The density of surface states as the total time delay

TL;DR

This paper establishes a generalized Levinson theorem linking the total density of surface states to the total time delay in a scattering problem involving weak random surface potentials, using advanced operator algebra techniques.

Contribution

It introduces a new Levinson theorem for surface states and time delay, with explicit wave operator formulas in the REI representation and an index theorem approach.

Findings

Proves the equality of surface state density and time delay density.

Develops explicit wave operator formulas in the REI representation.

Utilizes an index theorem for operator algebras in the proof.

Abstract

For a scattering problem of tight-binding Bloch electrons by a weak random surface potential, a generalized Levinson theorem is put forward showing the equality of the total density of surface states and the density of the total time delay. The proof uses explicit formulas for the wave operators in the new rescaled energy and interaction (REI) representation, as well as an index theorem for adequate associated operator algebras.