[Paper Review] On location of discrete spectrum for complex Jacobi matrices

Iryna Egorova, Leonid Golinskiĭ|ArXiv.org|Jun 21, 2004

Spectral Theory in Mathematical Physics4 references3 citations

TL;DR

This paper establishes sufficient conditions for the absence of discrete spectrum in complex Jacobi matrices that are compact perturbations of the discrete Laplacian. By analyzing the Jost function via integral equations and estimating its zeros in the unit disk, the authors derive explicit spectral inclusion regions and prove that the discrete spectrum is absent when the sum of certain coefficient deviations decays sufficiently fast—specifically, when $\sum m d_m < t \approx 0.567$ or $\sum d_m$ is small enough relative to $t$. The key result is a spectral gap condition in terms of the matrix entries.

ABSTRACT

We study spectrum inclusion regions for complex Jacobi matrices which are compact perturbations of the discrete laplacian. The condition sufficient for the lack of discrete spectrum for such matrices is given.

Motivation & Objective

• To determine conditions under which the discrete spectrum of complex Jacobi matrices is absent.
• To analyze the location of the discrete spectrum using the Jost function and its zeros in the unit disk.
• To establish explicit spectral inclusion regions free of discrete spectrum for compact perturbations of the discrete Laplacian.
• To provide sufficient conditions on matrix entries ensuring no eigenvalues exist in certain regions of the complex plane.

Proposed method

• Derive a discrete integral equation for the Jost solution using Green's kernel and recurrence relations.
• Introduce transformed variables $\tilde{v}_n(z) = v_n z^{-n}$ to simplify the integral equation and analyze the kernel $\tilde{J}(n,m;z)$.
• Establish uniform bounds on $\tilde{J}(n,m;z)$ using $|z| \leq 1$ and the decay of coefficient deviations $d_m = |b_m| + |1 - a_{m-1}c_{m-1}|$.
• Use iterative majorization and factorial-type estimates to prove convergence and analyticity of the Jost solution in $\mathbb{D}$.
• Apply the Lambert W function solution $t \approx 0.567$ to derive zero-free regions for the Jost function $v_0(z)$.
• Relate zeros of the Jost function in $\mathbb{D}$ to eigenvalues via $\lambda = z + z^{-1}$, mapping spectral regions to the complex plane.

Experimental results

Research questions

• RQ1 Under what conditions on the matrix entries $a_n, b_n, c_n$ does the discrete spectrum $\sigma_d(J)$ vanish entirely for a complex Jacobi matrix?
• RQ2 How can the location of the discrete spectrum be localized using the Jost function and its zero set in the unit disk?
• RQ3 What decay conditions on the deviations $d_m = |b_m| + |1 - a_{m-1}c_{m-1}|$ ensure that the Jost function has no zeros in a specified region of $\mathbb{D}$?
• RQ4 Can explicit spectral inclusion regions free of discrete spectrum be derived from estimates on the Jost function?

Key findings

• The Jost function $v_0(z)$ is analytic in $\mathbb{D}$ and continuous on $\overline{\mathbb{D}} \setminus \{\pm 1\}$ under $\sum d_m < \infty$, with the bound $|v_n - z^n| \leq |z|^n \left\{ \frac{2|z|}{|z^2 - 1|} \sum_{m=n+1}^\infty d_m \right\} \exp\left\{ \frac{2|z|}{|z^2 - 1|} \sum_{m=n+1}^\infty d_m \right\}$.
• Under $\sum m d_m < \infty$, the Jost function is analytic and continuous on $\overline{\mathbb{D}}$, with the bound $|v_n - z^n| \leq |z|^n \left\{ \sum_{m=n+1}^\infty m d_m \right\} \exp\left\{ \sum_{m=n+1}^\infty m d_m \right\}$.
• The Jost function $v_0(z)$ has no zeros in the domain $\Omega = \{ z \in \mathbb{D} : |z - z^{-1}| > 2t^{-1} \sum d_m \}$, where $t \approx 0.567$ solves $t e^t = 1$.
• If $\sum m d_m < t \approx 0.567$, then $v_0(z)$ has no zeros in $\mathbb{D}$, implying $\sigma_d(J) = \emptyset$.
• The discrete spectrum $\sigma_d(J)$ is contained in $\{ w : \sqrt{4 - c^2} < |\operatorname{Re} w| < \sqrt{4 + c^2}, \ |\operatorname{Im} w| < c^2/4 \}$ with $c = 2t^{-1} \sum d_m < 2$.
• The discrete spectrum $\sigma_d(J)$ is contained in the set $\{ z + z^{-1} : z \in \Omega \}$, where $\Omega$ is the zero-free region of the Jost function.