[Paper Review] The Case of Critical Coupling in a Class of Unbounded Jacobi Matrices Exhibiting a First-Order Phase Transition

David Damanik, Serguei Naboko|arXiv (Cornell University)|May 22, 2006

Spectral Theory in Mathematical Physics10 references3 citations

TL;DR

This paper investigates the spectral transition at critical coupling in a class of unbounded Jacobi matrices with periodically modulated coefficients, focusing on the case where the parameter governing the phase transition is set to zero. Using asymptotic analysis and test function constructions, the authors prove that for $\alpha > 2/3$, the spectrum is purely absolutely continuous on $(-\infty, 0)$, while the positive half-line supports purely discrete spectrum with eigenvalues growing as $n^\alpha$, resolving the open problem of spectral type at the critical point.

ABSTRACT

We consider a class of Jacobi matrices with unbounded coefficients. This class is known to exhibit a first-order phase transition in the sense that, as a parameter is varied, one has purely discrete spectrum below the transition point and purely absolutely continuous spectrum above the transition point. We determine the spectral type and solution asymptotics at the transition point.

Motivation & Objective

• To resolve the open problem of spectral type at the critical coupling point where $\tilde{b} = 0$ in a class of unbounded Jacobi matrices with $\alpha$-power growth coefficients.
• To determine the solution asymptotics of the associated difference equation at the critical point.
• To establish the spectral decomposition—specifically, the coexistence of absolutely continuous and discrete spectra—when $\alpha > 2/3$.
• To provide explicit bounds on the eigenvalues in the discrete spectrum, showing they scale as $n^\alpha$ with positive, finite constants.

Proposed method

• The authors analyze a specific class of Jacobi matrices with $a_n = n^\alpha$, $b_n = b n^\alpha$ for odd $n$, and $b_n = 0$ for even $n$, with $b \neq 0$ and $0 < \alpha \leq 1$.
• They use the method of test functions $f^{(k)}$ supported on odd sites within intervals $J_k$, designed to oscillate between $\pm 1$ to probe spectral properties.
• The key technique involves constructing sequences of test functions to verify the essential spectrum condition via the inequality $\|(J - aI)f\| \leq \varepsilon \|f\|$, ensuring $a$ is in the spectrum.
• The analysis relies on estimating the norm of $Jf - af$ by bounding the contributions from the diagonal $b_n$, off-diagonal $a_n$, and differences in $a_n$-terms using asymptotic expansions.
• The proof uses the fact that $\|f^{(k)}\|^2 \approx \Delta_n / 2$, where $\Delta_n$ is the number of sites in the interval, and controls the error via uniform bounds on $b_n$ and $a_n$ growth.
• The construction is tailored to the critical case $\tilde{b} = 0$, where previous results based on $|d(0)| \neq 2$ do not apply, requiring new analytical tools.

Experimental results

Research questions

• RQ1 What is the spectral type of the Jacobi matrix at the critical coupling point where $\tilde{b} = 0$?
• RQ2 How do the solutions to the associated difference equation behave asymptotically at the critical point?
• RQ3 What is the precise asymptotic behavior of the eigenvalues in the discrete spectrum for $E_n$ as $n \to \infty$?
• RQ4 Can the spectral decomposition be fully characterized when $\alpha > 2/3$, especially the coexistence of absolutely continuous and discrete spectra?
• RQ5 How does the spectral function behave as the parameter $\tilde{b}$ crosses zero, and is there concentration at emerging eigenvalues?

Key findings

• For $2/3 < \alpha \leq 1$, the spectrum of $J$ is purely absolutely continuous on $(-\infty, 0)$, with explicit solution asymptotics derived for all $E \leq 0$.
• For $0 < \alpha \leq 1$, zero is not an eigenvalue of $J$, ruling out a point mass at the origin.
• For $0 < \alpha \leq 1$, the spectrum of $J$ is purely discrete on $(0, \infty)$, with all eigenvalues being simple.
• The $n$-th eigenvalue $E_n$ satisfies the bounds $C_1(b) n^\alpha \leq E_n \leq C_2(b) n^\alpha$, where $C_1(b)$ and $C_2(b)$ are positive and finite constants depending on $b$.
• The proof relies on constructing test functions $f^{(k)}$ supported on odd sites within intervals $J_k$, with $\|f^{(k)}\|^2 \approx \Delta_n / 2$, to verify the essential spectrum condition via norm estimates.
• The method establishes that $a \geq C b^{1-2\alpha} n^\alpha$ with sufficiently large $C$ ensures the required spectral inclusion, valid uniformly for $\alpha \in (0,1]$.