Elementary linear algebra for advanced spectral problems

TL;DR

The paper presents Grushin problems as a versatile linear-algebraic tool to replace a (potentially non-invertible) operator by a well-posed invertible 2×2 system, with the effective Hamiltonian $E_{-+}$ encoding spectral information and invertibility. It develops general techniques for constructing, transforming, and iterating Grushin problems, and derives trace formulæ that connect quantum spectra to classical dynamics, including a rigorous derivation of the Poisson summation formula. The authors illustrate the method through simple algebraic examples (Moore–Penrose pseudoinverse, non-self-adjoint eigenvalues, Feshbach method, analytic Fredholm theory, boundary value problems) and extend it to advanced applications such as Lidskii perturbation theory, generalized Gutzwiller trace formulas, Peierls substitution under magnetic fields, and high-frequency scattering by convex obstacles. The framework thus provides both conceptual and computational tools for spectral theory, semiclassical analysis, and related numerical problems, with broad relevance to mathematical physics and PDEs.

Abstract

We discuss the general method of Grushin problems, closely related to Shur complements, Feshbach projections and effective Hamiltonians, and describe various appearances in spectral theory, pdes, mathematical physics and numerical problems.

Figures (2)

• Figure 1: Eigenvalues of a small random perturbation of a $200\times200$ Jordan block matrix (blue), and of the perturbation $Q = \epsilon e_- \otimes e_+$ (red)
• Figure 2: The distribution of resonances for a convex obstacle satisfying the pinched curvature assumption \ref{['eq:pinch']} with $j_0 = 1$.

Theorems & Definitions (33)

• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• Lemma 3.1
• proof