The fractional Schr\"{o}dinger operator and Toeplitz matrices

TL;DR

This paper explores the relationship between the one-dimensional fractional Schrödinger operator with Dirichlet boundary conditions and Toeplitz matrices, analyzing eigenvalue asymptotics using Szeg"o's theorem, with applications to quantum particle models.

Contribution

It establishes a connection between fractional Schrödinger operators and truncated Toeplitz matrices, providing asymptotic eigenvalue behavior analysis for fractional Laplacians.

Findings

Eigenvalue product asymptotics determined for fractional Schrödinger operators.

Application of Szeg"o's strong limit theorem to quantum models.

Insights into particle hopping probabilities in bounded intervals.

Abstract

Confining a quantum particle in a compact subinterval of the real line with Dirichlet boundary conditions, we identify the connection of the one-dimensional fractional Schr\"odinger operator with the truncated Toeplitz matrices. We determine the asymptotic behaviour of the product of eigenvalues for the $\alpha$-stable symmetric laws by employing the Szeg\"o's strong limit theorem. The results of the present work can be applied to a recently proposed model for a particle hopping on a bounded interval in one dimension whose hopping probability is given a discrete representation of the fractional Laplacian.