Resonance multiplicity of a perturbed periodic Schrödinger operator
Let \(V,W:\mathbb{R}^3\to \mathbb{R}\) be functions such that \(V\) is periodic, \(W\) is periodic in \(x_1, x_2\) and \(W\) decays exponentially as \(x_3\to\infty\). The author studies the structure of the resonances of the Schrödinger operator \(-\Delta+V+W\). It is shown that the set of resonances coincides with the set of zeros of a certain analytic function \(\alpha-1\) and that the resonance multiplicity coincides with the multiplicity of zeros of \(\alpha-1\). Under some hypothesis on \(\alpha\) it is also shown that (an analogue of) the forward scattering amplitude vanishes on the surface given by the zeros of the derivative of the eigenvalue of the periodic operator in a cell with respect to the third quasi-momentum component.
scientific article; zbMATH DE number 1465991