whose physical meaning and properties require further elucidation. In
this context we also obtain some unexpected Fibonacci sequences within the
weights of the combinatorial factors appearing in the counting of walks. | math-ph |
Casimir Energy for a Wedge with Three Surfaces and for a Pyramidal
Cavity: Casimir energy calculations for the conformally coupled massless scalar field
for a wedge defined by three intersecting planes and for a pyramid with four
triangular surfaces are presented. The group generated by reflections are
employed in the formulation of the required Green functions and the wave
functions. | math-ph |
Boundary Value Problem for $r^2 d^2 f/dr^2 + f = f^3$ (III): Global
Solution and Asymptotics: Based on the results in the previous papers that the boundary value problem
$y'' - y' + y = y^3, y(0) = 0, y(\infty) =1$ with the condition $y(x) > 0$ for
$0<x<\infty$ has a unique solution $y^*(x)$, and $a^*= y^{*^{'}}(0)$ satisfies
$0<a^*<1/4$, in this paper we show that $y'' - y' + y = y^3, -\infty < x < 0$,
with the initial conditions $ y(0) = 0, y'(0) = a^*$ has a unique solution by
using functional analysis method. So we get a globally well defined bounded
function $y^*(x), -\infty < x < +\infty$. The asymptotics of $y^*(x)$ as $x \to
- \infty$ and as $x \to +\infty$ are obtained, and the connection formulas for
the parameters in the asymptotics and the numerical simulations are also given.
Then by the properties of $y^*(x)$, the solution to the boundary value problem
$r^2 f'' + f = f^3, f(0)= 0, f(\infty)=1$ is well described by the asymptotics
and the connection formulas. | math-ph |
The Validity of the Local Density Approximation for Smooth Short Range
Interaction Potentials: In the full quantum theory, the energy of a many-body quantum system with a
given one-body density is described by the Levy-Lieb functional. It is exact,
but very complicated to compute. For practical computations, it is useful to
introduce the Local Density Approximation which is based on the local energy of
constant densities. The aim of this paper is to make a rigorous connection
between the Levy-Lieb functional theory and the Local Density Approximation.
Our justification is valid for fermionic systems with a general class of smooth
short range interaction potentials, in the regime of slowly varying densities.
We follow a general approach developed by Lewin, Lieb and Seiringer for Coulomb
potential, but avoid using any special properties of the potential including
the scaling property and screening effects for the localization of the energy. | math-ph |
Stationary solutions for a model of amorphous thin-film growth: We consider a class of stochastic partial differential equations arising as a
model for amorphous thin film growth. Using a spectral Galerkin method, we
verify the existence of stationary mild solutions, although the specific nature
of the nonlinearity prevents us from showing the uniqueness of the solutions as
well as their boundedness (in time). | math-ph |
A nonlocal formulation for the problem of microwave heating of material
with temperature dependent conductivity: Microwave electromagnetic heating are widely used in many industrial
processes. The mathematics involved is based on the Maxwell's equations coupled
with the heat equation. The thermal conductivity is strongly dependent on the
temperature, itself an unknown of the system of P.D.E. We propose here a model
which simplifies this coupling using a nonlocal term as the source of heating.
