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Eigenvalues of non-selfadjoint functional difference operators

Alexei Ilyin, Ari Laptev, Lukas Schimmer, Anna Zernova

TL;DR

The paper extends Davies–type methods to non-selfadjoint functional-difference operators W_V(b)=W_0(b)−V with complex potentials, deriving a sharp Lieb–Thirring–type bound that constrains complex eigenvalues via |\sin(ω)/ω| ≤ (1/(2π b))∫|V|. It develops the free resolvent kernel G_λ and crucial bounds, enabling a Birman–Schwinger analysis that yields the main inequality and its sharpness, demonstrated by a rank-one delta-potential. The work also clarifies resonance phenomena: complex potentials can produce resonances, and explicit examples with rank-one perturbations illustrate the eigenvalue–resonance structure and parameter dependence. Overall, the results provide precise spectral location information for a class of non-selfadjoint functional-difference operators with potential applications to lattice quantum models and related spectral problems.

Abstract

Using the well known approach developed in the papers of B. Davies and his co-authors we obtain inequalities for the location of possible complex eigenvalues of non-selfadjoint functional difference operators. When studying the sharpness of the main result we discovered that complex potentials can create resonances.

Eigenvalues of non-selfadjoint functional difference operators

TL;DR

The paper extends Davies–type methods to non-selfadjoint functional-difference operators W_V(b)=W_0(b)−V with complex potentials, deriving a sharp Lieb–Thirring–type bound that constrains complex eigenvalues via |\sin(ω)/ω| ≤ (1/(2π b))∫|V|. It develops the free resolvent kernel G_λ and crucial bounds, enabling a Birman–Schwinger analysis that yields the main inequality and its sharpness, demonstrated by a rank-one delta-potential. The work also clarifies resonance phenomena: complex potentials can produce resonances, and explicit examples with rank-one perturbations illustrate the eigenvalue–resonance structure and parameter dependence. Overall, the results provide precise spectral location information for a class of non-selfadjoint functional-difference operators with potential applications to lattice quantum models and related spectral problems.

Abstract

Using the well known approach developed in the papers of B. Davies and his co-authors we obtain inequalities for the location of possible complex eigenvalues of non-selfadjoint functional difference operators. When studying the sharpness of the main result we discovered that complex potentials can create resonances.

Paper Structure

This paper contains 5 sections, 3 theorems, 45 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Resonance state
  3. Free resolvent
  4. Proof of Theorem \ref{['1']}
  5. Examples

Key Result

Theorem 1.1

Let $V\in L^1(\Bbb R)$ be a complex-valued potential. Then the eigenvalues $\lambda\in \Bbb C\setminus [2,\infty)$ of the operator $W_V(b)$ satisfy the inequality where $\lambda = - 2\cos(\omega)$ and where $\omega \in \Omega$. The constant in this inequality is sharp in the sense that there are potentials $V$ such that inequality main becomes an equality.

Figures (3)

  • Figure 1: The solutions $\omega$ and $-2\cos\omega$ for $r/2\pi=2$.
  • Figure 2: The solutions $\omega$ and $-2\cos\omega$ for $r/2\pi=0.25$.
  • Figure 3: The solutions $\omega$ and $-2\cos\omega$ for $r/2\pi=0.2$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Proposition 3.1
  • proof