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Periodic striped states in Ising models with dipolar interactions: We review the problem of determining the ground states of 2D Ising models
with nearest neighbor ferromagnetic and dipolar interactions, and prove a new
result supporting the conjecture that, if the nearest neighbor coupling $J$ is
sufficiently large, the ground states are periodic and `striped'. More
precisely, we prove a restricted version of the conjecture, by constructing the
minimizers within the variational class of states whose domain walls are
arbitrary collections of horizontal and/or vertical straight lines. | math-ph |
Associated special functions and coherent states: A hypergeometric type equation satisfying certain conditions defines either a
finite or an infinite system of orthogonal polynomials. We present in a unified
and explicit way all these systems of orthogonal polynomials, the associated
special functions and some systems of coherent states. This general formalism
allows us to extend some results known only in particular cases. | math-ph |
A solution for the differences in the continuity of continuum among
mathematicians: There are the longstanding differences in the continuity of continuum among
mathematicians. Starting from studies on a mathematical model of contact, we
construct a set that is in contact everywhere by using the original idea of
Dedekind's cut and weakening Order axioms to violate Order axiom 1. It is
proved that the existence of the set constructed can eliminate the differences
in the continuity. | math-ph |
The classical spin triangle as an integrable system: The classical spin system consisting of three spins with Heisenberg
interaction is an example of a completely integrable mechanical system. In this
paper we explicitly calculate its time evolution and the corresponding
action-angle variables. This calculation is facilitated by splitting the six
degrees of freedom into three internal and three external variables, such that
the internal variables evolve autonomously. Their oscillations can be
explicitly calculated in terms of the Weierstrass elliptic function. We test
our results by means of an example and comparison with direct numerical
integration. A couple of special cases is analyzed where the general theory
does not apply, including the aperiodic limit case for special initial
conditions. The extension to systems with a time-depending magnetic field in a
constant direction is straightforward. | math-ph |
Emergent Behaviors of the generalized Lohe matrix model: We present a first-order aggregation model on the space of complex matrices
which can be derived from the Lohe tensor model on the space of tensors with
the same rank and size. We call such matrix-valued aggregation model as "the
generalized Lohe matrix model". For the proposed matrix model with two cubic
coupling terms, we study several structural properties such as the conservation
laws, solution splitting property. In particular, for the case of only one
coupling, we reformulate the reduced Lohe matrix model into the Lohe matrix
model with a diagonal frustration, and provide several sufficient frameworks
leading to the complete and practical aggregations. For the estimates of
collective dynamics, we use a nonlinear functional approach using an ensemble
diameter which measures the degree of aggregation. | math-ph |
Averaging versus Chaos in Turbulent Transport?: In this paper we analyze the transport of passive tracers by deterministic
stationary incompressible flows which can be decomposed over an infinite number
of spatial scales without separation between them. It appears that a low order
dynamical system related to local Peclet numbers can be extracted from these
flows and it controls their transport properties. Its analysis shows that these
flows are strongly self-averaging and super-diffusive: the delay $\tau(r)$ for
any finite number of passive tracers initially close to separate till a
distance $r$ is almost surely anomalously fast ($\tau(r)\sim r^{2-\nu}$, with
$\nu>0$). This strong self-averaging property is such that the dissipative
power of the flow compensates its convective power at every scale. However as
the circulation increase in the eddies the transport behavior of the flow may
(discontinuously) bifurcate and become ruled by deterministic chaos: the
self-averaging property collapses and advection dominates dissipation. When the
flow is anisotropic a new formula describing turbulent conductivity is
identified. | math-ph |
Noncommutative Ricci flow in a matrix geometry: We study noncommutative Ricci flow in a finite dimensional representation of
a noncommutative torus. It is shown that the flow exists and converges to the
flat metric. We also consider the evolution of entropy and a definition of
scalar curvature in terms of the Ricci flow. | math-ph |
Quadratic forms for Aharonov-Bohm Hamiltonians: We consider a charged quantum particle immersed in an axial magnetic field,
comprising a local Aharonov-Bohm singularity and a regular perturbation.
Quadratic form techniques are used to characterize different self-adjoint
realizations of the reduced two-dimensional Schr\"odinger operator, including
the Friedrichs Hamiltonian and a family of singular perturbations indexed by $2
\times 2$ Hermitian matrices. The limit of the Friedrichs Hamiltonian when the
Aharonov-Bohm flux parameter goes to zero is discussed in terms of $\Gamma$ -
convergence. | math-ph |
Classical scattering at low energies: For a class of negative slowly decaying potentials including the attractive
Coulombic one we study the classical scattering theory in the low-energy
regime. We construct a (continuous) family of classical orbits parametrized by
initial position $x\in \R^d$, final direction $\omega\in S^{d-1}$ of escape (to
infinity) and the energy $\lambda\geq 0$, yielding a complete classification of
the set of outgoing scattering orbits. The construction is given in the
outgoing part of phase-space (a similar construction may be done in the
incoming part of phase-space). For fixed $\omega\in S^{d-1}$ and $\lambda\geq
0$ the collection of constructed orbits constitutes a smooth manifold that we
show is Lagrangian. The family of those Lagrangians can be used to study the
quantum mechanical scattering theory in the low-energy regime for the class of
potentials considered here. We devote this study to a subsequent paper. | math-ph |
Anticoherent Subspaces: We extend the notion of anticoherent spin states to anticoherent subspaces.
An anticoherent subspace of order t, is a subspace whose unit vectors are all
anticoherent states of order t. We use Klein's description of algebras of
polynomials which are invariant under finite subgroups of SU(2) to provide
constructions of anticoherent subspaces. Furthermore, we show a connection
between the existence of these subspaces and the properties of the higher-rank
numerical range for a set of spin observables. We also note that these
constructions give us subspaces of spin states all of whose unit vectors have
Majorana representations which are spherical designs of order at least t. | math-ph |
Structure of the Electric Field in the Skin Effect Problem: The structure of the electric field in a plasma has been elucidated for the
skin effect problem. An expression for the distribution function in the
half-space and the electric field profile have been obtained in the explicit
form. The absolute value, the real part, and the imaginary part of the electric
filed have been analyzed in the case of the anomalous skin effect near to a
plasma resonance. It has been demonstrated that the electric field in the skin
effect problem is predominantly determined by the discrete spectrum, i.e., the
oscillation frequency of external field is the value of plasma frequency. | math-ph |
Nondecaying linear and nonlinear modes in a periodic array of spatially
localized dissipations: We demonstrate the existence of extremely weakly decaying linear and
nonlinear modes (i.e. modes immune to dissipation) in the one-dimensional
periodic array of identical spatially localized dissipations, where the
dissipation width is much smaller than the period of the array. We consider
wave propagation governed by the one-dimensional Schr\"odinger equation in the
array of identical Gaussian-shaped dissipations with three parameters, the
integral dissipation strength $\Gamma_0$, the width $\sigma$ and the array
period $d$. In the linear case, setting $\sigma\to0$, while keeping $\Gamma_0$
fixed, we get an array of zero-width dissipations given by the Dirac
delta-functions, i.e. the complex Kroning-Penney model, where an infinite
number of nondecaying modes appear with the Bloch index being either at the
center, $k= 0$, or at the boundary, $k= \pi/d $, of an analog of the Brillouin
zone. By using numerical simulations we confirm that the weakly decaying modes
persist for $\sigma$ such that $\sigma/d\ll1$ and have the same Bloch index.
The nondecaying modes persist also if a real-valued periodic potential is added
to the spatially periodic array of dissipations, with the period of the
dissipative array being multiple of that of the periodic potential. We also
consider evolution of the soliton-shaped pulses in the nonlinear Schr\"odinger
equation with the spatially periodic dissipative lattice and find that when the
pulse width is much larger than the lattice period and its wave number $k$ is
either at the center, $k= 2\pi/d$, or at the boundary, $k= \pi/d $, a
significant fraction of the pulse escapes the dissipation forming a stationary
nonlinear mode with the soliton shaped envelope and the Fourier spectrum
consisting of two peaks centered at $k $ and $-k$. | math-ph |
Convergence of the regularized Kohn-Sham iteration in Banach spaces: The Kohn-Sham iteration of generalized density-functional theory on Banach
spaces with Moreau-Yosida regularized universal Lieb functional and an adaptive
damping step is shown to converge to the correct ground-state density. This
result demands state spaces for (quasi)densities and potentials that are
uniformly convex with modulus of convexity of power type. The Moreau-Yosida
regularization is adapted to match the geometry of the spaces and some convex
analysis results are extended to this type of regularization. Possible
connections between regularization and physical effects are pointed out as
well. The proof of convergence presented here (Theorem 23) contains a critical
mistake that has been noted and fixed for the finite-dimensional case in
arXiv:1903.09579. Yet, the proposed correction is not straightforwardly
generalizable to a setting of infinite-dimensional Banach spaces. This means
the question of convergence in such a case is still left open. We present this
draft as a collection of techniques and ideas for a possible altered,
successful demonstration of convergence of the Kohn--Sham iteration in Banach
spaces. | math-ph |
Materials with a desired refraction coefficient can be made by embedding
small particles: A method is proposed to create materials with a desired refraction
coefficient, possibly negative one. The method consists of embedding into a
given material small particles. Given $n_0(x)$, the refraction coefficient of
the original material in a bounded domain $D \subset \R^3$, and a desired
refraction coefficient $n(x)$, one calculates the number $N(x)$ of small
particles, to be embedded in $D$ around a point $x \in D$ per unit volume of
$D$, in order that the resulting new material has refraction coefficient
$n(x)$. | math-ph |
Stability and Related Properties of Vacua and Ground States: We consider the formal non relativistc limit (nrl) of the :\phi^4:_{s+1}
relativistic quantum field theory (rqft), where s is the space dimension.
