Oscillation theory and the density of states for the Schrödinger operator in odd dimension
The purpose of the paper is to introduce a rotation number for the Schrödinger operator \[ h=-\Delta +q(x)=- \sum^{n}_{i=1}\frac{\partial^ 2}{\partial x_ i^ 2}+q(x_ 1,x_ 2,...,x_ n) \] in odd dimension \(n=1,3,5,... \). q(x) is a bounded \(C^{\infty}\) potential defined on \({\mathbb{R}}^ n\). This rotation number generalizes the one defined by Johnson-Moser when \(n=1\). The author introduces a measure of the number of ``changes of sign'' of the Dirichlet Green's function g of the operator h restricted to bounded smooth domains \(D\subset {\mathbb{R}}^ n\). Then the rotation number is defined by an averaging procedure, letting D tend to \({\mathbb{R}}^ n\). The main result of the paper is that the rotation number, thus defined, equals a constant times the integrated density of states of h, similar to that in one dimension. It starts with collecting some elementary facts about the Green's function for h on a bounded domain \(\bar D\subset {\mathbb{R}}^ n\) with \(C^{\infty}\) boundary \(\partial D\). This is followed by a geometric method of counting the eigenvalues of h. Finally the main result is obtained.
scientific article; zbMATH DE number 4214408
scientific article