Existence Of Eigenfunctions Research Articles

Multifractal Eigenfunctions for a Singular Quantum Billiard

Whereas much work in the mathematical literature on quantum chaos has focused on phenomena such as quantum ergodicity and scarring, relatively little is known at the rigorous level about the existence of eigenfunctions whose morphology is more complex. Quantum systems whose dynamics is intermediate between certain regimes - for example, at the transition between Anderson localized and delocalized eigenfunctions, or in systems whose classical dynamics is intermediate between integrability and chaos - have been conjectured in the physics literature to have eigenfunctions exhibiting multifractal, self-similar structure. To-date, no rigorous mathematical results have been obtained about systems of this kind in the context of quantum chaos. We give here the first rigorous proof of the existence of multifractal eigenfunctions for a widely studied class of intermediate quantum systems. Specifically, we derive an analytical formula for the Renyi entropy associated with the eigenfunctions of arithmetic Seba billiards, in the semiclassical limit, as the associated eigenvalues tend to infinity. We also prove multifractality of the ground state for more general, non-arithmetic billiards and show that the fractal exponent in this regime satisfies a symmetry relation, similar to the one predicted in the physics literature, by establishing a connection with the functional equation for Epstein's zeta function.

Semiclassical Eigenfunctions of the Schrödinger Operator on a Graph That Are Localized Near a Subgraph

We study semiclassical eigenvalues and eigenfunctions of the Schrodinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph.

Evolution of perturbations on a steady weakly inhomogeneous background. Complex Hamiltonian equations

Abstract The evolution of linear unidimensional perturbations on a steady weakly inhomogeneous background, that is, on a background that depends on the coordinate x through the ratio x / L where L is a large scale, is studied. The perturbation evolution time T is considered to be quite large such that the perturbations succeed in propagating over a distance comparable with L , and the inhomogeneity of the background is manifested in the behaviour of the perturbations. Perturbations, generated by a time-limited external action localized in a small domain, are considered in detail. It is assumed that local instability conditions are satisfied in the whole of the domain considered or a part of it, that is, it is assumed that, if the background parameters are “frozen” and assumed to be homogeneous, growing perturbations will exist for states corresponding to a certain domain of values of x / L . A procedure, based on a Fourier transform and the use of the steepest descent method, is formulated for finding the asymptotics of the perturbations at large values of L and T . The perturbations can be described by complex Hamiltonian equations in which the Hamiltonian is a frequency expressed from the dispersion equation as a function of the wave number and coordinate. In the case of instability, these quantities are complex. The relation between the asymptotics obtained and the eigenfunctions of the problem is considered. An example of constructing the asymptotics of the intensification factor is presented; they are identical, within the limits of the assumed accuracy, to the intensification factor found from the exact solution of the problem constructed. The existence of eigenfunctions is pointed out and the corresponding characteristic frequencies are estimated.

A class of multi-parameter eigenvalue problems for perturbed p-Laplacians

This paper is devoted to multi-parameter eigenvalue problems for one-dimensional perturbed p-Laplacians, modelling travelling waves for a class of nonlinear evolution PDE. Dispersion relations between the eigen-parameters, the existence of eigenfunctions and positive eigenfunctions, variational principles for eigenvalues and constructing solutions in the analytical and implicit forms are the main subject of this paper. We use both variational and analytical methods.

Existence of eigenfunctions of the Tricomi spectral problem for some classes of multidimensional mixed hyperbolic–parabolic equations

We show that there exists a countable set of eigenfunctions of the Tricomi spectral problem for multidimensional mixed hyperbolic–parabolic equations.

Eigenfunctions of Two-Scale Difference Equations with Dilation Parameter and Infinite Products

This paper deals with two-scale difference equations having an arbitrary dilation parameter and a formal power series as symbol. We investigate the equation concerning the existence of nonzero compactly supported distributional solutions. In order to include also continuous solutions it is advantageous to consider the two-scale difference equation as eigenvalue problem where the solutions are either compactly supported or integrals of compactly supported distributions. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure depending on the dilation parameter. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.

Eigenfunctions of Two-Scale Difference Equations with Rational Symbol

This paper deals with two-scale difference equations having a formal power series as symbol. We require that the equation has non-zero distributional solutions which are either compactly supported or integrals of compactly supported distributions with support bounded to the left. Such solutions are called eigenfunctions. As main result we determine the necessary and sufficient condition for the existence of eigenfunctions that the symbol must be a rational function with a special structure. We show that the eigenfunctions can be expressed by means of a finite sum of shifted eigenfunctions belonging to the case with a polynomial symbol (characteristic polynomial), which is well investigated.

On Eigenvalue Intervals and Eigenfunctions of Nonresonance Singular Dirichlet Boundary Value Problems

In this paper we shall consider the nonresonance Dirichlet boundary value problem $$ \left\{ {\begin{array}{*{20}l} {{ - {x}\ifmmode{''}\else$''$\fi + pp{\left( t \right)}x = \lambda f{\left( {t,x} \right)},} \hfill} & {{t \in {\left( {0,1} \right)},} \hfill} \\ {{x{\left( 0 \right)} = x{\left( 1 \right)} = 0,} \hfill} & {{} \hfill} \\ \end{array} } \right. $$ where λ>0 is a parameter, p>0 is a constant. Intervals of λ are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].

Boundary value contact acoustic problems

Boundary value contact problems are the problems of mathematical physics, whose statement contains some conditions at particular points or lines of the boundary. The purpose of this paper is to discuss some mathematical aspects of the BVCP, such as the correct functional classes and Fredholm type equations for the solution, the completeness of eigenfunctions, the connection between the BVCP and supersymmetric quantum mechanic, paritial radiation conditions, limiting absorption principle, existence of eigenfunctions in infinite domains, uniqueness of the solution, “optical” theorem, and so on.

Dynamics of Symmetrizable Nonconservative Systems

This work examines a subclass of nonconservative and nonself-adjoint linear distributed parameter systems modeled by partial differential equations subject to various boundary conditions. It has been previously illustrated that certain nonself-adjoint operators arising in mechanics can be shown to be self-adjoint with respect to a particular self-adjoint operator. The work here extends and formalizes this approach to include systems with viscous damping. The extension presented here follows the lumped parameter case presented earlier and places emphasis on the existence of eigenfunctions for use in performing modal analysis.

Loss in unstable resonators

We discuss the integral operator AF which models the behavior of an unstable resonator having rectangular symmetry. Interest focuses on modes of the resonator that exhibit low loss for sizable values of the Fresnel number F; in the mathematical description, these correspond to eigenfunctions of AF with eigenvalues of large modulus. A serious obstacle to a theoretical understanding of these entities is immediately presented by the non-Hermitian nature of the operator, which puts the very existence of eigenfunctions in doubt. Moreover, as the equation is hard to handle numerically, direct computational evidence has been sparse. In this paper we remove some of the conceptual difficulty by introducing a notion of approximate eigenvalues and eigenfunctions which, we argue, model the physical problem equally well. We then prove that each λ with |λ| = 1 becomes an approximate eigenvalue of AF for F sufficiently large, and that the number of mutually orthogonal approximate eigenfunctions corresponding to λ grows at least linearly with F. Interpreted in physical terms, this implies that for large Fresnel number, losses in the unstable resonator can be made arbitrarily low, but the resonator will not be capable of mode selection. These conclusions differ considerably from predictions contained in a recent paper on this subject.