Following work of R. Jackiw, we show that, for s=2 and a given value of the
ultraviolet cutoff \kappa, there are two ways to perform the nrl: i.) fixing
the renormalized mass m^2 equal to the bare mass m_0^2; ii.) keeping the
renormalized mass fixed and different from the bare mass m_0^2. In the
(infinite-volume) two-particle sector the scattering amplitude tends to zero as
\kappa -> \infty in case i.) and, in case ii.), there is a bound state,
indicating that the interaction potential is attractive. As a consequence,
stability of matter fails for our boson system. We discuss why both
alternatives do not reproduce the low-energy behaviour of the full rqft. The
singular nature of the nrl is also nicely illustrated for s=1 by a rigorous
stability/instability result of a different nature. | math-ph |
From Euler elements and 3-gradings to non-compactly causal symmetric
spaces: In this article we discuss the interplay between causal structures of
symmetric spaces and geometric aspects of Algebraic Quantum Field Theory
(AQFT). The central focus is the set of Euler elements in a Lie algebra, i.e.,
elements whose adjoint action defines a 3-grading. In the first half of this
article we survey the classification of reductive causal symmetric spaces from
the perspective of Euler elements. This point of view is motivated by recent
applications in AQFT. In the second half we obtain several results that prepare
the exploration of the deeper connection between the structure of causal
symmetric spaces and AQFT. In particular, we explore the technique of strongly
orthogonal roots and corresponding systems of sl_2-subalgebras. Furthermore, we
exhibit real Matsuki crowns in the adjoint orbits of Euler elements and we
describe the group of connected components of the stabilizer group of Euler
elements. | math-ph |
Construction of dynamics and time-ordered exponential for unbounded
non-symmetric Hamiltonians: We prove under certain assumptions that there exists a solution of the
Schrodinger or the Heisenberg equation of motion generated by a linear operator
H acting in some complex Hilbert space H, which may be unbounded, not
symmetric, or not normal. We also prove that, under the same assumptions, there
exists a time evolution operator in the interaction picture and that the
evolution operator enjoys a useful series expansion formula. This expansion is
considered to be one of the mathematically rigorous realizations of so called
"time-ordered exponential", which is familiar in the physics literature. We
apply the general theory to prove the existence of dynamics for the
mathematical model of Quantum Electrodynamics (QED) quantized in the Lorenz
gauge, the interaction Hamiltonian of which is not even symmetric or normal. | math-ph |
Noncommutative complex Grosse-Wulkenhaar model: This paper stands for an application of the noncommutative (NC) Noether
theorem, given in our previous work [AIP Proc 956 (2007) 55-60], for the NC
complex Grosse-Wulkenhaar model. It provides with an extension of a recent work
[Physics Letters B 653 (2007) 343-345]. The local conservation of
energy-momentum tensors (EMTs) is recovered using improvement procedures based
on Moyal algebraic techniques. Broken dilatation symmetry is discussed. NC
gauge currents are also explicitly computed. | math-ph |
Reduction and integrability: a geometric perspective: A geometric approach to integrability and reduction of dynamical system is
developed from a modern perspective. The main ingredients in such analysis are
the infinitesimal symmetries and the tensor fields that are invariant under the
given dynamics. Particular emphasis is given to the existence of invariant
volume forms and the associated Jacobi multiplier theory, and then the Hojman
symmetry theory is developed as a complement to Noether theorem and non-Noether
constants of motion. The geometric approach to Hamilton-Jacobi equation is
shown to be a particular example of the search for related field in a lower
dimensional manifold. | math-ph |
Quantum Newton duality: Newton revealed an underlying duality relation between power potentials in
classical mechanics. In this paper, we establish the quantum version of the
Newton duality. The main aim of this paper is threefold: (1) first generalizing
the original Newton duality to more general potentials, including general
polynomial potentials and transcendental-function potentials, 2) constructing a
quantum version of the Newton duality, including power potentials, general
polynomial potentials, transcendental-function potentials, and power potentials
in different spatial dimensions, and 3) suggesting a method for solving
eigenproblems in quantum mechanics based on the quantum Newton duality provided
in the paper. The classical Newton duality is a duality among orbits of
classical dynamical systems. Our result shows that the Newton duality is not
only limited to power potentials, but a more universal duality relation among
dynamical systems with various potentials. The key task of this paper is to
construct a quantum Newton duality, the quantum version of the classical Newton
duality. The quantum Newton duality provides a duality relations among wave
functions and eigenvalues. As applications, we suggest a method for solving
potentials from their Newtonianly dual potential: once the solution of a
potential is known, the solution of all its dual potentials can be obtained by
the duality transformation directly. Using this method, we obtain a series of
exact solutions of various potentials. In appendices, as preparations, we solve
the potentials which is solved by the Newton duality method in this paper by
directly solving the eigenequation. | math-ph |
The dressed mobile atoms and ions: We consider free atoms and ions in $\R^3$ interacting with the quantized
electromagnetic field. Because of the translation invariance we consider the
reduced hamiltonian associated with the total momentum. After introducing an
ultraviolet cutoff we prove that the reduced hamiltonian for atoms has a ground
state if the coupling constant and the total momentum are sufficiently small.
In the case of ions an extra infrared regularization is needed. We also
consider the case of the hydrogen atom in a constant magnetic field. Finally we
determine the absolutely continuous spectrum of the reduced hamiltonian. | math-ph |
Raise and Peel Models of fluctuating interfaces and combinatorics of
Pascal's hexagon: The raise and peel model of a one-dimensional fluctuating interface (model A)
is extended by considering one source (model B) or two sources (model C) at the
boundaries. The Hamiltonians describing the three processes have, in the
thermodynamic limit, spectra given by conformal field theory. The probability
of the different configurations in the stationary states of the three models
are not only related but have interesting combinatorial properties. We show
that by extending Pascal's triangle (which gives solutions to linear relations
in terms of integer numbers), to an hexagon, one obtains integer solutions of
bilinear relations. These solutions give not only the weights of the various
configurations in the three models but also give an insight to the connections
between the probability distributions in the stationary states of the three
models. Interestingly enough, Pascal's hexagon also gives solutions to a
Hirota's difference equation. | math-ph |
A generalization of Szebehely's inverse problem of dynamics in dimension
three: Extending a previous paper, we present a generalization in dimension 3 of the
traditional Szebehely-type inverse problem. In that traditional setting, the
data are curves determined as the intersection of two families of surfaces, and
the problem is to find a potential V such that the Lagrangian L = T - V, where
T is the standard Euclidean kinetic energy function, generates integral curves
which include the given family of curves. Our more general way of posing the
problem makes use of ideas of the inverse problem of the calculus of variations
and essentially consists of allowing more general kinetic energy functions,
with a metric which is still constant, but need not be the standard Euclidean
one. In developing our generalization, we review and clarify different aspects
of the existing literature on the problem and illustrate the relevance of the
newly introduced additional freedom with many examples. | math-ph |
Some integrals related to the Fermi function: Some elaborations regarding the Hilbert and Fourier transforms of Fermi
function are presented. The main result shows that the Hilbert transform of the
difference of two Fermi functions has an analytical expression in terms of the
$\Psi$ (digamma) function, while its Fourier transform is expressed by mean of
elementary functions. Moreover an integral involving the product of the
difference of two Fermi functions with its Hilbert transform is evaluated
analytically. These findings are of fundamental importance in discussing the
transport properties of electronic systems. | math-ph |
Spin models of Calogero-Sutherland type and associated spin chains: Several topics related to quantum spin models of Calogero-Sutherland type,
partially solvable spin chains and Polychronakos's "freezing trick" are
rigorously studied. | math-ph |
Non-Liouvillian solutions for second order linear ODEs: There exist sound literature and algorithms for computing Liouvillian
solutions for the important problem of linear ODEs with rational coefficients.
