Schrödinger operators with decaying potentials: some counterexamples
The Schrödinger equation is studied on the half-axis, with an asymptotically slowly decreasing potential \(V(x)\) which is asymptotically bounded by a negative power (between minus one and -- not included -- minus one half) of the coordinate. A mixed boundary condition in the origin is admitted. A review is given of the results concerning the essential spectrum (on the positive energy half-line) and its possible character (i.e., point or absolutely continuous or singularly continuous). The purpose is to construct examples which show that the results available in the literature are, in fact, optimal. An extremely well readable presentation of rigorous results (well motivated and complemented by heuristic arguments, e.g. in section 3) constructs most of its ``counterexample'' sample potentials by pasting together carefully chosen periodic pieces. Then, the bound on the embedded singular spectrum can only be shown to be optimal (in Theorem 1.3) for the asymptotic decay of \(V(x)\) between two thirds and one, which is a not entirely pleasant natural limitation of the constructive approach. Its strength, on the other hand, manifests itself at the Coulombic end (especially via sum rules of Theorem 1.5 and its Corollary 1.6). The proofs pay due attention to the effects of pasting (non-smoothness) and make ample use of periodicity of the pasted pieces, say, for related asymptotic estimates of the so called Lyapunov function (= trace of the transfer matrix), with considerations closely related to some recent Kiselev's results.
Hausdorff dimension of the exceptional set
contructive proof of optimality of its estimate
sum rules for asymptotically Coulomb potentials
scientific article; zbMATH DE number 1820768
scientific article