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Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. (English) Zbl 1180.35186

Adamyan, Vadim (ed.) et al., Modern analysis and applications. The Mark Krein centenary conference. Volume 2: Differential operators and mechanics. Papers based on invited talks at the international conference on modern analysis and applications, Odessa, Ukraine, April 9–14, 2007. Basel: Birkhäuser (ISBN 978-3-7643-9920-7/v. 2; 978-3-7643-9921-4/ebook; 978-3-7643-9924-5/set). Operator Theory: Advances and Applications 191, 81-113 (2009).
During the last ten years Gesztesy and collaborators published a number of papers on boundary value problems for differential operators, often Schrödinger operators on bounded domains \(\Omega\subset\mathbb{R}^n\), and discussed resolvent formulas. The present article continues the stream of interesting results though there is a slight change of direction: Schrödinger operators \(-\Delta +V\) with \(V\in L^\infty(\Omega, d^nx)\) are investigated assuming generalized Robin and Dirichlet boundary conditions under certain smoothness assumption on the domain \(\Omega\). As helpful tools the authors define Robin-to-Dirichlet and Dirichlet-to-Neumann maps.
The main new result is a derivation of Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains with two different generalized Robin boundary conditions. Potentials need not necessarily be real-valued. Also, the results extend naturally to potentials that are bounded in the Kato-Rellich sense with constant less than one. The list of 83 references demonstrates the huge amount of work that has gone into this field.
For the entire collection see [Zbl 1169.47002].

MSC:

35J10 Schrödinger operator, Schrödinger equation
35J25 Boundary value problems for second-order elliptic equations
35Q40 PDEs in connection with quantum mechanics
35P05 General topics in linear spectral theory for PDEs
47A10 Spectrum, resolvent
47F05 General theory of partial differential operators
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