Taking as sample the 363 second order equations of that type found in Kamke's
book, for instance, 51 % of them admit Liouvillian solutions and so are
solvable using Kovacic's algorithm. On the other hand, special function
solutions not admitting Liouvillian form appear frequently in mathematical
physics, but there are not so general algorithms for computing them. In this
paper we present an algorithm for computing special function solutions which
can be expressed using the 2F1, 1F1 or 0F1 hypergeometric functions. The
algorithm is easy to implement in the framework of a computer algebra system
and systematically solves 91 % of the 363 Kamke's linear ODE examples
mentioned. | math-ph |
Colombeau algebra as a mathematical tool for investigating step load and
step deformation of systems of nonlinear springs and dashpots: The response of mechanical systems composed of springs and dashpots to a step
input is of eminent interest in the applications. If the system is formed by
linear elements, then its response is governed by a system of linear ordinary
differential equations, and the mathematical method of choice for the analysis
of the response of such systems is the classical theory of distributions.
However, if the system contains nonlinear elements, then the classical theory
of distributions is of no use, since it is strictly limited to the linear
setting. Consequently, a question arises whether it is even possible or
reasonable to study the response of nonlinear systems to step inputs. The
answer is positive. A mathematical theory that can handle the challenge is the
so-called Colombeau algebra. Building on the abstract result by (Pr\r{u}\v{s}a
& Rajagopal 2016, Int. J. Non-Linear Mech) we show how to use the theory in the
analysis of response of a simple nonlinear mass--spring--dashpot system. | math-ph |
Uncertainty relations for a non-canonical phase-space noncommutative
algebra: We consider a non-canonical phase-space deformation of the Heisenberg-Weyl
algebra that was recently introduced in the context of quantum cosmology. We
prove the existence of minimal uncertainties for all pairs of non-commuting
variables. We also show that the states which minimize each uncertainty
inequality are ground states of certain positive operators. The algebra is
shown to be stable and to violate the usual Heisenberg-Pauli-Weyl inequality
for position and momentum. The techniques used are potentially interesting in
the context of time-frequency analysis. | math-ph |
Length and distance on a quantum space: This contribution is an introduction to the metric aspect of noncommutative
geometry, with emphasize on the Moyal plane. Starting by questioning "how to
define a standard meter in a space whose coordinates no longer commute?", we
list several recent results regarding Connes's spectral distance calculated
between eigenstates of the quantum harmonic oscillator arXiv:0912.0906, as well
as between coherent states arXiv:1110.6164. We also question the difference
(which remains hidden in the commutative case) between the spectral distance
and the notion of quantum length inherited from the length operator defined in
various models of noncommutative space-time (DFR and \theta-Minkowski). We
recall that a standard procedure in noncommutative geometry, consisting in
doubling the spectral triple, allows to fruitfully confront the spectral
distance with the quantum length. Finally we refine the idea of discrete vs.
continuous geodesics in the Moyal plane, introduced in arXiv:1106.0261. | math-ph |
Duality between Spin networks and the 2D Ising model: The goal of this paper is to exhibit a deep relation between the partition
function of the Ising model on a planar trivalent graph and the generating
series of the spin network evaluations on the same graph. We provide
respectively a fermionic and a bosonic Gaussian integral formulation for each
of these functions and we show that they are the inverse of each other (up to
some explicit constants) by exhibiting a supersymmetry relating the two
formulations. We investigate three aspects and applications of this duality.
First, we propose higher order supersymmetric theories which couple the
geometry of the spin networks to the Ising model and for which supersymmetric
localization still holds. Secondly, after interpreting the generating function
of spin network evaluations as the projection of a coherent state of loop
quantum gravity onto the flat connection state, we find the probability
distribution induced by that coherent state on the edge spins and study its
stationary phase approximation. It is found that the stationary points
correspond to the critical values of the couplings of the 2D Ising model, at
least for isoradial graphs. Third, we analyze the mapping of the correlations
of the Ising model to spin network observables, and describe the phase
transition on those observables on the hexagonal lattice. This opens the door
to many new possibilities, especially for the study of the coarse-graining and
continuum limit of spin networks in the context of quantum gravity. | math-ph |
A new method to generate superoscillating functions and supershifts: Superoscillations are band-limited functions that can oscillate faster than
their fastest Fourier component. These functions (or sequences) appear in weak
values in quantum mechanics and in many fields of science and technology such
as optics, signal processing and antenna theory.
In this paper we introduce a new method to generate superoscillatory
functions that allows us to construct explicitly a very large class of
superoscillatory functions. | math-ph |
Exceptional Lattice Green's Functions: The three exceptional lattices, $E_6$, $E_7$, and $E_8$, have attracted much
attention due to their anomalously dense and symmetric structures which are of
critical importance in modern theoretical physics. Here, we study the
electronic band structure of a single spinless quantum particle hopping between
their nearest-neighbor lattice points in the tight-binding limit. Using Markov
chain Monte Carlo methods, we numerically sample their lattice Green's
functions, densities of states, and random walk return probabilities. We find
and tabulate a plethora of Van Hove singularities in the densities of states,
including degenerate ones in $E_6$ and $E_7$. Finally, we use brute force
enumeration to count the number of distinct closed walks of length up to eight,
which gives the first eight moments of the densities of states. | math-ph |
On solutions of the 2D Navier-Stokes equations with constant energy and
enstrophy: It is not yet known if the global attractor of the space periodic 2D
Navier-Stokes equations contains nonstationary solutions $u(x,t)$ such that
their energy and enstrophy per unit mass are constant for every $t \in
(-\infty, \infty)$. The study of the properties of such solutions was initiated
in \cite{CMM13}, where, due to the hypothetical existence of such solutions,
they were called "ghost solutions". In this work, we introduce and study
geometric structures shared by all ghost solutions. This study led us to
consider a subclass of ghost solutions for which those geometric structures
have a supplementary stability property. In particular, we show that the wave
vectors of the active modes of this subclass of ghost solutions must satisfy
certain supplementary constraints. We also found a computational way to check
for the existence of these ghost solutions. | math-ph |
Harmonic Representation of Combinations and Partitions: In the present article a new method of deriving integral representations of
combinations and partitions in terms of harmonic products has been established.
This method may be relevant to statistical mechanics and to number theory. | math-ph |
Isodiametry, variance, and regular simplices from particle interactions: Consider a collection of particles interacting through an
attractive-repulsive potential given as a difference of power laws and
normalized so that its unique minimum occurs at unit separation. For a range of
exponents corresponding to mild repulsion and strong attraction, we show that
the minimum energy configuration is uniquely attained -- apart from
translations and rotations -- by equidistributing the particles over the
vertices of a regular top-dimensional simplex (i.e. an equilateral triangle in
two dimensions and regular tetrahedron in three). If the attraction is not
assumed to be strong, we show these configurations are at least local energy
minimizers in the relevant $d_\infty$ metric from optimal transportation, as
are all of the other uncountably many unbalanced configurations with the same
support. We infer the existence of phase transitions. The proof is based on a
simple isodiametric variance bound which characterizes regular simplices: it
shows that among probability measures on ${\mathbf R}^n$ whose supports have at
most unit diameter, the variance around the mean is maximized precisely by
those measures which assign mass $1/(n+1)$ to each vertex of a (unit-diameter)
regular simplex. | math-ph |
Threshold singularities in the correlators of the one-dimensional models: We calculate the threshold singularities in one-dimensional models using the
universal low-energy formfactors obtained in the framework of the non-linear
Luttinger liquid model. We find the reason why the simplified picture of the
impurity moving in the Luttinger liquid leads to the correct results. We obtain
the prefactors of the singularities including their $k$- dependence at small
$k<<p_F$. | math-ph |
Group classification of variable coefficient K(m,n) equations: Lie symmetries of K(m,n) equations with time-dependent coefficients are
classified. Group classification is presented up to widest possible equivalence
groups, the usual equivalence group of the whole class for the general case and
conditional equivalence groups for special values of the exponents m and n.
