Perron-Frobenius operators and their eigendecompositions are increasingly being used as tools of global analysis for higher dimensional systems. The numerical computation of large, isolated eigenvalues and their corresponding eigenfunctions can reveal important persistent structures such as almostinvariant sets, however, often little can be said rigorously about such calculations. We attempt to...

Riemann hypothesis is proven by reducing the vanishing of Riemann Zeta to an orthogonality condition for the eigenfunctions of a nonHermitian operator having the zeros of Riemann Zeta as its eigenvalues. Eigenfunctions are analogous to the so called coherent states and in general not orthogonal to each other. The construction of the operator is inspired by the conviction that Riemann Zeta is as...

The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely dense and the eigenfunctions become concentrated in the vicinity of the intermittent fixed point. Analytical considerations generalize the results to a broad...

Michael A. Stroscio Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, IL 60607 , USA Abstract This note describes the solution of the Helmholtz equation inside a nanotorus with uniform Dirichlet boundary conditions. The eigenfunction symmetry is discussed and the lower-order eigenvalues and eigenfunctions are shown. The similarity with the case of a ...

In this study, singular diffusion operator with jump conditions is considered. Integral representations have been derived for solutions that satisfy boundary and conditions. Some properties of eigenvalues eigenfunctions are investigated. Asymtotic representation eigenfunction obtained. Reconstruction the shown by Weyl function.

In this paper we solve problems of eigenvalues of stochastic Hamiltonian systems with boundary conditions and construct the corresponding eigenfunctions. This is a sort of forward–backward stochastic di erential equations (FBSDE) parameterized by ∈ R. The problem is to nd non-trivial solutions while the trivial solution 0 exists. We show that, as the classical cases, the phenomenon of statistic...

A method to compute the bound state eigenvalues and eigenfunctions of a Schrödinger equation or a spinless Salpeter equation with central interaction is presented. This method is the generalization to the three-dimensional case of the Fourier grid Hamiltonian method for one-dimensional Schrödinger equation. It requires only the evaluation of the potential at equally spaced grid points and yield...

This note starts from work done by Dai, Geary, and Kadanoff[1] on exact eigenfunctions for Toeplitz operators. It builds methods for finding convergent expansions for eigenvectors and eigenvalues of singular, largen Toeplitz matrices, using the infinite-n case[1] as a starting point. One expansion is derived from operator equations having a two-dimensional continuous spectrum of right eigenvalu...

In this paper, we study a class of degenerate unperturbed problems. We first investigate some properties eigenvalues and eigenfunctions for the strongly elliptic operator then obtain two existence theorems nontrivial solutions when nonlinearity is function with an asymptotically condition.

In this paper we address the problem of determining and efficiently computing an approximation to the eigenvalues of the negative Laplacian −" on general domain Ω ⊂ R2 subject to homogeneous Dirichlet or Neumann boundary conditions. The basic idea is to look for eigenfunctions as the superposition of generalized eigenfunctions of the corresponding free space operator in the spirit of the classi...