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

Abstract : This paper is devoted to the asymptotics of eigenvalues for a Schrö-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 , (Université 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.
Document type :
Journal articles
Complete list of metadatas

Cited literature [19 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00350061
Contributor : Francoise Truc <>
Submitted on : Monday, January 5, 2009 - 5:43:18 PM
Last modification on : Wednesday, September 26, 2018 - 10:44:03 AM
Long-term archiving on : Tuesday, June 8, 2010 - 6:34:31 PM

Files

suede.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00350061, version 1
  • ARXIV : 0901.0539

Collections

Citation

Francoise Truc. Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area. Operator Theory: Advances and Applications, 2008, 186, pp.403-413. ⟨hal-00350061⟩

Share

Metrics

Record views

263

Files downloads

109