Semi-excited states in nondegenerate potential wells. (English) Zbl 0782.35050

Sjöstrand, Johannes

Asymptotic Anal. 6, No. 1, 29-43 (1992).

This article is devoted to the study of the Schrödinger operator \(-h^ 2 \Delta+V(x)\) on \(\mathbb{R}^ n\), where \(V\) is a smooth real-valued potential which has a unique non-degenerate minimum at the origin: \(V(0)=\nabla V(0)=0\), \(\text{Hess }V(0)>0\). Under the condition \(\liminf_{| x|\to\infty} V>0\) the spectrum is discrete near 0 and for any fixed \(\delta>0\) the author gets uniform asymptotic formulae for the eigenvalues up to \(h^ \delta\) when \(h\to 0\). This improves a lot the preceding results of Helffer-Sjöstrand [B. Helffer and J. Sjöstrand, Commun. Partial Differ. Equations 9, 337-408 (1984; Zbl 0546.35053)] or B. Simon [Ann. Inst. Henri Poincaré, Sect. A 38, 295-308 (1983; Zbl 0526.35027)] which correspond to asymptotic formulae for the eigenvalues up to \(Ch\), when \(h\to 0\) (where \(C\) is a fixed constant). The proofs use Birkhoff normal forms and a pseudodifferential calculus for \(h\)-pseudodifferential operators.

Reviewer: B.Helffer (Paris)

MSC:

35P20	Asymptotic distributions of eigenvalues in context of PDEs
35Q40	PDEs in connection with quantum mechanics
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

Schrödinger operator; uniform asymptotic formulas for the eigenvalues

Citations:

Zbl 0546.35053; Zbl 0526.35027