[Paper Review] On the spectrum of Bargmann-Toeplitz operators with symbols of a variable sign

Alexander Pushnitski, Grigori Rozenblum|ArXiv.org|Dec 22, 2009

Spectral Theory in Mathematical Physics15 references7 citations

TL;DR

This paper investigates the spectral asymptotics of Bargmann-Toeplitz operators with real-valued, compactly supported symbols of variable sign. It establishes that the eigenvalue counting function for such operators exhibits a logarithmic-logarithmic decay rate, revealing that the interplay between positive and negative parts of the symbol leads to a non-trivial, geometry-dependent eigenvalue distribution, with applications to the perturbed Landau Hamiltonian showing two infinite sequences of eigenvalues accumulating at zero.

ABSTRACT

The paper discusses the spectrum of Toeplitz operators in Bargmann spaces. Our Toeplitz operators have real symbols with a variable sign and a compact support. A class of examples is considered where the asymptotics of the eigenvalues of such operators can be computed. These examples show that this asymptotics depends on the geometry of the supports of the positive and negative parts of the symbol. Applications to the perturbed Landau Hamiltonian are given.

Motivation & Objective

• To analyze the asymptotic distribution of eigenvalues for Bargmann-Toeplitz operators with compactly supported, real-valued symbols of variable sign.
• To understand how the geometry of the supports of the positive and negative parts of the symbol influences the spectral behavior.
• To establish precise asymptotic estimates for the eigenvalue counting function in the case of compactly supported symbols, contrasting with the non-compact case.
• To apply the results to the spectral analysis of the perturbed Landau Hamiltonian, particularly the splitting of the lowest Landau level.

Proposed method

• The authors use variational principles and min-max techniques to derive upper and lower bounds on the eigenvalue counting function.
• They employ spectral shift function theory and the monotonicity properties of the spectral shift function to compare the perturbed operator with auxiliary operators involving positive and negative parts of the symbol.
• The analysis relies on unitary equivalence between the restriction $ P_0 V P_0 $ and the Toeplitz operator $ T_V $, enabling transfer of spectral results.
• The method involves constructing regularized versions $ V_ ho^ ho $ of the symbol to ensure boundedness and separation from zero, facilitating application of known spectral asymptotics.
• The key technical tool is the eigenvalue counting function $ N(( ho, ho); T_V) $, whose asymptotic behavior is linked to the logarithmic-logarithmic decay rate.
• The authors use the spectral shift function identity $ \Xi(\lambda; H, H_0) = N((\lambda, \infty); H) - N((\lambda, \infty); H_0) $ to relate the perturbation to the counting function.

Experimental results

Research questions

• RQ1 How does the spectral asymptotics of Bargmann-Toeplitz operators with compactly supported, variable-sign symbols differ from the non-negative symbol case?
• RQ2 What role does the geometric configuration of the supports of $ V_+ $ and $ V_- $ play in determining the eigenvalue distribution?
• RQ3 Can the eigenvalue counting function for such operators be asymptotically characterized, and if so, in what form?
• RQ4 How does the spectral behavior of the perturbed Landau Hamiltonian relate to the spectral properties of the associated Toeplitz operator?
• RQ5 What is the precise asymptotic behavior of the number of eigenvalues accumulating at zero for operators with compactly supported, sign-changing symbols?

Key findings

• For compactly supported, sign-changing symbols, the eigenvalue counting function satisfies $ N(( ho, ho); T_V) = \frac{|\log \rho|}{\log |\log \rho|}(1+o(1)) $ as $ \rho \to 0^+ $, indicating super-exponential decay of eigenvalues.
• The asymptotic behavior is not determined solely by the positive or negative parts of the symbol, unlike in the non-compact case, due to lack of asymptotic orthogonality.
• The number of eigenvalues in $ (-\infty, -\lambda) \cup (\lambda, a) $ for the perturbed Landau Hamiltonian satisfies $ N((- ho, -\lambda); H) + N((\lambda, a); H) = \frac{|\log \lambda|}{\log |\log \lambda|}(1+o(1)) $ as $ \lambda \to 0^+ $, for any $ a \in (0,2) $.
• Both the positive and negative parts of the spectrum contribute infinitely many eigenvalues accumulating at zero, with each sequence satisfying $ N((\lambda, \infty); \pm T_V) \geq c \frac{|\log \lambda|}{\log |\log \lambda|} $ for some $ c > 0 $.
• The results are derived via spectral shift function techniques and comparison with auxiliary operators $ H_\epsilon^\pm $, which are perturbations of the Landau Hamiltonian by regularized versions of the symbol.
• The key insight is that the spectral asymptotics depend critically on the geometry of the supports of $ V_+ $ and $ V_- $, not just their size or regularity.