On the semi-classical analysis of Schr\"odinger operators with purely imaginary electric potentials in a bounded domain

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator ${\mathcal{A}}_h=-h^2\mathrm{\Delta }+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$, as $h\to 0$. $V$ is assumed to be a Morse function without critical point at the boundary of $\mathrm{\Omega }$. More precisely, we compare $\inf \mathrm{\Re }\sigma ({\mathcal{A}}_h)$ with the minimum of the spectrum’s real part for some model operator. In the case where $V$ has no critical point, the spectrum is determined by the boundary points where $\mathrm{\nabla }V$ is orthogonal, and the model operator involves a $1$-dimensional complex Airy operator in ${\mathbb{R}}^{+}$. If $V$ is a Morse function with critical points in $\mathrm{\Omega }$, the behavior of the operator near the critical points prevails, and the model operator is a complex harmonic oscillator. This question is related to the decay of associated semigroups. In particular, it allows to recover, in a simplified setting, some stability results by Almog in superconductivity theory.