Anisotropic Lavine's formula and symmetrised time delay in scattering theory

TL;DR

This paper establishes the existence of symmetrised time delay in quantum scattering for rapidly decaying potentials and demonstrates it satisfies an anisotropic Lavine's formula, highlighting the role of anisotropic dilations.

Contribution

It introduces an anisotropic Lavine's formula for symmetrised time delay in quantum scattering with potentials decaying faster than |x|^{-4}.

Findings

Existence of symmetrised time delay for certain potentials.

Derivation of an anisotropic Lavine's formula.

Identification of anisotropic dilations' significance.

Abstract

We consider, in quantum scattering theory, symmetrised time delay defined in terms of sojourn times in arbitrary spatial regions symmetric with respect to the origin. For potentials decaying more rapidly than $|x|^{-4}$ at infinity, we show the existence of symmetrised time delay, and prove that it satisfies an anisotropic version of Lavine's formula. The importance of an anisotropic dilations-type operator is revealed in our study.