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On the singularities of the spectral shift function for some tight-binding models

Marouane Assal, Olivier Bourget, Diomba Sambou, Amal Taarabt

TL;DR

The paper analyzes the spectral shift function for perturbed discrete tight-binding models on $\ell^2(\mathbb{Z}_h,\mathcal{G})$, where the perturbations are self-adjoint matrix-valued potentials with polynomial decay. It develops a rigorous SSF framework via Pushnitski’s representation, distinguishing finite- and infinite-dimensional internal spaces $\mathcal{G}$ and showing bounded SSF behavior away from thresholds or possible singularities at thresholds, respectively. The authors identify the leading threshold behavior through explicit Berezin–Toeplitz type operators and derive comprehensive threshold asymptotics, Levinson-type formulas, and eigenvalue-counting results for both power-like and exponential decay potentials. The results extend analogous continuous-model phenomena to the discrete setting, providing precise mechanisms for spectral accumulation and sharp asymptotics that are relevant for tight-binding and related quantum lattice systems.

Abstract

We consider perturbed discrete tight-binding models in $\ell^2(\mathbb{Z_h},\mathcal{G})$ describing union of quantum particles with localized interactions, where $\mathbb{Z_h}$ is the 1D lattice $h\mathbb{Z_h}$, $h > 0$, and $\mathcal G$ is a separable Hilbert space. The perturbations play the role of self-adjoint relatively compact (matrix-valued) electric potentials with $\mathcal B(\mathcal G)$-valued coefficients decaying polynomially at infinity. We analyze the Spectral Shift Function (SSF) associated to the pair of the perturbed and the unperturbed operators. On the one hand, we show that the SSF is bounded near the spectral thresholds of the essential spectrum if $\dim(\mathcal G) < +\infty$. On the other hand, if $\dim(\mathcal G) = +\infty$, we show that it may have singularities at some thresholds points $μ$ of the essential spectrum. In particular, new mechanisms allowing the SSF to have singularities at the thresholds are exhibited, based on the degeneracy of the spectrum of the unperturbed operator. Moreover, we give the main terms of the asymptotic behaviors of the SSF near $μ$ described in terms of some explicit effective Berezin-Toeplitz type operators. These results are completed by Levinson type formulas and examples of eigenvalues asymptotics for power-like and exponential decay potentials.

On the singularities of the spectral shift function for some tight-binding models

TL;DR

The paper analyzes the spectral shift function for perturbed discrete tight-binding models on , where the perturbations are self-adjoint matrix-valued potentials with polynomial decay. It develops a rigorous SSF framework via Pushnitski’s representation, distinguishing finite- and infinite-dimensional internal spaces and showing bounded SSF behavior away from thresholds or possible singularities at thresholds, respectively. The authors identify the leading threshold behavior through explicit Berezin–Toeplitz type operators and derive comprehensive threshold asymptotics, Levinson-type formulas, and eigenvalue-counting results for both power-like and exponential decay potentials. The results extend analogous continuous-model phenomena to the discrete setting, providing precise mechanisms for spectral accumulation and sharp asymptotics that are relevant for tight-binding and related quantum lattice systems.

Abstract

We consider perturbed discrete tight-binding models in describing union of quantum particles with localized interactions, where is the 1D lattice , , and is a separable Hilbert space. The perturbations play the role of self-adjoint relatively compact (matrix-valued) electric potentials with -valued coefficients decaying polynomially at infinity. We analyze the Spectral Shift Function (SSF) associated to the pair of the perturbed and the unperturbed operators. On the one hand, we show that the SSF is bounded near the spectral thresholds of the essential spectrum if . On the other hand, if , we show that it may have singularities at some thresholds points of the essential spectrum. In particular, new mechanisms allowing the SSF to have singularities at the thresholds are exhibited, based on the degeneracy of the spectrum of the unperturbed operator. Moreover, we give the main terms of the asymptotic behaviors of the SSF near described in terms of some explicit effective Berezin-Toeplitz type operators. These results are completed by Levinson type formulas and examples of eigenvalues asymptotics for power-like and exponential decay potentials.
Paper Structure (28 sections, 35 theorems, 258 equations, 1 figure)

This paper contains 28 sections, 35 theorems, 258 equations, 1 figure.

Table of Contents

  1. Introduction
  2. General setting and motivations
  3. Description of the main results
  4. Comments on the literature
  5. Spectral properties of the hamiltonian $H_0$
  6. The electric potentials
  7. Representation of the spectral shift function
  8. Main results
  9. Statement of the main results
  10. The case $\lambda \nearrow \mu_s$ and $\lambda \searrow \frac{4}{h^2}+\mu_s$, $0 \le s \le d$
  11. The case $\lambda \searrow \mu_s$ and $\lambda \nearrow \frac{4}{h^2}+\mu_s$, $0 \le s \le d$
  12. Corollaries
  13. Boundedness of the SSF at the spectral thresholds
  14. Thresholds singularities and asymptotic behaviors of the SSF
  15. Levinson type formulas
  16. ...and 13 more sections

Key Result

Theorem 4.1

Let Assumption eq:hyppert holds. Then, for a.e. $\lambda \in \mathbb R$, the SSF $\xi(\cdot;H^\pm,H_Q)$ admits the representation via the converging integral

Figures (1)

  • Figure 1: Illustration of the Hubbard model.

Theorems & Definitions (65)

  • Definition 1.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.5
  • Theorem 4.1
  • Lemma 4.2: Lemma 2.1 of pus
  • Theorem 5.1
  • Definition 5.2
  • Theorem 5.3
  • Remark 5.4
  • ...and 55 more