Operator theoretic methods for the eigenvalue counting function in spectral gaps

Abstract

Using the notion of spectral flow, we suggest a simple approach to various asymptotic problems involving eigenvalues in the gaps of the essential spectrum of self-adjoint operators. Our approach uses some elements of the spectral shift function theory. Using this approach, we provide generalisations and streamlined proofs of two results in this area already existing in the literature. We also give a new proof of the generalised Birman-Schwinger principle.

Latex, 25 pages

References in corpus (5)

• On the eigenvalues of operators with gaps. Application to Dirac operators
• Eigenvalue bounds in the gaps of Schrodinger operators and Jacobi matrices
• The Birman-Schwinger principle in von Neumann algebras of finite type
• Discrete spectrum distribution of the Landau Operator Perturbed by an Expanding Electric Potential
• Operator integrals, spectral shift and spectral flow

Cited by in corpus (8)

• A variational approach to dislocation problems for periodic Schrödinger operators
• Critical Lieb-Thirring Bounds in Gaps and the Generalized Nevai Conjecture for Finite Gap Jacobi Matrices
• On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign
• Spectral shift via "lateral" perturbation
• The index formula and the spectral shift function for relatively trace class perturbations
• An integer-valued version of the Birman-Krein formula
• The Birman-Schwinger principle on the essential spectrum
• Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator