Coupling constant thresholds of perturbed periodic Hamiltonians

Abstract

We consider Schrödinger operators of the form Hλ=−Δ+V+λW on L2(Rν) (ν=1, 2, or 3) with V periodic, W short range, and λ a real non-negative parameter. Then the continuous spectrum of Hλ has the typical band structure consisting of intervals, separated by gaps. In the gaps there may be discrete eigenvalues of Hλ that are functions of the parameter λ. Let (a,b) be a gap and E(λ)∈(a,b) an eigenvalue of Hλ. We study the asymptotic behavior of E(λ) as λ approaches a critical value λ0, called a coupling constant threshold, at which the eigenvalue either emerges from or is absorbed into the continuous spectrum. A typical question is the following: Assuming E(λ)↓a as λ↓λ0, is E(λ)−a∼c(λ−λ0)α for some α>0 and c≠0, or is there an expansion in some other quantity? As one expects from previous work in the case V=0, the answer strongly depends on ν.