Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere

Abstract

We describe the spectral series of the Schrodinger operator H = −(h2/2)Δ + V(x) + αδ(x−x0), α ∈ ℝ, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as h → 0. We consider a smooth potential V(x) such that lim|x|→∞V(x)=+∞ in the first case and V(x) = 0 in the last two cases. In the semiclassical limit in each case, we describe the classical trajectories corresponding to the quantum problem with a delta potential.