@article{2e6dd0fe105944f29727b29643831737,

title = "THE GENERALIZED BIRMAN-SCHWINGER PRINCIPLE",

abstract = "We prove a generalized Birman-Schwinger principle in the nonself- 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 Schr{\"o}dinger 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 operatorvalued 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.",

keywords = "Algebraic and geometric multiplicities, Birman-Schwinger principle, Jordan chains, The index of meromorphic operator-valued functions, The Weinstein-Aronszajn formula",

author = "Jussi Behrndt and {Ter Elst}, {A. F.M.} and Fritz Gesztesy",

note = "Funding Information: We are indebted to Rupert Frank, Yuri Latushkin, and Alim Sukhtayev for very helpful discussions. We are also grateful to the anonymous referee for an exceptionally careful reading of our manuscript and for many helpful suggestions. J.B. gratefully acknowledges support for the Distinguished Visiting Austrian Chair at Stanford University by the Europe Center and the Freeman Spogli Institute for International Studies, where this work was completed in the spring of 2020. J.B. is also most grateful for the stimulating research stay and the hospitality at the University of Auckland, where some parts of this paper were written. F.G. gratefully acknowledges kind invitations to the Institute for Applied Mathematics at the Graz University of Technology, Austria, for parts of June 2018 and June 2019. The extraordinary hospitality as well as the stimulating atmosphere at the Graz University of Technology is greatly appreciated. Funding Information: Received by the editors May 3, 2020, and, in revised form, January 3, 2021, and February 5, 2021. 2020 Mathematics Subject Classification. Primary: 47A53, 47A56 Secondary: 47A10, 47B07. Key words and phrases. Birman–Schwinger principle, Jordan chains, algebraic and geometric multiplicities, the index of meromorphic operator-valued functions, the Weinstein–Aronszajn formula. This work is supported by the Marsden Fund Council from Government funding, administered by the Royal Society of New Zealand. Publisher Copyright: {\textcopyright} 2022 American Mathematical Society. All rights reserved.",

year = "2022",

doi = "10.1090/tran/8401",

language = "English",

volume = "375",

pages = "799--845",

journal = "Transactions of the American Mathematical Society",

issn = "0002-9947",

publisher = "American Mathematical Society",

number = "2",

}