We prove that the corresponding mathematical initial-boundary value problem has
solutions using the Schauder's fixed point theorem. | math-ph |
Rényi entropies and nonlinear diffusion equations: Since their introduction in the early sixties, the R\'enyi entropies have
been used in many contexts, ranging from information theory to astrophysics,
turbulence phenomena and others. In this note, we enlighten the main
connections between R\'enyi entropies and nonlinear diffusion equations. In
particular, it is shown that these relationships allow to prove various
functional inequalities in sharp form. | math-ph |
Discrete velocity Boltzmann eqations in the plane:stationary solutions
for a generic class: The paper proves existence of renormalized stationary solutions for a dense
class of discrete velocity Boltzmann equations in the plane with given ingoing
boundary values. The proof is based on the construction of a sequence of
approximations with L1 compactness for the integrated collision frequency and
gain term. Compactness is obtained using the Kolmogorov-Riesz theorem. | math-ph |
Non-Lyapunov annealed decay for 1d Anderson eigenfunctions: In [10] Jitomirskaya, Kr\"uger and Liu analysed the dynamical decay in
expectation for the super-critical almost-Mathieu operator in function of the
coupling parameter , showing that it is equal to the Lyapunov exponent of its
transfer matrix cocycle, and asked whether the same is true for the 1d Anderson
model. We show that this is essentially never true when the disorder parameter
is sufficiently large. | math-ph |
How to commute: A simple exposition of the rarely discussed fact that a set of free boson
fields describing different, i.e. kinematically different particle types can be
quantized with mutual anticommutation relations is given by the explicit
construction of the Klein transformations changing anticommutation relations
into commutation relations. The q-analog of the presented results is also
treated. The analogous situation for two independent free fermion fields with
mutual commutation or anticommutation relations is briefly investigated. | math-ph |
Eigenvalue Separation in Some Random Matrix Models: The eigenvalue density for members of the Gaussian orthogonal and unitary
ensembles follows the Wigner semi-circle law. If the Gaussian entries are all
shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in
the large N limit a single eigenvalue will separate from the support of the
Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis
of the secular equation for the eigenvalue condition, we compare this effect to
analogous effects occurring in general variance Wishart matrices and matrices
from the shifted mean chiral ensemble. We undertake an analogous comparative
study of eigenvalue separation properties when the size of the matrices are
fixed and c goes to infinity, and higher rank analogues of this setting. This
is done using exact expressions for eigenvalue probability densities in terms
of generalized hypergeometric functions, and using the interpretation of the
latter as a Green function in the Dyson Brownian motion model. For the shifted
mean Gaussian unitary ensemble and its analogues an alternative approach is to
use exact expressions for the correlation functions in terms of classical
orthogonal polynomials and associated multiple generalizations. By using these
exact expressions to compute and plot the eigenvalue density, illustrations of
the various eigenvalue separation effects are obtained. | math-ph |
Longitudinal permeability of collisional plasmas under arbitrary degree
of degeneration of electron gas: Electric conductivity and dielectric permeability of the non-degenerate
electronic gas for the collisional plasmas under arbitrary degree of
degeneration of electron gas is found. The kinetic equation of Wigner - Vlasov
- Boltzmann with collision integral in relaxation form BGK (Bhatnagar, Gross
and Krook) in coordinate space is used. Dielectric permeability with using of
the relaxation equation in the momentum space has been received by Mermin.
Comparison with Mermin's formula has been realized. It is shown, that in the
limit when Planck's constant tends to zero expression for dielectric
permeability passes in the classical. | math-ph |
Toward a classification of semidegenerate 3D superintegrable systems: Superintegrable systems of 2nd order in 3 dimensions with exactly 3-parameter
potentials are intriguing objects. Next to the nondegenerate 4-parameter
potential systems they admit the maximum number of symmetry operators but their
symmetry algebras don't close under commutation and not enough is known about
their structure to give a complete classification. Some examples are known for
which the 3-parameter system can be extended to a 4th order superintegrable
system with a 4-parameter potential and 6 linearly independent symmetry
generators. In this paper we use B\^ocher contractions of the conformal Lie
algebra $so(5,C)$ to itself to generate a large family of 3-parameter systems
with 4th order extensions, on a variety of manifolds, and all from B\^ocher
contractions of a single "generic" system on the 3-sphere. We give a
contraction scheme relating these systems. The results have myriad applications
