Semiclassical spectral invariants for Schrödinger operators
The paper is essentially devoted to the study of the semiclassical spectral measure of the semiclassical Schrödinger operator \(S_{h}= -(h^{2}/2)\Delta + V(x)\), where \(V\) is a smooth nonnegative potential and \(V^{-1}([0,a])\) is compact for some \(a>0\). A semiclassical spectral theorem for elliptic operators (Theorem 4.1 from [\textit{B. Helffer} and \textit{D. Robert}, J. Funct. Anal. 53, 246--268 (1983; Zbl 0524.35103)]) is used to compute the first two terms of the asymptotic expansion for \((2\pi h)^{n} \sum {f(\lambda_{i}(h))}\), \(\lambda_{i}(h)\), \(1\leq i\leq N(h)\), being the eigenvalues of \(S_{h}\) in \([0,a)\). The algorithm for computing higher order terms is also given. The method is next applied to prove inverse spectral results for single well potentials \(V\), for the perturbed Schrödinger operator \(P_{h} = -(h^{2}/2)\Delta + V(x) + h^{2}V_{1}(x)\) and for the magnetic Schrödinger operator in dimension 2 with radially symmetric electric potential and radially symmetric magnetic field. In the last two sections of the paper the relation between the spectral measure of \(S_{h}\) and the Birkhoff canonical form in one dimension is studied.
scientific article; zbMATH DE number 6081390