Spectral sum rules for the Schrödinger equation

• Published in: Annals of physics 2020-12, Vol.423, p.168334, Article 168334
• Main Author: Amore, Paolo
• Format: Article
• Language: English
• Subjects: Perturbation theory; Quantum sum rule; Schrödinger equation
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2020.168334

We study the sum rules of the form Z(s)=∑nEn−s, where En are the eigenvalues of the time-independent Schrödinger 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 Padé-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 Schrödinger equation into the Helmholtz equation with a suitable density. •We study the spectral sum rules associated to the eigenvalues of the Schrödinger equation.•We obtain an explicit spectral decomposition for the Green’s function of order 1∕N, N=2,3…•We have obtained exact results for a number of examples.