Related papers: A nodal domain theorem for integrable billiards in…
We characterize quantum dynamics in triangular billiards in terms of five properties: (1) the level spacing ratio (LSR), (2) spectral complexity (SC), (3) Lanczos coefficient variance, (4) energy eigenstate localisation in the Krylov basis,…
For an integral domain $R$ and a commutative cancellative monoid $M$, the ring consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$ is called the monoid ring of $M$ over $R$. An integral domain is called…
One of the most interesting problems in the theory of Birkhoff billiards is the problem of integrability. In all known examples of integrable billiards, the billiard tables are either conics, quadrics (closed ellipsoids as well as unclosed…
In this article we are interested for the numerical study of nonlinear eigenvalue problems. We begin with a review of theoretical results obtained by functional analysis methods, especially for the Schrodinger pencils. Some recall are given…
Two super-integrable and super-separable classical systems which can be considered as deformations of the harmonic oscillator and the Smorodinsky-Winternitz in two dimensions are studied and identified with motions in spaces of constant…
We use the semiclassical quantization scheme of Bogomolny to calculate eigenvalues of the Lima\c con quantum billiard corresponding to a conformal map of the circle billiard. We use the entire billiard boundary as the chosen surface of…
Inequalities are derived for sums and quotients of eigenvalues of magnetic Schroedinger operators with non-negative electric potentials in domains. The bounds reflect the correct order of growth in the semi-classical limit.
In this paper the domain wall solutions of a Ginzburg-Landau non-linear $\mathbb{S}^2$-sigma hybrid model are exactly calculated. There exist two types of basic domain walls and two families of composite domain walls. The domain wall…
We develop powerful numerical and analytical techniques for the solution of the Helmholtz equation on general domains. We prove two theorems: the first theorem provides an exact formula for the ground state of an arbirtrary membrane, while…
A short overview of the billiard approach for cosmological-type models with n Einstein factor-spaces is presented. We start with the billiard representation for pseudo-Euclidean Toda-like systems of cosmological origin. Then we consider…
Dynamical billiards, or the behavior of a particle traveling in a planar region $D$ undergoing elastic collisions with the boundary, has been extensively studied and is used to model many physical phenomena such as a Boltzmann gas. Of…
We consider the distribution of eigenvalues for the wave equation in annular (electromagnetic or acoustic) ray-splitting billiards. These systems are interesting in that the derivation of the associated smoothed spectral counting function…
This work presents a framework for billiards in convex domains on two dimensional Riemannian manifolds. These domains are contained in connected, simply connected open subsets which are totally normal. In this context, some basic properties…
Polygonalization of any smooth billiard boundary can be carried out in several ways. We show here that the semiclassical description depends on the polygonalization process and the results can be inequivalent. We also establish that…
In the present work we explore the concept of solitary wave billiards. I.e., instead of a point particle, we examine a solitary wave in an enclosed region and explore its collision with the boundaries and the resulting trajectories in cases…
We extend our finite difference time domain method for numerical solution of the Schrodinger equation to cases where eigenfunctions are complex-valued. Illustrative numerical results for an electron in two dimensions, subject to a confining…
We study the nodal deficiency of pairs of Neumann eigenfunctions defined over symmetric dumbbell domains. As the width of the connecting neck shrinks, these eigenfunctions converge to Neumann eigenfunctions defined over the ends of the…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
We introduce an analogue of Payne's nodal line conjecture, which asserts that the nodal (zero) set of any eigenfunction associated with the second eigenvalue of the Dirichlet Laplacian on a bounded planar domain should reach the boundary of…
Existence of amplitude independent frequencies of oscillation is an unusual property for a nonlinear oscillator. We find that a class of N coupled nonlinear Li\'enard type oscillators exhibit this interesting property. We show that a…