for finding explicit solutions for both quantum and classical systems. | math-ph |
Nodal domains of a non-separable problem - the right angled isosceles
triangle: We study the nodal set of eigenfunctions of the Laplace operator on the right
angled isosceles triangle. A local analysis of the nodal pattern provides an
algorithm for computing the number of nodal domains for any eigenfunction. In
addition, an exact recursive formula for the number of nodal domains is found
to reproduce all existing data. Eventually we use the recursion formula to
analyse a large sequence of nodal counts statistically. Our analysis shows that
the distribution of nodal counts for this triangular shape has a much richer
structure than the known cases of regular separable shapes or completely
irregular shapes. Furthermore we demonstrate that the nodal count sequence
contains information about the periodic orbits of the corresponding classical
ray dynamics. | math-ph |
Spontaneous Resonances and the Coherent States of the Queuing Networks: We present an example of a highly connected closed network of servers, where
the time correlations do not go to zero in the infinite volume limit. This
phenomenon is similar to the continuous symmetry breaking at low temperatures
in statistical mechanics. The role of the inverse temperature is played by the
average load. | math-ph |
The spectral gap of a fractional quantum Hall system on a thin torus: We study a fractional quantum Hall system with maximal filling $ \nu = 1/3 $
in the thin torus limit. The corresponding Hamiltonian is a truncated version
of Haldane's pseudopotential, which upon a Jordan-Wigner transformation is
equivalent to a one-dimensional quantum spin chain with periodic boundary
conditions. Our main result is a lower bound on the spectral gap of this
Hamiltonian, which is uniform in the system size and total particle number. The
gap is also uniform with respect to small values of the coupling constant in
the model. The proof adapts the strategy of individually estimating the gap in
invariant subspaces used for the bosonic $ \nu = 1/2 $ model to the present
fermionic case. | math-ph |
Integrable spin-1/2 Richardson-Gaudin XYZ models in an arbitrary
magnetic field: We establish the most general class of spin-1/2 integrable Richardson-Gaudin
models including an arbitrary magnetic field, returning a fully anisotropic
(XYZ) model. The restriction to spin-1/2 relaxes the usual integrability
constraints, allowing for a general solution where the couplings between spins
lack the usual antisymmetric properties of Richardson-Gaudin models. The full
set of conserved charges are constructed explicitly and shown to satisfy a set
of quadratic equations, allowing for the numerical treatment of a fully
anisotropic central spin in an external magnetic field. While this approach
does not provide expressions for the exact eigenstates, it allows their
eigenvalues to be obtained, and expectation values of local observables can
then be calculated from the Hellmann-Feynman theorem. | math-ph |
Reduction of multidimensional non-linear d'Alembert equations to
two-dimensional equations: ansatzes, compatibility of reduction conditions: We study conditions of reduction of multidimensional wave equations - a
system of d'Alembert and Hamilton equations. Necessary conditions for
compatibility of such reduction conditions are proved. Possible types of the
reduced equations and ansatzes are described. We also provide a brief review of
the literature with respect to compatibility of the system of d'Alembert and
Hamilton equations and construction of solutions for the nonlinear d'Alembert
equation. | math-ph |
Coherent states for the supersymmetric partners of the truncated
oscillator: We build the coherent states for a family of solvable singular Schr\"odinger
Hamiltonians obtained through supersymmetric quantum mechanics from the
truncated oscillator. The main feature of such systems is the fact that their
eigenfunctions are not completely connected by their natural ladder operators.
We find a definition that behaves appropriately in the complete Hilbert space
of the system, through linearised ladder operators. In doing so, we study basic
properties of such states like continuity in the complex parameter, resolution
of the identity, probability density, time evolution and possibility of
entanglement. | math-ph |
Symmetry of Differential Equations and Quantum Theory: The symmetry study of main differential equations of mechanics and
electrodynamics has shown, that differential equations, which are invariant
under transformations of groups, which are symmetry groups of mathematical
numbers (considered within the frames of the number theory) determine the
mathematical nature of the quantities, incoming in given equations. It allowed
to proof the main postulate of quantum mechanics, consisting in that, that to
any mechanical quantity can be set up into the correspondence the Hermitian
matrix by quantization.
High symmetry of Maxwell equations allows to show, that to quantities,
incoming in given equations can be set up into the correspondence the
Quaternion (twice-Hermitian) matrix by their quantization.
