Affiliations: CNRS URA 742, Universite Paris-Nord, Institut Galilee, Department de Mathematiques, Av. J.B. Clement - 93430, Villetaneuse, France
Note: [] Permanent address: Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40127 Bologna, Italy. E-mail: [email protected]. Author partially supported by the CNR grant n.203.01.60.
Abstract: In this paper we study the bottom of the spectrum of the semiclassical anharmonic oscillator Pm(h)=−h2Δm+Vm(x) where Vm(x)=μΣj=1mx2j+((g)/(mn−1))(Σj=1mxj2)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 Pm. Moreover, we consider integrals of the form I(β,m)=∫Rm=e−βVm(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.
