Anharmonic oscillator in high dimension

In this paper we study the bottom of the spectrum of the semiclassical anharmonic oscillator P-m(h)=-h(2) Delta(m)+ V-m(x) where V-m(x) = mu Sigma(j=1)(m)x(j)(2)+g/(m)n-1)(n), mu is an element of R, g is an element of R(+) and n is an element of N, n>1, when the number m of interacting particles is large. Denoting by lambda(m,h) its lowest eigenvalue, we prove that lim(m-->infinity) lambda(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(beta,m) = integral(Rm) e(-beta Vm(x)) dx where beta is a large parameter and we prove the existence of the limit, as m--> +infinity, of the quantity (1/m)1mI(beta,m) and that this limit has an asymptotic expansion in power of beta(-1) for large values of beta.