It is concluded, that the equations of the dynamics of mechanical systems are
not invariant under transformations of quaternion multiplicative group and,
consecuently, direct application of quaternions with usually used basis \{e, i,
j, k \} to build the new version of quantum mechanics, which was undertaken in
the number of modern publications, is incorrect. It is the consequence of
non-abelian character of given group. At the same time we have found the
correct ways for the creation of the new versions of quantum mechanics on the
quaternion base by means of choice of new bases in quaternion ring, from which
can be formed the bases for complex numbers under multiplicative groups of
which the equations of the dynamics of mechanical systems are invariant. | math-ph |
Scattering of EM waves by many small perfectly conducting or impedance
bodies: A theory of electromagnetic (EM) wave scattering by many small particles of
an arbitrary shape is developed. The particles are perfectly conducting or
impedance. For a small impedance particle of an arbitrary shape an explicit
analytical formula is derived for the scattering amplitude. The formula holds
as $a\to 0$, where $a$ is a characteristic size of the small particle and the
wavelength is arbitrary but fixed. The scattering amplitude for a small
impedance particle is shown to be proportional to $a^{2-\kappa}$, where
$\kappa\in [0,1)$ is a parameter which can be chosen by an experimenter as
he/she wants. The boundary impedance of a small particle is assumed to be of
the form $\zeta=ha^{-\kappa}$, where $h=$const, Re$h\ge 0$. The scattering
amplitude for a small perfectly conducting particle is proportional to $a^3$,
it is much smaller than that for the small impedance particle. The many-body
scattering problem is solved under the physical assumptions $a\ll d\ll
\lambda$, where $d$ is the minimal distance between neighboring particles and
$\lambda$ is the wavelength. The distribution law for the small impedance
particles is $\mathcal{N}(\delta)\sim\int_{\delta}N(x)dx$ as $a\to 0$. Here
$N(x)\ge 0$ is an arbitrary continuous function that can be chosen by the
experimenter and $\mathcal{N}(\delta)$ is the number of particles in an
arbitrary sub-domain $\Delta$. It is proved that the EM field in the medium
where many small particles, impedance or perfectly conducting, are distributed,
has a limit, as $a\to 0$ and a differential equation is derived for the
limiting field. On this basis the recipe is given for creating materials with a
desired refraction coefficient by embedding many small impedance particles into
a given material. | math-ph |
Construction of two-dimensional quantum field models through
Longo-Witten endomorphisms: We present a procedure to construct families of local, massive and
interacting Haag-Kastler nets on the two-dimensional spacetime through an
operator-algebraic method. An existence proof of local observable is given
without relying on modular nuclearity.
By a similar technique, another family of wedge-local nets is constructed
using certain endomorphisms of conformal nets recently studied by Longo and
Witten. | math-ph |
On the Wigner function of the relativistic finite-difference oscillator
in an external field: The phase-space representation for a relativistic linear oscillator in a
homogeneous external field expressed through the finite-difference equation is
constructed. Explicit expressions of the relativistic oscillator Wigner
quasi-distribution function for the stationary states as well as of states of
thermodynamical equilibrium are obtained and their correct limits are shown. | math-ph |
Spectral equations for the modular oscillator: Motivated by applications for non-perturbative topological strings in toric
Calabi--Yau manifolds, we discuss the spectral problem for a pair of commuting
modular conjugate (in the sense of Faddeev) Harper type operators,
corresponding to a special case of the quantized mirror curve of local
$\mathbb{P}^1\times\mathbb{P}^1$ and complex values of Planck's constant. We
illustrate our analytical results by numerical calculations. | math-ph |
On the existence of stable charged Q-balls: This paper concerns hylomorphic solitons, namely stable, solitary waves whose
existence is related to the ratio energy/charge. In theoretical physics, the
name Q-ball refers to a type of hylomorphic solitons or soli- tary waves
relative to the Nonlinear Klein-Gordon equation (NKG). We are interested in the
existence of charged Q-balls, namely Q-balls for the Nonlinear Klein-Gordon
equation coupled with the Maxwell equations (NKGM). In this case the charge
reduces to the electric charge. The main result of this paper establishes that
stable, charged Q-balls exist provided that the interaction between matter and
the gauge field is sufficiently small. | math-ph |
Analytic Bethe ansatz and functional equations associated with any
simple root systems of the Lie superalgebra sl(r+1|s+1): The Lie superalgebra sl(r+1|s+1) admits several inequivalent choices of
simple root systems. We have carried out analytic Bethe ansatz for any simple
root systems of sl(r+1|s+1). We present transfer matrix eigenvalue formulae in
dressed vacuum form, which are expressed as the Young supertableaux with some
semistandard-like conditions. These formulae have determinant expressions,
which can be viewed as quantum analogue of Jacobi-Trudi and Giambelli formulae
for sl(r+1|s+1). We also propose a class of transfer matrix functional
relations, which is specialization of Hirota bilinear difference equation.
