[Paper Review] On Perturbations of Quasiperiodic Schroedinger Operators

Helge Krueger|ArXiv.org|Nov 12, 2007

Spectral Theory in Mathematical Physics8 references3 citations

TL;DR

This paper establishes relative oscillation criteria and eigenvalue asymptotics for perturbations of quasiperiodic Schrödinger operators using Eliasson's reducibility result and relative oscillation theory. It extends classical Weyl and Kneser-type results to quasiperiodic potentials, deriving precise conditions under which eigenvalues accumulate at spectral edges and quantifying their rate of convergence, particularly for critical potentials like $ \mu/x^2 $. The key contribution is a spectral threshold condition involving the Prüfer angle and potential perturbations, generalizing known results from the periodic and decaying cases to the quasiperiodic setting.

ABSTRACT

Using relative oscillation theory and the reducibility result of Eliasson, we study perturbations of quasiperiodic Schroedinger operators. In particular, we derive relative oscillation criteria and eigenvalue asymptotics for critical potentials.

Motivation & Objective

• To generalize classical perturbation results on eigenvalue accumulation and asymptotics to quasiperiodic Schrödinger operators.
• To determine conditions under which eigenvalues accumulate at spectral edges of quasiperiodic operators, particularly for critical potentials.
• To derive precise eigenvalue asymptotics for perturbations near spectral boundaries using relative oscillation theory.
• To extend Rofe-Beketov's gap-multiplicity result to the quasiperiodic case, analyzing how many spectral gaps can contain infinitely many eigenvalues under critical perturbations.

Proposed method

• Uses Eliasson's reducibility theorem to transform the Schrödinger equation into a linear system with a nilpotent matrix exponent, enabling control over solution behavior near spectral edges.
• Applies relative oscillation theory via Prüfer angles to compare solutions of perturbed and unperturbed operators, introducing a weighted oscillation count.
• Derives a differential equation for the Prüfer angle of the Wronskian of two solutions, which governs relative oscillation behavior.
• Employs averaging techniques for ordinary differential equations to analyze long-term behavior of oscillation phases under slowly varying weights.
• Introduces a criterion based on the limsup/liminf of time-averaged integrals involving the perturbation $ \Delta q $, the fundamental solution $ u_0 $, and weight functions $ \alpha, \beta $, to determine relative oscillation.
• Applies the theory to quasiperiodic potentials $ q_0(x) = Q(\omega x) $ with Diophantine frequency $ \omega $, leveraging the rotation number and spectral properties from Johnson-Moser theory.

Experimental results

Research questions

• RQ1 Under what conditions does the essential spectrum of a quasiperiodic Schrödinger operator remain purely absolutely continuous under perturbations?
• RQ2 When does the eigenvalue sequence accumulate at a spectral edge for critical potentials such as $ \mu/x^2 $, and how fast does it do so?
• RQ3 How many spectral gaps can contain infinitely many eigenvalues under a critical perturbation in the quasiperiodic setting?
• RQ4 What is the precise relative oscillation criterion for perturbations of quasiperiodic operators, and how does it generalize the classical $ -1/4 $ threshold?
• RQ5 How do eigenvalue asymptotics for $ -d^2/dx^2 + \mu/x^2 $ extend from the half-line to the quasiperiodic case?

Key findings

• For quasiperiodic operators with Diophantine frequency, the essential spectrum is purely absolutely continuous, and the rotation number determines the spectrum via $ \rho(E) = \frac{1}{2}\langle \omega, n \rangle $.
• A relative oscillation criterion is established: eigenvalues accumulate at a spectral edge $ E $ if $ \limsup_{x\to\infty} \frac{1}{\ell} \int_x^{x+\ell} \frac{\beta(t)^2}{\beta'(t)} u_0(t)^2 \Delta q(t) dt < -\frac{1}{4} $, and do not accumulate if the sup is greater than $ -\frac{1}{4} $.
• For the critical potential $ \mu/x^2 $, the number of eigenvalues below $ \lambda \uparrow 0 $ satisfies $ N(\lambda) = \frac{1}{4\pi} \sqrt{\frac{\mu}{\mu_{\text{crit}}} - 1} |\ln|\lambda|| (1+o(1)) $, with $ \mu_{\text{crit}} = -\frac{1}{4} $, matching known half-line results.
• The paper shows that only finitely many spectral gaps can contain infinitely many eigenvalues under a critical perturbation $ \Delta q = \mu/x^2 $, extending Rofe-Beketov's periodic result to quasiperiodic operators.
• Eigenvalue asymptotics for $ -d^2/dx^2 + \mu/x^\gamma $ with $ 0 < \gamma < 2 $ are derived as $ N(\lambda) \sim \frac{\sqrt{\mu/\mu_{\text{crit}}}}{\pi(2-\gamma)} \left| \frac{\mu}{\lambda} \right|^{(2-\gamma)/(2\gamma)} $ as $ \lambda \uparrow 0 $, confirming the expected power-law scaling.
• The method successfully generalizes classical results from the decaying and periodic cases to quasiperiodic potentials by combining reducibility, oscillation theory, and averaging techniques for ODEs.