The fractional Schrödinger operator and Toeplitz matrices

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 Schrödinger 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 Szegö'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.

10 pages, 2 figures

References in corpus (2)

• Fractional Laplacian in Bounded Domains
• One dimensional stable probability density functions for rational index $0<\alpha \le 2$