Schrödinger operator eigenvalue (resonance) on a zone boundary (Q5959433)

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 operator \(H_0 = -\Delta+V\) defined on Bloch functions has discrete spectrum \(\{E_n\}_{n=1,2,...}\). The essential spectrum of the perturbed operator \(H=H_0 + W\) has a zone structure. The author finds conditions under which eigenvalue or resonance of the operator \(H\) falls on a zone boundary. For the operator \(H=H_0+\mu W\) with coupling constant \(\mu\) the passage of the eigenvalue or the resonance through the zone boundary under variation of \(\mu\) is discussed.