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Aaron Sümnick: Accumulation of Complex Eigenvalues of Schrödinger-Operators with Non-Real Potentials

Time: Mon 2021-05-31 13.30 - 14.30

Location: Meeting ID: 640 6342 1581

Respondent: Aaron Sümnick

Abstract

In this thesis we consider Schrödingeroperators \(-\Delta +V\) with complex valued potentials \(V\) on both the entire euclidian space \(\mathbb{R}^d\) and \(\mathbb{Z}^d\)-periodic metric graphs \(\Gamma\). We construct like in [Bö I] for \(p>d\) a potential \(V\in L^\infty(\mathbb{R}^d)\cap L^p(\mathbb{R}^d)\), decaying at infinity such that the non-real eigenvalues of the operator \(-\Delta+V\) accumulate at all points of \([0,\infty)\). Moreover for certain periodic graphs \(\Gamma\) whose Kirchhoff-Laplacian \(\mathcal{H}\) has non-empty point spectrum we construct for \(p\geq 1\) potentials \(V\in L^\infty(\Gamma)\cap L^p(\Gamma)\), decaying at infinity such that the non-real eigenvalues of \(\mathcal{H} +V\) accumulate at eigenvalues of \(\mathcal{H}\).

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