Examples on reduction of K(m,n) equations (with initial and boundary
conditions) to nonlinear ordinary differential equations (with initial
conditions) are presented. | math-ph |
The Hartree and Vlasov equations at positive density: We consider the nonlinear Hartree and Vlasov equations around a
translation-invariant (homogeneous) stationary state in infinite volume, for a
short range interaction potential. For both models, we consider time-dependent
solutions which have a finite relative energy with respect to the reference
translation-invariant state. We prove the convergence of the Hartree solutions
to the Vlasov ones in a semi-classical limit and obtain as a by-product global
well-posedness of the Vlasov equation in the (relative) energy space. | math-ph |
A graphical calculus for integration over random diagonal unitary
matrices: We provide a graphical calculus for computing averages of tensor network
diagrams with respect to the distribution of random vectors containing
independent uniform complex phases. Our method exploits the order structure of
the partially ordered set of uniform block permutations. A similar calculus is
developed for random vectors consisting of independent uniform signs, based on
the combinatorics of the partially ordered set of even partitions. We employ
our method to extend some of the results by Johnston and MacLean on the family
of local diagonal unitary invariant matrices. Furthermore, our graphical
approach applies just as well to the real (orthogonal) case, where we introduce
the notion of triplewise complete positivity to study the condition for
separability of the relevant bipartite matrices. Finally, we analyze the
twirling of linear maps between matrix algebras by independent diagonal unitary
matrices, showcasing another application of our method. | math-ph |
Liouville theorems and spectral edge behavior on abelian coverings of
compact manifolds: The paper describes relations between Liouville type theorems for solutions
of a periodic elliptic equation (or a system) on an abelian cover of a compact
Riemannian manifold and the structure of the dispersion relation for this
equation at the edges of the spectrum. Here one says that the Liouville theorem
holds if the space of solutions of any given polynomial growth is finite
dimensional. The necessary and sufficient condition for a Liouville type
theorem to hold is that the real Fermi surface of the elliptic operator
consists of finitely many points (modulo the reciprocal lattice). Thus, such a
theorem generically is expected to hold at the edges of the spectrum. The
precise description of the spaces of polynomially growing solutions depends
upon a `homogenized' constant coefficient operator determined by the analytic
structure of the dispersion relation. In most cases, simple explicit formulas
are found for the dimensions of the spaces of polynomially growing solutions in
terms of the dispersion curves. The role of the base of the covering (in
particular its dimension) is rather limited, while the deck group is of the
most importance.
The results are also established for overdetermined elliptic systems, which
in particular leads to Liouville theorems for polynomially growing holomorphic
functions on abelian coverings of compact analytic manifolds.
Analogous theorems hold for abelian coverings of compact combinatorial or
quantum graphs. | math-ph |
Complementarity problems for two pairs of charged bodies: We consider an interaction of charged bodies under the following simplified
conditions: the distribution of charge over each body is stable; the
interaction of bodies is governed by electrical forces only. Physically, these
assumptions can be treated as the following decomposition of charges: the
structure of each body is assumed to be stable due to inner forces (say,
quantum forces [1]), which do not influence the interaction of the bodies; the
bodies interact due to the classical electrical forces [2] only. In this model,
the role of inner forces is to create a specific stable distribution of the
charge over a body. We assume that the charge distribution over a body can be
described by the density of the charge. In our model, the distribution of the
charge is the property of a body and does not change in the process of the
bodies' interaction. For the simplicity we assume that the bodies are similar
in the sense of geometry, say, occupy domain $Q$ and have a preferable
direction of interaction denoted by $Ox_3$. | math-ph |
Homogenized boundary conditions and resonance effects in Faraday cages: We present a mathematical study of two-dimensional electrostatic and
electromagnetic shielding by a cage of conducting wires (the so-called `Faraday
cage effect'). Taking the limit as the number of wires in the cage tends to
infinity we use the asymptotic method of multiple scales to derive continuum
models for the shielding, involving homogenized boundary conditions on an
effective cage boundary. We show how the resulting models depend on key cage
parameters such as the size and shape of the wires, and, in the electromagnetic
case, on the frequency and polarisation of the incident field. In the
electromagnetic case there are resonance effects, whereby at frequencies close
to the natural frequencies of the equivalent solid shell, the presence of the
cage actually amplifies the incident field, rather than shielding it. By
appropriately modifying the continuum model we calculate the modified resonant
frequencies, and their associated peak amplitudes. We discuss applications to
radiation containment in microwave ovens and acoustic scattering by perforated
shells. | math-ph |
Data characterization in dynamical inverse problem for the 1d wave
equation with matrix potential: The dynamical system under consideration is \begin{align*} &
u_{tt}-u_{xx}+Vu=0,\qquad x>0,\,\,\,t>0;\\ &
u|_{t=0}=u_t|_{t=0}=0,\,\,x\geqslant 0;\quad u|_{x=0}=f,\,\,t\geqslant 0,
\end{align*} where $V=V(x)$ is a matrix-valued function ({\it potential});
$f=f(t)$ is an $\mathbb R^N$-valued function of time ({\it boundary control});
$u=u^f(x,t)$ is a {\it trajectory} (an $\mathbb R^N$-valued function of $x$ and
$t$). The input/output map of the system is a {\it response operator}
$R:f\mapsto u^f_x(0,\cdot),\,\,\,t\geqslant0$.
The {\it inverse problem} is to determine $V$ from given $R$. To characterize
its data is to provide the necessary and sufficient conditions on $R$ that
ensure its solvability.
The procedure that solves this problem has long been known and the
characterization has been announced (Avdonin and Belishev, 1996). However, the
proof was not provided and, moreover, it turned out that the formulation must
be corrected. Our paper fills this gap. | math-ph |
The Wigner function of a q-deformed harmonic oscillator model: The phase space representation for a q-deformed model of the quantum harmonic
oscillator is constructed. We have found explicit expressions for both the
Wigner and Husimi distribution functions for the stationary states of the
$q$-oscillator model under consideration. The Wigner function is expressed as a
basic hypergeometric series, related to the Al-Salam-Chihara polynomials. It is
shown that, in the limit case $h \to 0$ ($q \to 1$), both the Wigner and Husimi
distribution functions reduce correctly to their well-known non-relativistic
analogues. Surprisingly, examination of both distribution functions in the
q-deformed model shows that, when $q \ll 1$, their behaviour in the phase space
is similar to the ground state of the ordinary quantum oscillator, but with a
displacement towards negative values of the momentum. We have also computed the
mean values of the position and momentum using the Wigner function. Unlike the
ordinary case, the mean value of the momentum is not zero and it depends on $q$
and $n$. The ground-state like behaviour of the distribution functions for
excited states in the q-deformed model opens quite new perspectives for further
experimental measurements of quantum systems in the phase space. | math-ph |
Mean-field dynamics for mixture condensates via Fock space methods: We consider a mean-field model to describe the dynamics of $N_1$ bosons of
species one and $N_2$ bosons of species two in the limit as $N_1$ and $N_2$ go
to infinity. We embed this model into Fock space and use it to describe the
time evolution of coherent states which represent two-component condensates.
Following this approach, we obtain a microscopic quantum description for the
dynamics of such systems, determined by the Schr\"{o}dinger equation.
Associated to the solution to the Schr\"{o}dinger equation, we have a reduced
density operator for one particle in the first component of the condensate and
one particle in the second component. In this paper, we estimate the difference
between this operator and the projection onto the tensor product of two
functions that are solutions of a system of equations of Hartree type. Our
results show that this difference goes to zero as $N_1$ and $N_2$ go to
infinity. Our hypotheses allow the Coulomb interaction. | math-ph |
Fractional supersymmetric Quantum Mechanics as a set of replicas of
ordinary supersymmetric Quantum Mechanics: A connection between fractional supersymmetric quantum mechanics and ordinary
supersymmetric quantum mechanics is established in this Letter. | math-ph |
φ^4 Solitary Waves in a Parabolic Potential: Existence, Stability,
and Collisional Dynamics: We explore a {\phi}^4 model with an added external parabolic potential term.
This term dramatically alters the spectral properties of the system. We
identify single and multiple kink solutions and examine their stability
features; importantly, all of the stationary structures turn out to be
unstable. We complement these with a dynamical study of the evolution of a
single kink in the trap, as well as of the scattering of kink and anti-kink
solutions of the model. We see that some of the key characteristics of
kink-antikink collisions, such as the critical velocity and the multi-bounce
windows, are sensitively dependent on the trap strength parameter, as well as
the initial displacement of the kink and antikink. | math-ph |
Tautological Tuning of the Kostant-Souriau Quantization Map with
Differential Geometric Structures: For decades, mathematical physicists have searched for a coordinate
independent quantization procedure to replace the ad hoc process of canonical
quantization. This effort has largely coalesced into two distinct research
programs: geometric quantization and deformation quantization. Though both of
these programs can claim numerous successes, neither has found mainstream
acceptance within the more experimentally minded quantum physics community,
owing both to their mathematical complexities and their practical failures as
empirical models. This paper introduces an alternative approach to
coordinate-independent quantization called tautologically tuned quantization.
