[Paper Review] The Generalized Birman-Schwinger Principle

Jussi Behrndt, Tom Ter Elst+1 more|arXiv (Cornell University)|May 3, 2020

Spectral Theory in Mathematical Physics67 references4 citations

TL;DR

This paper establishes a generalized Birman–Schwinger principle for non-self-adjoint operators, extending the classical framework to include detailed analysis of algebraic and geometric multiplicities and Jordan chains of generalized eigenvectors. The key contribution is a rigorous correspondence between eigenvalues and generalized eigenvectors of a perturbed operator and those of its associated Birman–Schwinger operator, supported by a new Weinstein–Aronszajn formula for non-self-adjoint settings.

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.

Motivation & Objective

• To extend the Birman–Schwinger principle beyond self-adjoint operators to the non-self-adjoint setting.
• To analyze the correspondence between algebraic and geometric multiplicities of eigenvalues for both the original operator and its Birman–Schwinger counterpart.
• To study Jordan chains of generalized eigenvectors in the context of strongly analytic operator-valued functions.
• To relate algebraic multiplicities to the index of meromorphic operator-valued functions.
• To derive a generalized Weinstein–Aronszajn formula for non-self-adjoint operator pairs.

Proposed method

• Formulate the generalized Birman–Schwinger principle via factorization of the perturbation as $ V = V_2^*V_1 $, linking eigenvalue problems of $ H = H_0 + V $ to those of the Birman–Schwinger operator.
• Analyze the structure of Jordan chains and generalized eigenvectors for unbounded non-self-adjoint operators, particularly in the context of Schrödinger operators.
• Study the zeros of strongly analytic operator-valued functions, characterizing their algebraic and geometric multiplicities via Laurent expansions and Fredholm theory.
• Apply the analytic Fredholm theorem and meromorphic Fredholm theorem to operator families, ensuring the existence and structure of resolvent singularities.
• Relate the algebraic multiplicity of eigenvalues to the index of meromorphic operator-valued functions using residue theory.
• Derive a general Weinstein–Aronszajn formula for non-self-adjoint operators by linking spectral data to the index of the associated operator family.

Experimental results

Research questions

• RQ1 How does the Birman–Schwinger principle extend to non-self-adjoint operators, particularly in terms of eigenvalue and generalized eigenvector correspondence?
• RQ2 What is the precise relationship between the algebraic and geometric multiplicities of eigenvalues for a non-self-adjoint Schrödinger operator and its associated Birman–Schwinger operator?
• RQ3 How do Jordan chains of generalized eigenvectors behave under this generalized principle, and how are they related across the two operators?
• RQ4 Can the algebraic multiplicity of a zero of a strongly analytic operator-valued function be characterized via the index of the function?
• RQ5 What is the form of the generalized Weinstein–Aronszajn formula in the non-self-adjoint case, and how does it relate to spectral data and Fredholm theory?

Key findings

• The generalized Birman–Schwinger principle establishes a one-to-one correspondence between Jordan chains of generalized eigenvectors of the perturbed operator $ H = H_0 + V_2^*V_1 $ and those of the associated Birman–Schwinger operator.
• Algebraic and geometric multiplicities of eigenvalues are preserved under the generalized correspondence, with explicit formulas derived for their equality in the non-self-adjoint setting.
• The paper proves that the algebraic multiplicity of a zero of a strongly analytic operator-valued function equals the index of the function at that point, providing a spectral-theoretic interpretation.
• A new Weinstein–Aronszajn formula is derived for non-self-adjoint operators, linking the spectral shift to the index of the associated operator family.
• The meromorphic Fredholm theorem is applied to show that the resolvent of the Birman–Schwinger operator is finitely meromorphic, with poles corresponding to eigenvalues and their multiplicities.
• The analysis confirms that the generalized eigenvector structure, including Jordan chains, is preserved under the generalized Birman–Schwinger correspondence, even in the absence of self-adjointness.