WKB expansions for systems of Schrödinger operators with crossing eigenvalues

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 Y_k,ε=∂^kY_ε/∂ε^k of suitable parabolic cylinder functions Y_ε. 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.