L\'evy flights in the infinite potential well as the hypersingular Fredholm problem

TL;DR

This paper investigates Le9vy flights within an infinite potential well by transforming the fractional Schrf6dinger eigenvalue problem into a hypersingular Fredholm integral equation and solving it numerically.

Contribution

It introduces a novel approach to solve fractional quantum problems by converting them into hypersingular Fredholm equations and analyzing their spectra.

Findings

Eigenvalues and eigenfunctions obtained numerically.

Analytical insights into the spectral structure.

Applicable to various physical systems with fractional dynamics.

Abstract

We study L\'evy flights {{with arbitrary index $0< \mu \leq 2$}} inside a potential well of infinite depth. Such problem appears in many physical systems ranging from stochastic interfaces to fracture dynamics and multifractality in disordered quantum systems. The major technical tool is a transformation of the eigenvalue problem for initial fractional Schr\"odinger equation into that for Fredholm integral equation with hypersingular kernel. The latter equation is then solved by means of expansion over the complete set of orthogonal functions in the domain $D$, reducing the problem to the spectrum of a matrix of infinite dimensions. The eigenvalues and eigenfunctions are then obtained numerically with some analytical results regarding the structure of the spectrum.