WKB expansions for systems of Schrödinger operators with crossing eigenvalues

Article type: Research Article

Authors: Pettersson, Peter

Affiliations: Department of Mathematics, University of Lund, Box 118, S-221 00 Lund, Sweden

Note: [] This paper is a condensation of the thesis [13] of the author dated 28 May 1993.

Abstract: Let M be a compact, connected C∞ manifold with a C∞ Riemannian metric or let M=Rn with Euclidean metric. Let g denote the metric. On M consider the 2×2 system of Schrödinger operators \[P=-h^{2}\Delta +V+h^{2}\mathcal{W},\qquadV=\left(\begin{array}[cc]v_{1}&0\\0&v_{2}\end{array}\right)\] where v1,v2∈C∞(M) are non-negative, and W is a C∞, first order formally selfadjoint differential operator with real coefficients. We study the eigenfunctions of P corresponding to the lowest eigenvalues in the semi-classical limit h→0. They are concentrated near a minimal geodesic γ with respect to the Agmon metric vg, v = min(v1, v2), connecting two non-degenerate zeros of v (wells). This metric is Lipschitz continuous but not C2 at $\varGamma =\{x\in M;\ v_{1}(x)=v_{2}(x)\}$. When the derivative of v1 - v2 along γ does not vanish on $\gamma \cap \varGamma $ and the intersection is transversal we obtain WKB expansions of the eigenfunctions. At $\gamma \cap \varGamma $ they are expressed in terms of derivatives Yk,ε=∂kYε/∂εk of suitable parabolic cylinder functions Yε. As an application of the WKB constructions we compute the splitting due to tunnelling of the lowest eigenvalues of P under a strong symmetry condition.

DOI: 10.3233/ASY-1997-14101

Journal: Asymptotic Analysis, vol. 14, no. 1, pp. 1-48, 1997