Schrödinger operators in high dimension: convex potentials

Abstract

In this paper we study the bottom of the spectrum of a semiclassical Schrödinger operator P_m(h)=−(h²/2)Δ_m+V_m in high dimension m. We assume that V_m is convex and satisfies some conditions uniformly with respect to m. We get a complete asymptotic expansion in powers of h with an explicit control of the coefficients and of the remainder terms with respect to m of its lowest eigenvalue and we show that its first eigenfunctions decays exponentially outside a ℓ²-ball of radius $\sqrt{m}$ centered at the point where V_m reaches its minimum, as h→0.