Using the particle-hole transformation, relations among the Bethe ansatz
equations for various kinds of simple root systems are discussed. | math-ph |
Asymptotical study of two-layered discrete waveguide with a weak
coupling: A thin two-layered waveguide is considered. The governing equations for this
waveguide is a matrix Klein--Gordon equation of dimension~2. A formal solution
of this system in the form of a double integral can be obtained by using
Fourier transformation. Then, the double integral can be reduced to a single
integral with the help of residue integration with respect to the time
frequency. However, such an integral can be difficult to estimate since it
involves branching and oscillating functions. This integral is studied
asymptotically. A zone diagram technique is proposed to represent the set of
possible asymptotic formulae. The zone diagram generalizes the concept of
far-field and near-field zones. | math-ph |
Exact solutions of the Liénard and generalized Liénard type ordinary
non-linear differential equations obtained by deforming the phase space
coordinates of the linear harmonic oscillator: We investigate the connection between the linear harmonic oscillator equation
and some classes of second order nonlinear ordinary differential equations of
Li\'enard and generalized Li\'enard type, which physically describe important
oscillator systems. By using a method inspired by quantum mechanics, and which
consist on the deformation of the phase space coordinates of the harmonic
oscillator, we generalize the equation of motion of the classical linear
harmonic oscillator to several classes of strongly non-linear differential
equations. The first integrals, and a number of exact solutions of the
corresponding equations are explicitly obtained. The devised method can be
further generalized to derive explicit general solutions of nonlinear second
order differential equations unrelated to the harmonic oscillator. Applications
of the obtained results for the study of the travelling wave solutions of the
reaction-convection-diffusion equations, and of the large amplitude free
vibrations of a uniform cantilever beam are also presented. | math-ph |
Extrapolation of perturbation-theory expansions by self-similar
approximants: The problem of extrapolating asymptotic perturbation-theory expansions in
powers of a small variable to large values of the variable tending to infinity
is investigated. The analysis is based on self-similar approximation theory.
