On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator $\mathcal{A}_h = -h^2Δ+iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $Ω\subset\mathbb{R}^n\,$, as $h\rightarrow0\,$. $V$ is assumed to be a Morse function without critical point at the boundary of $Ω\,$. More precisely, we compare $\inf\Reσ(\mathcal{A}_h)$ 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 $\nabla V$ 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 $Ω\,$, 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.

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