Spectral sum rules for the Schr\"odinger equation

TL;DR

This paper investigates spectral sum rules for the Schrödinger equation, deriving explicit formulas using perturbation theory and Padé approximants, and computes exact sum rules for specific impurity configurations in one and two dimensions.

Contribution

It introduces a method to compute spectral sum rules for Schrödinger operators, including perturbative and non-perturbative approaches, with exact results for certain impurity models.

Findings

Derived explicit formulas for sum rules up to second order.

Extended sum rules non-perturbatively using Padé approximants.

Calculated exact sum rules for specific impurity configurations in 1D and 2D.

Abstract

We study the sum rules of the form $Z(s) = \sum_n E_n^{-s}$, where $E_n$ are the eigenvalues of the time--independent Schr\"odinger equation (in one or more dimensions) and $s$ is a rational number for which the series converges. We have used perturbation theory to obtain an explicit formula for the sum rules up to second order in the perturbation and we have extended it non--perturbatively by means of a Pad\'e--approximant. For the special case of a box decorated with one impurity in one dimension we have calculated the first few sum rules of integer order exactly; the sum rule of order one has also been calculated exactly for the problem of a box with two impurities. In two dimensions we have considered the case of an impurity distributed on a circle of arbitrary radius and we have calculated the exact sum rules of order two. Finally we show that exact sum rules can be obtained, in one dimension, by transforming the Schr\"odinger equation into the Helmholtz equation with a suitable density.