Several types of self-similar approximants are considered and their use in
different problems of applied mathematics is illustrated. Self-similar
approximants are shown to constitute a powerful tool for extrapolating
asymptotic expansions of different natures. | math-ph |
Nonlinearly-PT-symmetric systems: spontaneous symmetry breaking and
transmission resonances: We introduce a class of PT-symmetric systems which include mutually matched
nonlinear loss and gain (inother words, a class of PT-invariant Hamiltonians in
which both the harmonic and anharmonic parts are non-Hermitian). For a basic
system in the form of a dimer, symmetric and asymmetric eigenstates, including
multistable ones, are found analytically. We demonstrate that, if coupled to a
linear chain, such a nonlinear PT-symmetric dimer generates new types of
nonlinear resonances, with the completely suppressed or greatly amplified
transmission, as well as a regime similar to the electromagnetically-induced
transparency (EIT). The implementation of the systems is possible in various
media admitting controllable linear and nonlinear amplification of waves. | math-ph |
Instability of an inverse problem for the stationary radiative transport
near the diffusion limit: In this work, we study the instability of an inverse problem of radiative
transport equation with angularly averaged measurement near the diffusion
limit, i.e. the normalized mean free path (the Knudsen number) $0 < \eps \ll
1$. It is well-known that there is a transition of stability from H\"{o}lder
type to logarithmic type with $\eps\to 0$, the theory of this transition of
stability is still an open problem. In this study, we show the transition of
stability by establishing the balance of two different regimes depending on the
relative sizes of $\eps$ and the perturbation in measurements. When $\eps$ is
sufficiently small, we obtain exponential instability, which stands for the
diffusive regime, and otherwise we obtain H\"{o}lder instability instead, which
stands for the transport regime. | math-ph |
Mathematical predominance of Dirichlet condition for the one-dimensional
Coulomb potential: We restrict a quantum particle under a coulombian potential (i.e., the
Schr\"odinger operator with inverse of the distance potential) to three
dimensional tubes along the x-axis and diameter $\varepsilon$, and study the
confining limit $\varepsilon\to0$. In the repulsive case we prove a strong
resolvent convergence to a one-dimensional limit operator, which presents
Dirichlet boundary condition at the origin. Due to the possibility of the
falling of the particle in the center of force, in the attractive case we need
to regularize the potential and also prove a norm resolvent convergence to the
Dirichlet operator at the origin. Thus, it is argued that, among the infinitely
many self-adjoint realizations of the corresponding problem in one dimension,
the Dirichlet boundary condition at the origin is the reasonable
one-dimensional limit. | math-ph |
Level-rank duality via tensor categories: We give a new way to derive branching rules for the conformal embedding
$$(\asl_n)_m\oplus(\asl_m)_n\subset(\asl_{nm})_1.$$ In addition, we show that
the category $\Cc(\asl_n)_m^0$ of degree zero integrable highest weight
$(\asl_n)_m$-representations is braided equivalent to $\Cc(\asl_m)_n^0$ with
the reversed braiding. | math-ph |
Subgroup type coordinates and the separation of variables in
Hamilton-Jacobi and Schrődinger equations: Separable coordinate systems are introduced in the complex and real
four-dimensional flat spaces. We use maximal Abelian subgroups to generate
coordinate systems with a maximal number of ignorable variables. The results
are presented (also graphically) in terms of subgroup chains. Finally, the
explicit solutions of the Schr\H{o}dinger equation in the separable coordinate
systems are computed. | math-ph |
Localization in Abelian Chern-Simons Theory: Chern-Simons theory on a closed contact three-manifold is studied when the
Lie group for gauge transformations is compact, connected and abelian. A
rigorous definition of an abelian Chern-Simons partition function is derived
using the Faddeev-Popov gauge fixing method. A symplectic abelian Chern-Simons
partition function is also derived using the technique of non-abelian
localization. This physically identifies the symplectic abelian partition
function with the abelian Chern-Simons partition function as rigorous
topological three-manifold invariants. This study leads to a natural
identification of the abelian Reidemeister-Ray-Singer torsion as a specific
multiple of the natural unit symplectic volume form on the moduli space of flat
abelian connections for the class of Sasakian three-manifolds. The torsion part
of the abelian Chern-Simons partition function is computed explicitly in terms
of Seifert data for a given Sasakian three-manifold. | math-ph |
An algebraic theory of infinite classical lattices I: General theory: We present an algebraic theory of the states of the infinite classical
lattices. The construction follows the Haag-Kastler axioms from quantum field
theory. By comparison, the *-algebras of the quantum theory are replaced here
with the Banach lattices (MI-spaces) to have real-valued measurements, and the