This approach uses only differential geometric structures from symplectic and
Riemannian geometry, especially the tautological one form and vector field
(hence the name). In its focus on physically important functions,
tautologically tuned quantization hews much more closely to the ad hoc approach
of canonical quantization than either traditional geometric quantization or
deformation quantization and thereby avoid some of the mathematical challenges
faced by those methods. Given its focus on standard differential geometric
structures, tautologically tuned quantization is also a better candidate than
either traditional geometric or deformation quantization for application to
covariant Hamiltonian field theories, and therefore may pave the way for the
geometric quantization of classical fields. | math-ph |
Calculating the algebraic entropy of mappings with unconfined
singularities: We present a method for calculating the dynamical degree of a mapping with
unconfined singularities. It is based on a method introduced by Halburd for the
computation of the growth of the iterates of a rational mapping with confined
singularities. In particular, we show through several examples how simple
calculations, based on the singularity patterns of the mapping, allow one to
obtain the exact value of the dynamical degree for nonintegrable mappings that
do not possess the singularity confinement property. We also study linearisable
mappings with unconfined singularities to show that in this case our method
indeed yields zero algebraic entropy. | math-ph |
Approximate formulas for moderately small eikonal amplitudes: The eikonal approximation for moderately small scattering amplitudes is
considered. With the purpose of using for their numerical estimations, the
formulas are derived which contain no Bessel functions, and, hence, no rapidly
oscillating integrands. To obtain these formulas, the improper integrals of the
first kind which contain products of the Bessel functions J_0(z) are studied.
The expression with four functions J_0(z) is generalized. The expressions for
the integrals with the product of five and six Bessel functions J_0(z) are also
found. The known formula for the improper integral with two functions J_nu(z)
is generalized for non-integer nu. | math-ph |
Motif based hierarchical random graphs: structural properties and
critical points of an Ising model: A class of random graphs is introduced and studied. The graphs are
constructed in an algorithmic way from five motifs which were found in [Milo
R., Shen-Orr S., Itzkovitz S., Kashtan N., Chklovskii D., Alon U., Science,
2002, 298, 824-827]. The construction scheme resembles that used in [Hinczewski
M., A. Nihat Berker, Phys. Rev. E, 2006, 73, 066126], according to which the
short-range bonds are non-random, whereas the long-range bonds appear
independently with the same probability. A number of structural properties of
the graphs have been described, among which there are degree distributions,
clustering, amenability, small-world property. For one of the motifs, the
critical point of the Ising model defined on the corresponding graph has been
studied. | math-ph |
Ergodic properties of random billiards driven by thermostats: We consider a class of mechanical particle systems interacting with
thermostats. Particles move freely between collisions with disk-shaped
thermostats arranged periodically on the torus. Upon collision, an energy
exchange occurs, in which a particle exchanges its tangential component of the
velocity for a randomly drawn one from the Gaussian distribution with the
variance proportional to the temperature of the thermostat. In the case when
all temperatures are equal one can write an explicit formula for the stationary
distribution. We consider the general case and show that there exists a unique
absolutely continuous stationary distribution. Moreover under rather mild
conditions on the initial distribution the corresponding Markov dynamics
converges to the equilibrium with exponential rate. One of the main technical
difficulties is related to a possible overheating of moving particle. However
as we show in the paper non-compactness of the particle velocity can be
effectively controlled. | math-ph |
The quenched central limit theorem for a model of random walk in random
environment: A short proof of the quenched central limit theorem for the random walk in
random environment introduced by Boldrighini, Minlos, and Pellegrinotti is
given. | math-ph |
Crossover phenomena in the critical behavior for long-range models with
power-law couplings: This is a short review of the two papers on the $x$-space asymptotics of the
critical two-point function $G_{p_c}(x)$ for the long-range models of
self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$, defined
by the translation-invariant power-law step-distribution/coupling
$D(x)\propto|x|^{-d-\alpha}$ for some $\alpha>0$. Let $S_1(x)$ be the
random-walk Green function generated by $D$. We have shown that
$\bullet~~S_1(x)$ changes its asymptotic behavior from Newton ($\alpha>2$) to
Riesz ($\alpha<2$), with log correction at $\alpha=2$;
$\bullet~~G_{p_c}(x)\sim\frac{A}{p_c}S_1(x)$ as $|x|\to\infty$ in dimensions
higher than (or equal to, if $\alpha=2$) the upper critical dimension $d_c$
(with sufficiently large spread-out parameter $L$). The model-dependent $A$ and
$d_c$ exhibit crossover at $\alpha=2$.
The keys to the proof are (i) detailed analysis on the underlying random walk
to derive sharp asymptotics of $S_1$, (ii) bounds on convolutions of power
functions (with log corrections, if $\alpha=2$) to optimally control the
lace-expansion coefficients $\pi_p^{(n)}$, and (iii) probabilistic
interpretation (valid only when $\alpha\le2$) of the convolution of $D$ and a
function $\varPi_p$ of the alternating series
$\sum_{n=0}^\infty(-1)^n\pi_p^{(n)}$. We outline the proof, emphasizing the
above key elements for percolation in particular. | math-ph |
Defect lines, dualities, and generalised orbifolds: Defects are a useful tool in the study of quantum field theories. This is
illustrated in the example of two-dimensional conformal field theories. We
describe how defect lines and their junction points appear in the description
of symmetries and order-disorder dualities, as well as in the orbifold
construction and a generalisation thereof that covers exceptional modular
invariants. | math-ph |
The resistive state in a superconducting wire: Bifurcation from the
normal state: We study formally and rigorously the bifurcation to steady and time-periodic
states in a model for a thin superconducting wire in the presence of an imposed
current. Exploiting the PT-symmetry of the equations at both the linearized and
nonlinear levels, and taking advantage of the collision of real eigenvalues
leading to complex spectrum, we obtain explicit asymptotic formulas for the
stationary solutions, for the amplitude and period of the bifurcating periodic
solutions and for the location of their zeros or "phase slip centers" as they
are known in the physics literature. In so doing, we construct a center
manifold for the flow and give a complete description of the associated
finite-dimensional dynamics. | math-ph |
Superdiffusion in the periodic Lorentz gas: We prove a superdiffusive central limit theorem for the displacement of a
test particle in the periodic Lorentz gas in the limit of large times $t$ and
low scatterer densities (Boltzmann-Grad limit). The normalization factor is
$\sqrt{t\log t}$, where $t$ is measured in units of the mean collision time.
This result holds in any dimension and for a general class of finite-range
scattering potentials. We also establish the corresponding invariance
principle, i.e., the weak convergence of the particle dynamics to Brownian
motion. | math-ph |
Umbral methods and operator ordering: By using methods of umbral nature, we discuss new rules concerning the
operator ordering. We apply the technique of formal power series to take
advantage from the wealth of properties of the exponential operators. The
usefulness of the obtained results in quantum field theory is discussed. | math-ph |
On solutions of the Schlesinger Equations in Terms of $Θ$-Functions: In this paper we construct explicit solutions and calculate the corresponding
$\tau$-function to the system of Schlesinger equations describing isomonodromy
deformations of $2\times 2$ matrix linear ordinary differential equation whose
coefficients are rational functions with poles of the first order; in
particular, in the case when the coefficients have four poles of the first
order and the corresponding Schlesinger system reduces to the sixth Painlev\'e
equation with the parameters $1/8, -1/8, 1/8, 3/8$, our construction leads to a
new representation of the general solution to this Painlev\'e equation obtained
earlier by K. Okamoto and N. Hitchin, in terms of elliptic theta-functions. | math-ph |
Homogenization for Inertial Particles in a Random Flow: We study the problem of homogenization for inertial particles moving in a
time dependent random velocity field and subject to molecular diffusion. We
show that, under appropriate assumptions on the velocity field, the
large--scale, long--time behavior of the inertial particles is governed by an
effective diffusion equation for the position variable alone. This is achieved
by the use of a formal multiple scales expansion in the scale parameter. The
expansion relies on the hypoellipticity of the underlying diffusion. An
expression for the diffusivity tensor is found and various of its properties
are studied. The results of the formal multiscale analysis are justified
rigorously by the use of the martingale central limit theorem. Our theoretical
findings are supported by numerical investigations where we study the
parametric dependence of the effective diffusivity on the various
non--dimensional parameters of the problem. | math-ph |
Hamiltonian formulation of systems with linear velocities within
Riemann-Liouville fractional derivatives: The link between the treatments of constrained systems with fractional
derivatives by using both Hamiltonian and Lagrangian formulations is studied.
