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

Tid: Må 2021-05-31 kl 13.30 - 14.30

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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Tillhör: Institutionen för matematik