Spectral asymptotics for Schrödinger operators with a degenerate potential

This paper is devoted to the asymptotic growth of the number of eigenvalues less than an energy À of a Schrödinger operator Hh = h^ + V on L(R) , in the case when the potential V does notfulfUl the non degen era cy condition: V(x) — +00 as |jc| — +00. For such a model, the point is that the set defined in the phase space by: Hh ^ A may no longer have an a finite volume, so that the Weyl formula has no sense. We present a survey of various results in this area. in the classical context (h = 1 andA — +oo),aswellasinthesemi-classicalone(fc — 0) and comment the different methods. The last resuit presented deals with a gênerai potential of the form: V (x) = f{y)g(z), f e C(R;R+),g € C(RP;R+) , ghomogeneous and ƒ +00 as Ivl +00.