The generalized Birman–Schwinger principle
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- by Jussi Behrndt, A. F. M. ter Elst and Fritz Gesztesy PDF
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Abstract:
We prove a generalized Birman–Schwinger principle in the non-self-adjoint context. In particular, we provide a detailed discussion of geometric and algebraic multiplicities of eigenvalues of the basic operator of interest (e.g., a Schrödinger operator) and the associated Birman–Schwinger operator, and additionally offer a careful study of the associated Jordan chains of generalized eigenvectors of both operators. In the course of our analysis we also study algebraic and geometric multiplicities of zeros of strongly analytic operator-valued functions and the associated Jordan chains of generalized eigenvectors. We also relate algebraic multiplicities to the notion of the index of analytic operator-valued functions and derive a general Weinstein–Aronszajn formula for a pair of non-self-adjoint operators.References
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Additional Information
- Jussi Behrndt
- Affiliation: Institut für Angewandte Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria; and Department of Mathematics, Stanford University, 450 Jane Stanford Way, Stanford, California 94305-2125
- MR Author ID: 760074
- Email: behrndt@tugraz.at
- A. F. M. ter Elst
- Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland 1142, New Zealand
- MR Author ID: 63185
- Email: terelst@math.auckland.ac.nz
- Fritz Gesztesy
- Affiliation: Department of Mathematics, Baylor University, Sid Richardson Bldg., 1410 S. 4th Street, Waco, Texas 76706
- MR Author ID: 72880
- Email: fritz_gesztesy@baylor.edu
- Received by editor(s): May 3, 2020
- Received by editor(s) in revised form: January 3, 2021, and February 5, 2021
- Published electronically: November 29, 2021
- Additional Notes: This work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 799-845
- MSC (2020): Primary 47A53, 47A56; Secondary 47A10, 47B07
- DOI: https://doi.org/10.1090/tran/8401
- MathSciNet review: 4369236