Anharmonic oscillator in high dimension

Abstract

In this paper we study the bottom of the spectrum of the semiclassical anharmonic oscillator P_m(h)=−h²Δ_m+V_m(x) where V_m(x)=μΣ_j=1^mx²_j+((g)/(mⁿ⁻¹))(Σ_j=1^mx_j²)ⁿ, μ∈R, g∈R₊ and n∈N, n>1, when the number m of interacting particles is large. Denoting by λ(m,h) its lowest eigenvalue, we prove that lim _m→+∞λ(m,h)/m exists and has a complete asymptotic expansion in powers of h, when the Planck's constant h tends to 0. For h fixed, we also obtain an expansion in powers of m⁻¹ for the first eigenvalues of P_m. Moreover, we consider integrals of the form I(β,m)=∫_R^m=e^−βV_m(x) dx where β is a large parameter and we prove the existence of the limit, as m→+∞, of the quantity (1/m)ln I(β,m) and that this limit has an asymptotic expansion in power of β⁻¹ for large values of β.