Friedel Sum Rule as a Trace Formula

TL;DR

This paper proves the Friedel sum rule for excess charge in metals with impurities by defining it directly in the infinite volume limit, addressing previous issues with boundedness in finite volume.

Contribution

It introduces a new definition of excess charge in infinite volume and proves the Friedel sum rule under this framework.

Findings

Friedel sum rule holds for the defined excess charge in infinite volume.

Addresses boundedness issues of excess charge in finite volume.

Provides a rigorous proof of the sum rule in the infinite volume limit.

Abstract

We examine the Friedel sum rule which states that the "excess charge" due to a single impurity potential in a metal is equal to a sum of phase shifts for scatterings of electrons by the impurity. For finite volume, the ``excess charge" is given by the difference between total numbers of levels in the Fermi sea with and without the impurity potential. However, a sequence of the "excess charge" for finite volume is not necessarily bounded in the infinite volume limit, as was pointed out by Kirsch. In order to circumvent this difficulty, we define "excess charge" directly for the infinite volume. The Friedel sum rule is proven to hold for the "excess charge" thus defined.