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

We consider the non-selfadjoint, semiclassical Schrödinger operator \mathscr{L}(h) := -h^{2}\partial_{x}^{2}+e^{i\alpha}V , where \alpha \in (-\pi,\pi) and V\colon \R\to \R_{+} is even and vanishes at exactly two (symmetric) non-degenerate minima. We establish a semiclassical tunneling result: the spectrum of \mathscr{L}(h) near the origin is given by a sequence of algebraically simple eigenvalues which come in exponentially close pairs (within a \mathscr{O}(e^{-S/h}) distance where S > 0 is explicit), each pair being separated from the others by a distance \mathscr{O}(h) . A one-term estimate of the gap between the two smallest eigenvalues in magnitude is derived; it reveals that, when \alpha \neq 0 , they quickly rotate around each other as h goes to 0 .