Gelfand-Naimark-Segal construction with the structure theorem for MI-spaces to
represent the Segal algebra as C(X). The theory represents any compact convex
set of states as a decomposition problem of states on an abstract Segal algebra
C(X), where X is isomorphic with the space of extremal states of the set. Three
examples are treated, the study of groups of symmetries and symmetry breakdown,
the Gibbs states, and the set of all stationary states on the lattice. For
relating the theory to standard problems of statistical mechanics, it is shown
that every thermodynamic-limit state is uniquely identified by expectation
values with an algebraic state. | math-ph |
A tale of two Nekrasov's integral equations: Just 100 years ago, Nekrasov published the widely cited paper \cite{N1}, in
which he derived the first of his two integral equations describing steady
periodic waves on the free surface of water. We examine how Nekrasov arrived at
these equations and his approach to investigating their solutions. In this
connection, Nekrasov's life after 1917 is briefly outlined, in particular, how
he became a victim of Stalin's terror. Further results concerning Nekrasov's
equations and related topicz are surveyed. | math-ph |
The Density-Potential Mapping in Quantum Dynamics: This work studies in detail the possibility of defining a one-to-one mapping
from charge densities as obtained by the time-dependent Schr\"odinger equation
to external potentials. Such a mapping is provided by the Runge-Gross theorem
and lies at the very core of time-dependent density functional theory. After
introducing the necessary mathematical concepts, the usual mapping "there" -
from potentials to wave functions as solutions to the Schr\"odinger equation -
is revisited paying special attention to Sobolev regularity. This is
scrutinised further when the question of functional differentiability of the
solution with respect to the potential arises, a concept related to linear
response theory. Finally, after a brief introduction to general density
functional theory, the mapping "back again" - from densities to potentials
thereby inverting the Schr\"odinger equation for a fixed initial state - is
defined. Apart from utilising the original Runge-Gross proof this is achieved
through a fixed-point procedure. Both approaches give rise to mathematical
issues, previously unresolved, which however could be dealt with to some extent
within the framework at hand. | math-ph |
Bulk Universality for Unitary Matrix Models: We give a proof of universality in the bulk of spectrum of unitary matrix
models, assuming that the potential is globally $C^{2}$ and locally $C^{3}$
function. The proof is based on the determinant formulas for correlation
functions in terms of polynomials orthogonal on the unit circle. We do not use
asymptotics of orthogonal polynomials. We obtain the $sin$-kernel as a unique
solution of a certain non-linear integro-differential equation. | math-ph |
A hyperbolic problem with non-local constraint describing
ion-rearrangement in a model for ion-lithium batteries: In this paper we study the Fokker-Plank equation arising in a model which
describes the charge and discharge process of ion-lithium batteries. In
particular we focus our attention on slow reaction regimes with non-negligible
entropic effects, which triggers the mass-splitting transition. At first we
prove that the problem is globally well-posed. After that we prove a stability
result under some hypothesis of improved regularity and a uniqueness result for
the stability under some additional condition of | math-ph |
Degenerate Spin Structures and the Levy-Leblond Equation: Newton-Cartan manifolds and the Galilei group are defined by the use of
co-rank one degenerate metric tensor. Newton-Cartan connection is lifted to the
degenerate spinor bundle over a Newton-Cartan 4-manifold by the aid of
degenerate spin group. Levy-Leblond equation is constructed with the lifted
connection. | math-ph |
Semiclassical energy formulas for power-law and log potentials in
quantum mechanics: We study a single particle which obeys non-relativistic quantum mechanics in
R^N and has Hamiltonian H = -Delta + V(r), where V(r) = sgn(q)r^q. If N \geq 2,
then q > -2, and if N = 1, then q > -1. The discrete eigenvalues E_{n\ell} may
be represented exactly by the semiclassical expression E_{n\ell}(q) =
min_{r>0}\{P_{n\ell}(q)^2/r^2+ V(r)}. The case q = 0 corresponds to V(r) =
ln(r). By writing one power as a smooth transformation of another, and using
envelope theory, it has earlier been proved that the P_{n\ell}(q) functions are
monotone increasing. Recent refinements to the comparison theorem of QM in
which comparison potentials can cross over, allow us to prove for n = 1 that
Q(q)=Z(q)P(q) is monotone increasing, even though the factor Z(q)=(1+q/N)^{1/q}
is monotone decreasing. Thus P(q) cannot increase too slowly. This result
yields some sharper estimates for power-potential eigenvlaues at the bottom of
each angular-momentum subspace. | math-ph |
Conserved currents of massless fields of spin s>0: A complete and explicit classification of all locally constructed conserved
currents and underlying conserved tensors is obtained for massless linear
symmetric spinor fields of any spin s>0 in four dimensional flat spacetime.
