T. A. Filatova, A. I. Shafarevich, “Semiclassical spectral series of the Schrödinger operator with a delta potential on a straight line and on a sphere”, TMF, 2010, Volume 164, Number 2,Pages <nobr>279

This article is cited in 14 papers

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

T. A. Filatova^ab, A. I. Shafarevich^cab

^a Ishlinskii Institute for Problems in Mechanics, RAS, Moscow, Russia
^b Moscow Institute of Physics and Technology, Moscow, Russia
^c Lomonosov Moscow State University, Moscow, Russia

Abstract: We describe the spectral series of the Schrödinger operator $H=-(h^2/2) \Delta+V(x)+\alpha\delta(x-x_0)$, $\alpha\in\mathbb R$, with a delta potential on the real line and on the three- and two-dimensional standard spheres in the semiclassical limit as $h\to0$. We consider a smooth potential $V(x)$ such that $\lim_{|x|\to\infty}V(x)=+\infty$ 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.

Keywords: semiclassical spectrum, Schrödinger operator, delta potential, Lagrangian manifold, Maslov canonical operator.

Received: 13.02.2010

DOI: 10.4213/tmf6539