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

Abstract

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