Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials

We consider the non-selfadjoint, semiclassical Schrödinger operator L(h) := -h²∂ₓ²+eiαV , where α ∈ (-π,π) and V → ₊ is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of L(h) near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a O(e-S/h) distance where S > 0 is explicit), each pair being separated from the others by a distance O(h) . A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when α ≠ 0 , they quickly rotate around each other as h goes to 0 .