Schrödinger operators in high dimension: convex potentials

Article type: Research Article

Authors: Sordoni, Vania

Affiliations: CNRS URA 742, Universite Paris-Nord, Institut Galilee, Departement 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. Email: sordoni@dm.unibo.it. Author partially supported by the CNR grant n.203.01.60.

Abstract: In this paper we study the bottom of the spectrum of a semiclassical Schrödinger operator Pm(h)=−(h2/2)Δm+Vm in high dimension m. We assume that Vm 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 Vm reaches its minimum, as h→0.

DOI: 10.3233/ASY-1996-13201

Journal: Asymptotic Analysis, vol. 13, no. 2, pp. 109-129, 1996