It is shown that both treatments for systems with linear velocities are
equivalent. | math-ph |
Lower bounds for resonance counting functions for obstacle scattering in
even dimensions: In even dimensional Euclidean scattering, the resonances lie on the
logarithmic cover of the complex plane. This paper studies resonances for
obstacle scattering in ${\mathbb R}^d$ with Dirchlet or admissable Robin
boundary conditions, when $d$ is even. Set $n_m(r)$ to be the number of
resonances with norm at most $r$ and argument between $m\pi$ and $(m+1)\pi$.
Then $\lim\sup _{r\rightarrow \infty}\frac{\log n_m(r)}{\log r}=d$ if $m\in
{\mathbb Z}\setminus \{ 0\}$. | math-ph |
Jacobi - type identities in algebras and superalgebras: We introduce two remarkable identities written in terms of single commutators
and anticommutators for any three elements of arbitrary associative algebra.
One is a consequence of other (fundamental identity). From the fundamental
identity, we derive a set of four identities (one of which is the Jacobi
identity) represented in terms of double commutators and anticommutators. We
establish that two of the four identities are independent and show that if the
fundamental identity holds for an algebra, then the multiplication operation in
that algebra is associative. We find a generalization of the obtained results
to the super case and give a generalization of the fundamental identity in the
case of arbitrary elements. For nondegenerate even symplectic (super)manifolds,
we discuss analogues of the fundamental identity. | math-ph |
The Fragmentation Kernel in Multinary/Multicomponent Fragmentation: The fragmentation equation is commonly expressed in terms of two functions,
the rate of fragmentation and the mean number of fragments. In the case of
binary fragmentation an alternative description is possible based on the
fragmentation kernel, a function from which the rate of fragmentation and the
mean distribution of fragments can be obtained. We extend the fragmentation
kernel to multinary/multicomponent fragmentation and derive expressions for
certain special cases of random and non random fragmentation. | math-ph |
Spectral asymptotics of a strong $δ'$ interaction on a planar loop: We consider a generalized Schr\"odinger operator in $L^2(\R^2)$ with an
attractive strongly singular interaction of $\delta'$ type characterized by the
coupling parameter $\beta>0$ and supported by a $C^4$-smooth closed curve
$\Gamma$ of length $L$ without self-intersections. It is shown that in the
strong coupling limit, $\beta\to 0_+$, the number of eigenvalues behaves as
$\frac{2L}{\pi\beta} + \OO(|\ln\beta|)$, and furthermore, that the asymptotic
behaviour of the $j$-th eigenvalue in the same limit is $-\frac{4}{\beta^2}
+\mu_j+\OO(\beta|\ln\beta|)$, where $\mu_j$ is the $j$-th eigenvalue of the
Schr\"odinger operator on $L^2(0,L)$ with periodic boundary conditions and the
potential $-\frac14 \gamma^2$ where $\gamma$ is the signed curvature of
$\Gamma$. | math-ph |
Nonpolynomial vector fields under the Lotka-Volterra normal form: We carry out the generalization of the Lotka-Volterra embedding to flows not
explicitly recognizable under the Generalized Lotka-Volterra format. The
procedure introduces appropiate auxiliary variables, and it is shown how, to a
great extent, the final Lotka-Volterra system is independent of their specific
definition. Conservation of the topological equivalence during the process is
also demonstrated. | math-ph |
Cohomologie De Hochschild Des Surfaces De Klein: Given a mechanical system $(M, \mathcal{F}(M))$, where $M$ is a Poisson
manifold and $\mathcal{F}(M)$ the algebra of regular functions on $M$, it is
important to be able to quantize it, in order to obtain more precise results
than through classical mechanics. An available method is the deformation
quantization, which consists in constructing a star-product on the algebra of
formal power series $\mathcal{F}(M)[[\hbar]]$. A first step toward study of
star-products is the calculation of Hochschild cohomology of $\mathcal{F}(M)$.
The aim of this article is to determine this Hochschild cohomology in the case
of singular curves of the plane -- so we rediscover, by a different way, a
result proved by Fronsdal and make it more precise -- and in the case of Klein
surfaces. The use of a complex suggested by Kontsevich and the help of
Gr\"obner bases allow us to solve the problem. | math-ph |
The point scatterer approximation for wave dynamics: Given an open, bounded and connected set $\Omega\subset\mathbb{R}^{3}$ and
its rescaling $\Omega_{\varepsilon}$ of size $\varepsilon\ll 1$, we consider
the solutions of the Cauchy problem for the inhomogeneous wave equation $$
(\varepsilon^{-2}\chi_{\Omega_{\varepsilon}}+\chi_{\mathbb{R}^{3}\backslash\Omega_{\varepsilon}})\partial_{tt}u=\Delta
u+f $$ with initial data and source supported outside $\Omega_{\varepsilon}$;
here, $\chi_{S}$ denotes the characteristic function of a set $S$. We provide
the first-order $\varepsilon$-corrections with respect to the solutions of the
inhomogeneous free wave equation and give space-time estimates on the
remainders in the $L^{\infty}((0,1/\varepsilon^{\tau}),L^{2}(\mathbb{R}^{3}))
$-norm. Such corrections are explicitly expressed in terms of the eigenvalues
and eigenfunctions of the Newton potential operator in $L^{2}(\Omega)$ and
provide an effective dynamics describing a legitimate point scatterer
approximation in the time domain. | math-ph |
On local equivalence problem of spacetimes with two orthogonally
transitive commuting Killing fields: Considered is the problem of local equivalence of generic four-dimensional
metrics possessing two commuting and orthogonally transitive Killing vector
fields. A sufficient set of eight differential invariants is explicitly
constructed, among them four of first order and four of second order in terms
of metric coefficients. In vacuum case the four first-order invariants suffice
to distinguish generic metrics. | math-ph |
Graded Geometric Structures Underlying F-Theory Related Defect Theories: In the context of F-theory, we study the related eight dimensional
super-Yang-Mills theory and reveal the underlying supersymmetric quantum
mechanics algebra that the fermionic fields localized on the corresponding
defect theory are related to. Particularly, the localized fermionic fields
constitute a graded vector space, and in turn this graded space enriches the
geometric structures that can be built on the initial eight-dimensional space.
We construct the implied composite fibre bundles, which include the graded
affine vector space and demonstrate that the composite sections of this fibre
bundle are in one-to-one correspondence to the sections of the square root of
the canonical bundle corresponding to the submanifold on which the zero modes
are localized. | math-ph |
Entanglement of vortices in the Ginzburg--Landau equations for
superconductors: In 1988, Nelson proposed that neighboring vortex lines in high-temperature
superconductors may become entangled with each other. In this article we
construct solutions to the Ginzburg--Landau equations which indeed have this
property, as they exhibit entangled vortex lines of arbitrary topological
complexity. | math-ph |
Conformal maps in periodic flows and in suppression of stretch-twist and
fold on Riemannian manifolds: Examples of conformal dynamo maps have been presented earlier [Phys Plasmas
\textbf{14}(2007)] where fast dynamos in twisted magnetic flux tubes in
Riemannian manifolds were obtained. This paper shows that conformal maps, under
the Floquet condition, leads to coincidence between exponential stretching or
Lyapunov exponent, conformal factor of fast dynamos. Unfolding conformal dynamo
maps can be obtained in Riemann-flat manifolds since here, Riemann curvature
plays the role of folding. Previously, Oseledts [Geophys Astrophys Fluid Dyn
\textbf{73} (1993)] has shown that the number of twisted and untwisting orbits
in a two torus on a compact Riemannian manifold induces a growth of fast dynamo
action. In this paper, the stretching of conformal thin magnetic flux tubes is
constrained to vanish, in order to obtain the conformal factor for
non-stretching non-dynamos. Since thin flux tube can be considered as a twisted
or untwisting two-torus map, it is shown that the untwisting, weakly torsion,
and non-stretching conformal torus map cannot support a fast dynamo action, a
marginal dynamo being obtained. This is an example of an anti-fast dynamo
theorem besides the ones given by Vishik and Klapper and Young [Comm Math Phys
\textbf{173}(1996)] in ideally high conductive flow. From the Riemann curvature
tensor it is shown that new conformal non-dynamo, is actually singular as one
approaches the magnetic flux tube axis. Thus conformal map suppresses the
stretching directions and twist, leading to the absence of fast dynamo action
while Riemann-flat unfolding manifolds favors non-fast dynamos. | math-ph |
Hidden quartic symmetry in N=2 supersymmetry: It is shown that for N=2 supersymmetry a hidden symmetry arises from the
hybrid structure of a quartic algebra. The implications for invariant
Lagrangians and multiplets are explored. | math-ph |
Vision-based macroscopic pedestrian models: We propose a hierarchy of kinetic and macroscopic models for a system
consisting of a large number of interacting pedestrians. The basic interaction
rules are derived from earlier work where the dangerousness level of an
interaction with another pedestrian is measured in terms of the derivative of
the bearing angle (angle between the walking direction and the line connecting
the two subjects) and of the time-to-interaction (time before reaching the
closest distance between the two subjects). A mean-field kinetic model is
derived. Then, three different macroscopic continuum models are proposed. The
