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The Generalized Birman-Schwinger Principle

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arxiv 2005.01195 v4 pith:7Z6EWLI4 submitted 2020-05-03 math.SP math-phmath.MP

The Generalized Birman-Schwinger Principle

classification math.SP math-phmath.MP
keywords generalizedalgebraicassociatedbirman-schwingermultiplicitiesoperatoranalyticchains
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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 Schr\"odinger 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.

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  1. Birman-Schwinger Formulation of the Faddeev-Popov Zero-Mode Problem

    hep-th 2026-07 conditional novelty 6.0

    The first Gribov horizon of a transverse gauge background equals the first appearance of −1 in the spectrum of a normalized Birman-Schwinger operator, via an inertia-preserving congruence rather than a similarity.