Autocorrelation functions for quantum particles in supersymmetric P\"oschl-Teller Potentials

TL;DR

This paper analyzes the long-term behavior of autocorrelation functions in supersymmetric quantum systems with P"oschl-Teller potentials, revealing their limiting distribution via Jacobi theta functions.

Contribution

It introduces a novel analysis of autocorrelation functions in supersymmetric quantum mechanics with P"oschl-Teller potentials, connecting to theta functions and homogeneous spaces.

Findings

Limiting autocorrelation distribution described by Jacobi theta functions.

Applicable to large classes of P"oschl-Teller partner potentials.

Provides insights into quantum localization and dynamics.

Abstract

We consider autocorrelation functions for supersymmetric quantum mechanical systems (consisting of a fermion and a boson) confined in trigonometric P\"oschl-Teller partner potentials. We study the limit of rescaled autocorrelation functions (at random time) as the localization of the initial state goes to infinity. The limiting distribution can be described using pairs of Jacobi theta functions on a suitably defined homogeneous space, as a corollary of the work of Cellarosi and Marklof. A construction by Contreras-Astorga and Fern\'andez provides large classes of P\"oschl-Teller partner potentials to which our analysis applies.