first two ones rely on two different closure assumptions of the kinetic model,
respectively based on a monokinetic and a von Mises-Fisher distribution. The
third one is derived through a hydrodynamic limit. In each case, we discuss the
relevance of the model for practical simulations of pedestrian crowds. | math-ph |
Bosons in a Trap: Asymptotic Exactness of the Gross-Pitaevskii Ground
State Energy Formula: Recent experimental breakthroughs in the treatment of dilute Bose gases have
renewed interest in their quantum mechanical description, respectively in
approximations to it. The ground state properties of dilute Bose gases confined
in external potentials and interacting via repulsive short range forces are
usually described by means of the Gross-Pitaevskii energy functional. In joint
work with Elliott H. Lieb and Jakob Yngvason its status as an approximation for
the quantum mechanical many-body ground state problem has recently been
rigorously clarified. We present a summary of this work, for both the two- and
three-dimensional case. | math-ph |
Infinitesimal Legendre symmetry in the Geometrothermodynamics programme: The work within the Geometrothermodynamics programme rests upon the metric
structure for the thermodynamic phase-space. Such structure exhibits discrete
Legendre symmetry. In this work, we study the class of metrics which are
invariant along the infinitesimal generators of Legendre transformations. We
solve the Legendre-Killing equation for a $K$-contact general metric. We
consider the case with two thermodynamic degrees of freedom, i.e. when the
dimension of the thermodynamic phase-space is five. For the generic form of
contact metrics, the solution of the Legendre-Killing system is unique, with
the sole restriction that the only independent metric function -- $\Omega$ --
should be dragged along the orbits of the Legendre generator. We revisit the
ideal gas in the light of this class of metrics. Imposing the vanishing of the
scalar curvature for this system results in a further differential equation for
the metric function $\Omega$ which is not compatible with the Legendre
invariance constraint. This result does not allow us to use the regular
interpretation of the curvature scalar as a measure of thermodynamic
interaction for this particular class. | math-ph |
Boltzmann limit for a homogenous Fermi gas with dynamical Hartree-Fock
interactions in a random medium: We study the dynamics of the thermal momentum distribution function for an
interacting, homogenous Fermi gas on $\Z^3$ in the presence of an external weak
static random potential, where the pair interactions between the fermions are
modeled in dynamical Hartree-Fock theory. We determine the Boltzmann limits
associated to different scaling regimes defined by the size of the random
potential, and the strength of the fermion interactions. | math-ph |
On Malyshev's method of automorphic functions in diffraction by wedges: We describe Malyshev's method of automorphic functions in application to
boundary value problems in angles and to diffraction by wedges. We give a
consize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller,
G.E. Shilov and others. | math-ph |
Extended Hamiltonians, Coupling-Constant Metamorphosis and the
Post-Winternitz System: The coupling-constant metamorphosis is applied to modified extended
Hamiltonians and sufficient conditions are found in order that the transformed
high-degree first integral of the transformed Hamiltonian is determined by the
same algorithm which computes the corresponding first integral of the original
extended Hamiltonian. As examples, we consider the Post-Winternitz system and
the 2D caged anisotropic oscillator. | math-ph |
Functional Classical Mechanics and Rational Numbers: The notion of microscopic state of the system at a given moment of time as a
point in the phase space as well as a notion of trajectory is widely used in
classical mechanics. However, it does not have an immediate physical meaning,
since arbitrary real numbers are unobservable. This notion leads to the known
paradoxes, such as the irreversibility problem. A "functional" formulation of
classical mechanics is suggested. The physical meaning is attached in this
formulation not to an individual trajectory but only to a "beam" of
trajectories, or the distribution function on phase space. The fundamental
equation of the microscopic dynamics in the functional approach is not the
Newton equation but the Liouville equation for the distribution function of the
single particle. The Newton equation in this approach appears as an approximate
equation describing the dynamics of the average values and there are
corrections to the Newton trajectories. We give a construction of probability
density function starting from the directly observable quantities, i.e., the
results of measurements, which are rational numbers. | math-ph |
Geometric Mean of States and Transition Amplitudes: The transition amplitude between square roots of states, which is an analogue
of Hellinger integral in classical measure theory, is investigated in
connection with operator-algebraic representation theory. A variational
expression based on geometric mean of positive forms is utilized to obtain an
approximation formula for transition amplitudes. | math-ph |
Co-primeness preserving higher dimensional extension of q-discrete
Painleve I, II equations: We construct the q-discrete Painleve I and II equations and their higher
order analogues by virtue of periodic cluster algebras. Using particular (k,k)
exchange matrices, we show that the cluster algebras corresponding to k=4 and 5
give the q-discrete Painleve I and II equations respectively. For k=6,7,..., we
have the higher order discrete equations that satisfy an integrable criterion,
the co-primeness property. | math-ph |
Energy extremals and Nonlinear Stability in a Variational theory of
Barotropic Fluid - Rotating Sphere System: A new variational principle - extremizing the fixed frame kinetic energy
under constant relative enstrophy - for a coupled barotropic flow - rotating
solid sphere system is introduced with the following consequences. In
particular, angular momentum is transfered between the fluid and the solid
sphere through a modelled torque mechanism. The fluid's angular momentum is
therefore not fixed but only bounded by the relative enstrophy, as is required
of any model that supports super-rotation.
The main results are: At any rate of spin $\Omega $ and relative enstrophy,
the unique global energy maximizer for fixed relative enstrophy corresponds to
solid-body super-rotation; the counter-rotating solid-body flow state is a
constrained energy minimum provided the relative enstrophy is small enough,
otherwise, it is a saddle point.
For all energy below a threshold value which depends on the relative
enstrophy and solid spin $\Omega $, the constrained energy extremals consist of
only minimizers and saddles in the form of counter-rotating states$.$ Only when
the energy exceeds this threshold value can pro-rotating states arise as global
maximizers.
Unlike the standard barotropic vorticity model which conserves angular
momentum of the fluid, the counter-rotating state is rigorously shown to be
nonlinearly stable only when it is a local constrained minima. The global
constrained maximizer corresponding to super-rotation is always nonlinearly
stable. | math-ph |
Scattering phase shift for relativistic separable potential with
Laguerre-type form factors: As an extension of earlier work [J. Phys. A: Math. Gen. 34 (2001) 11273] we
obtain analytic expressions for the scattering phase shift of M-term
relativistic separable potential with Laguerre-type form factors and for M = 1,
2, and 3. We take the Dirac Hamiltonian as the reference Hamiltonian. Just like
in the cited article, the tools of the relativistic J-matrix method of
scattering will be used. However, the results obtained here are for a general
angular momentum, which is in contrast to the previous work where only S-wave
scattering could be calculated. An exact numerical evaluation for higher order
potentials (M >= 4) can be obtained in a simple and straightforward way. | math-ph |
Fourier--Bessel functions of singular continuous measures and their many
asymptotics: We study the Fourier transform of polynomials in an orthogonal family, taken
with respect to the orthogonality measure. Mastering the asymptotic properties
of these transforms, that we call Fourier--Bessel functions, in the argument,
the order, and in certain combinations of the two is required to solve a number
of problems arising in quantum mechanics. We present known results, new
approaches and open conjectures, hoping to justify our belief that the
importance of these investigations extends beyond the application just
mentioned, and may involve interesting discoveries. | math-ph |
On a Schrödinger operator with a purely imaginary potential in the
semiclassical limit: We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical
$h\rightarrow 0$, where $V$ is a smooth real potential with no critical points.
We obtain both the left margin of the spectrum, as well as resolvent estimates
on the left side of this margin. We extend here previous results obtained for
the Dirichlet realization of ${\mathcal A}_h$ by removing significant
limitations that were formerly imposed on $V$. In addition, we apply our
techniques to the more general Robin boundary condition and to a transmission
problem which is of significant interest in physical applications. | math-ph |
Exact and approximate solutions of Schrödinger's equation with
hyperbolic double-well potentials: Analytic and approximate solutions for the energy eigenvalues generated by
the hyperbolic potentials
$V_m(x)=-U_0\sinh^{2m}(x/d)/\cosh^{2m+2}(x/d),\,m=0,1,2,\dots$ are constructed.
