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Potential wells in high dimensions I

Motivated by a problem with a large number of interacting particles with a strong exterior potential, we consider the bottom of the spectrum of the semiclassical Schrodinger operator −h 2 Δ+V in high dimension N. We assume that V satisfies certain conditions uniformly with respect to N and in particular that V has non degenerate local minima. Assuming that N = O h −N0 ) for some fixed N 0 , we are able to describe a low part of the spectrum. For instance, in the case of one potential well, we get a complete asymptotic expansion in powers of h valid uniformly with respect to N of the lowest eigenvalue and we show that this eigenvalue is simple and separated from the rest of the spectrum by a distance ≥ h/Const