Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area

TL;DR

This paper reviews recent advances in the asymptotics of eigenvalues for Schrödinger operators with degenerate potentials, including new semi-classical and low eigenvalue estimates using Born-Oppenheimer approximations.

Contribution

It introduces novel semi-classical asymptotics and sharp eigenvalue estimates for degenerate potentials, extending classical results to more complex cases.

Findings

Semi-classical asymptotics for eigenvalues with degenerate potentials

Sharp estimates of low eigenvalues using Born-Oppenheimer approximation

Localization of higher energies for specific potential classes

Abstract

This paper is devoted to the asymptotics of eigenvalues for a Schr\"o-dinger operator in the case when the potential V does not tend to infinity at infinity. Such a potential is called degenerate. The point is that the set in the phase space where the associated hamiltonian is smaller than a fixed energy E may have an infinite volume, so that the Weyl formula which gives the behaviour of the counting function has to be revisited. We recall various results in this area, in the classical context as well as in the semi-classical one and comment the different methods. In sections 3, 4 we present our joint works with A Morame, (Universit\'e de Nantes),concerning a degenerate potential V(x) =f(y) g(z), where g is assumed to be a homogeneous positive function of m variables, and f is a smooth and strictly positive function of n variables, with a minimum in 0. In the case where f tends to infinity at infinity, we give the semi-classical asymptotic behaviour of the number of eigenvalues less than a fixed energy . Then we give a sharp estimate of the low eigenvalues, using a Born Oppenheimer approximation. With a refined approach we localize also higher energies . Finally we apply the previous methods to a class of potentials which vanish on a regular hypersurface.