A byproduct of this work is the construction of polynomial solutions for the
confluent Heun equation along with necessary and sufficient conditions for the
existence of such solutions based on the evaluation of a three-term recurrence
relation. Very accurate approximate solutions for the general problem with
arbitrary potential parameters are found by use of the {\it asymptotic
iteration method}. | math-ph |
On a novel iterative method to compute polynomial approximations to
Bessel functions of the first kind and its connection to the solution of
fractional diffusion/diffusion-wave problems: We present an iterative method to obtain approximations to Bessel functions
of the first kind $J_p(x)$ ($p>-1$) via the repeated application of an integral
operator to an initial seed function $f_0(x)$. The class of seed functions
$f_0(x)$ leading to sets of increasingly accurate approximations $f_n(x)$ is
considerably large and includes any polynomial. When the operator is applied
once to a polynomial of degree $s$, it yields a polynomial of degree $s+2$, and
so the iteration of this operator generates sets of increasingly better
polynomial approximations of increasing degree. We focus on the set of
polynomial approximations generated from the seed function $f_0(x)=1$. This set
of polynomials is not only useful for the computation of $J_p(x)$, but also
from a physical point of view, as it describes the long-time decay modes of
certain fractional diffusion and diffusion-wave problems. | math-ph |
Quantum Field Theories and Prime Numbers Spectrum: The Riemann hypothesis states that all nontrivial zeros of the zeta function
lie on the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested a possible
approach to prove it, based on spectral theory. Within this context, some
authors formulated the question: is there a quantum mechanical system related
to the sequence of prime numbers? In this Letter we show that such a sequence
is not zeta regularizable. Therefore, there are no physical systems described
by self-adjoint operators with countably infinite number of degrees of freedom
with spectra given by the sequence of primes numbers. | math-ph |
Quasi-normal modes for de Sitter-Reissner-Nordström Black Holes: The quasi-normal modes for black holes are the resonances for the scattering
of incoming waves by black holes. Here we consider scattering of massless
uncharged Dirac fields propagating in the outer region of de
Sitter-Reissner-Nordstr{\"o}m black hole, which is spherically symmetric
charged exact solution of the Einstein-Maxwell equations. Using the spherical
symmetry of the equation and restricting to a fixed harmonic the problem is
reduced to a scattering problem for the 1D massless Dirac operator on the line.
The resonances for the problem are related to the resonances for a certain
semiclassical Schr{\"o}dinger operator with exponentially decreasing positive
potential. We give exact relation between the sets of Dirac and Schr{\"o}dinger
resonances. The asymptotic distribution of the resonances is close to the
lattice of pseudopoles associated to the non-degenerate maxima of the
potentials.
Using the techniques of quantum Birkhoff normal form we give the complete
asymptotic formulas for the resonances. In particular, we calculate the first
three leading terms in the expansion. Moreover, similar results are obtained
for the de Sitter-Schwarzschild quasi-normal modes, thus improving the result
of S\'a Barreto and Zworski from 1997. | math-ph |
Foldy-Wouthuysen transformation for relativistic particles in external
fields: A method of Foldy-Wouthuysen transformation for relativistic spin-1/2
particles in external fields is proposed. It permits determination of the
Hamilton operator in the Foldy-Wouthuysen representation with any accuracy.
Interactions between a particle having an anomalous magnetic moment and
nonstationary electromagnetic and electroweak fields are investigated. | math-ph |
Conservation laws, symmetries, and line soliton solutions of generalized
KP and Boussinesq equations with p-power nonlinearities in two dimensions: Nonlinear generalizations of integrable equations in one dimension, such as
the KdV and Boussinesq equations with $p$-power nonlinearities, arise in many
physical applications and are interesting in analysis due to critical
behaviour. This paper studies analogous nonlinear $p$-power generalizations of
the integrable KP equation and the Boussinesq equation in two dimensions.
Several results are obtained. First, for all $p\neq 0$, a Hamiltonian
formulation of both generalized equations is given. Second, all Lie symmetries
are derived, including any that exist for special powers $p\neq0$. Third,
Noether's theorem is applied to obtain the conservation laws arising from the
Lie symmetries that are variational. Finally, explicit line soliton solutions
are derived for all powers $p>0$, and some of their properties are discussed. | math-ph |
Nonlinear integrable couplings of a generalized super
Ablowitz-Kaup-Newell-Segur hierarchy and its super bi-Hamiltonian structures: In this paper, a new generalized $5\times5$ matrix spectral problem of
Ablowitz-Kaup-Newell-Segur(AKNS) type associated with the enlarged matrix Lie
super algebra is proposed and its corresponding super soliton hierarchy is
established. The super variational identities is used to furnish
super-Hamiltonian structures for the resulting super soliton hierarchy. | math-ph |
The Graded Differential Geometry of Mixed Symmetry Tensors: We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a
non-trivial generalisation of supermanifolds - may be useful in a geometrical
approach to mixed symmetry tensors such as the dual graviton. The geometric
aspects of such tensor fields on both flat and curved space-times are
discussed. | math-ph |
Estimating complex eigenvalues of non-self-adjoint Schrödinger
operators via complex dilations: The phenomenon "hypo-coercivity," i.e., the increased rate of contraction for
a semi-group upon adding a large skew-adjoint part to the generator, is
considered for 1D semigroups generated by the Schr\"odinger operators
$-\partial^2_x + x^2 + i{\gamma} f (x)$ with a complex potential. For $f$ of
the special form$ f (x) = 1/(1 + |x|^\kappa)$, it is shown using complex
dilations that the real part of eigenvalues of the operator are larger than a
constant times $|\gamma|^{2/(\kappa+2)}$. | math-ph |
Effective dislocation lines in continuously dislocated crystals. I.
Material anholonomity: A continuous geometric description of Bravais monocrystals with many
dislocations and secondary point defects created by the distribution of these
dislocations is proposed. Namely, it is distinguished, basing oneself on Kondo
and Kroners Gedanken Experiments for dislocated bodies, an anholonomic triad of
linearly independent vector fields. The triad defines local crystallographic
directions of the defective crystal as well as a continuous counterpart of the
Burgers vector for single dislocations. Next, the influence of secondary point
defects on the distribution of many dislocations is modeled by treating these
local crystallographic directions as well as Burgers circuits as those located
in such a Riemannian material space that becomes an Euclidean 3-manifold when
dislocations are absent. Some consequences of this approach are discussed. | math-ph |
Modes and quasi-modes on surfaces: variation on an idea of Andrew
Hassell: This paper is inspired from the nice result of Andrew Hassell on the
eigenfunctions in the stadium billiard. From a classical paper of V. Arnol'd,
we know that quasi-modes are not always close to exact modes. We show that, for
almost all Riemannian metrics on closed surfaces with an elliptic generic
closed geodesic C, there exists exact modes located on C. Related problems in
the integrable case are discussed in several papers of John Toth and Steve
Zelditch. | math-ph |
Development of the method of quaternion typification of Clifford algebra
elements: In this paper we further develop the method of quaternion typification of
Clifford algebra elements suggested by the author in the previous paper. On the
basis of new classification of Clifford algebra elements it is possible to
reveal and prove a number of new properties of Clifford algebra. We use k-fold
commutators and anticommutators. In this paper we consider Clifford and
exterior degrees and elementary functions of Clifford algebra elements. | math-ph |
Bounds on the spectral shift function and the density of states: We study spectra of Schr\"odinger operators on $\RR^d$. First we consider a
pair of operators which differ by a compactly supported potential, as well as
the corresponding semigroups. We prove almost exponential decay of the singular
values $\mu_n$ of the difference of the semigroups as $n\to \infty$ and deduce
bounds on the spectral shift function of the pair of operators.
Thereafter we consider alloy type random Schr\"odinger operators. The single
site potential $u$ is assumed to be non-negative and of compact support. The
distributions of the random coupling constants are assumed to be H\"older
continuous. Based on the estimates for the spectral shift function, we prove a
Wegner estimate which implies H\"older continuity of the integrated density of
states. | math-ph |
New multisymplectic approach to the Metric-Affine (Einstein-Palatini)
action for gravity: We present a covariant multisymplectic formulation for the Einstein-Palatini
(or Metric-Affine) model of General Relativity (without energy-matter sources).
As it is described by a first-order affine Lagrangian (in the derivatives of
the fields), it is singular and, hence, this is a gauge field theory with
constraints. These constraints are obtained after applying a constraint
algorithm to the field equations, both in the Lagrangian and the Hamiltonian
formalisms. In order to do this, the covariant field equations must be written
in a suitable geometrical way, using integrable distributions which are
represented by multivector fields of a certain type. We obtain and explain the
geometrical and physical meaning of the Lagrangian constraints and we construct
the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of
the model are discussed in both formalisms and, from them, the equivalence with
the Einstein-Hilbert model is established. | math-ph |
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