These results generalize the recent classification in the spin s=1 case of all
conserved currents locally constructed from the electromagnetic spinor field.
The present classification yields spin s>0 analogs of the well-known
electromagnetic stress-energy tensor and Lipkin's zilch tensor, as well as a
spin s>0 analog of a novel chiral tensor found in the spin s=1 case. The chiral
tensor possesses odd parity under a duality symmetry (i.e., a phase rotation)
on the spin s field, in contrast to the even parity of the stress-energy and
zilch tensors. As a main result, it is shown that every locally constructed
conserved current for each s>0 is equivalent to a sum of elementary linear
conserved currents, quadratic conserved currents associated to the
stress-energy, zilch, and chiral tensors, and higher derivative extensions of
these currents in which the spin s field is replaced by its repeated
conformally-weighted Lie derivatives with respect to conformal Killing vectors
of flat spacetime. Moreover, all of the currents have a direct, unified
characterization in terms of Killing spinors. The cases s=2, s=1/2 and s=3/2
provide a complete set of conserved quantities for propagation of gravitons
(i.e., linearized gravity waves), neutrinos and gravitinos, respectively, on
flat spacetime. The physical meaning of the zilch and chiral quantities is
discussed. | math-ph |
Unfolding the conical zones of the dissipation-induced subcritical
flutter for the rotationally symmetrical gyroscopic systems: Flutter of an elastic body of revolution spinning about its axis of symmetry
is prohibited in the subcritical spinning speed range by the Krein theorem for
the Hamiltonian perturbations. Indefinite damping creates conical domains of
the subcritical flutter (subcritical parametric resonance) bifurcating into the
pockets of two Whitney's umbrellas when non-conservative positional forces are
additionally taken into account. This explains why in contrast to the common
intuition, but in agreement with experience, symmetry-breaking stiffness
variation can promote subcritical friction-induced oscillations of the rotor
rather than inhibit them. | math-ph |
Spectral analysis of the 2+1 fermionic trimer with contact interactions: We qualify the main features of the spectrum of the Hamiltonian of point
interaction for a three-dimensional quantum system consisting of three
point-like particles, two identical fermions, plus a third particle of
different species, with two-body interaction of zero range. For arbitrary
magnitude of the interaction, and arbitrary value of the mass parameter (the
ratio between the mass of the third particle and that of each fermion) above
the stability threshold, we identify the essential spectrum, localise the
discrete spectrum and prove its finiteness, qualify the angular symmetry of the
eigenfunctions, and prove the increasing monotonicity of the eigenvalues with
respect to the mass parameter. We also demonstrate the existence or absence of
bound states in the physically relevant regimes of masses. | math-ph |
Baxter equations and Deformation of Abelian Differentials: In this paper the proofs are given of important properties of deformed
Abelian differentials introduced earlier in connection with quantum integrable
systems. The starting point of the construction is Baxter equation. In
particular, we prove Riemann bilinear relation. Duality plays important role in
our consideration. Classical limit is considered in details. | math-ph |
Nonholonomic Clifford and Finsler Structures, Non-Commutative Ricci
Flows, and Mathematical Relativity: In this summary of Habilitation Thesis, it is outlined author's 18 years
research activity on mathematical physics, geometric methods in particle
physics and gravity, modifications and applications (after defending his PhD
thesis in 1994). Ten most relevant publications are structured conventionally
into three "strategic directions": 1) nonholonomic geometric flows evolutions
and exact solutions for Ricci solitons and field equations in (modified)
gravity theories; 2) geometric methods in quantization of models with nonlinear
dynamics and anisotropic field interactions; 3) (non) commutative geometry,
almost Kaehler and Clifford structures, Dirac operators and effective
Lagrange-Hamilton and Riemann-Finsler spaces. | math-ph |
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