The Floquet theory for the Schrödinger operator with a perturbed periodic potential and precise spectral asymptotics. (Q1595626)

Let \(H=-\Delta+H\), where \(W\in L_\infty(\mathbb R^n,\mathbb R)\) is periodic. Suppose that \(V\) is a potential that decreases at infinity. The author asserts that the study of the perturbed Schrödinger operator \(H+V\) can be replaced with the study of a pseudodifferential operator on a torus. This fact enables one to obtain new information with the aid of the technique of pseudodifferential operators. Let \(({\mathcal E}^-,{\mathcal E}^+)\) be a lacuna in \(\sigma_{\text{ess}}(H)= \sigma(H)= \sigma_{\text{ess}}(H+V)\) and \(E\in({\mathcal E}^-,{\mathcal E}^+)\). The author obtains the asymptotic behaviour of the numbers of the eigenvalues of \(H+V\) in the intervals \((E,{\mathcal E}^+-\nu)\) and \(({\mathcal E}^-+\nu,E)\) as \(\nu\to+0\), and that as \(t\to\infty\) of \(N_\pm(t;H-E;V)\) which is the number of the eigenvalues \(\mu\) of the eigenvalue problem \((H-E)u= \pm V\mu u\) with \(\mu\in(0,t)\). One of the results obtained is the following: \[ N_-(t;H-E;V)= c(H-E;V_{-m})t^{n/m}+ O(t^{(n-\omega)/m}), \] where \(\omega>0\) if \(V(x)= V_{-m}(x)+ (|x|^{-m-1})\) as \(x\to\infty\) and \(V_{-m}\) is positive homogeneous of degree \(-m\). Analogous results are established for \(N_+(t;H-E;V)\).