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.
