[Paper Review] Discrete spectrum distribution of the Landau Operator Perturbed by an Expanding Electric Potential

Grigori Rozenblum, Alexander V. Sobolev|ArXiv.org|Nov 14, 2007

Quantum Mechanics and Non-Hermitian Physics4 citations

TL;DR

This paper establishes a quasi-classical asymptotic formula for the counting function of discrete eigenvalues of the Landau Hamiltonian perturbed by an expanding electric potential $ V(t^{-1}x) $ as $ t \to \infty $. Using spectral analysis of Toeplitz-type operators and Birman-Schwinger principles, it proves that the rescaled eigenvalue count $ t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) $ converges to a limit involving the measure of level sets of $ V $, matching a canonical semi-classical expression under genericity conditions on $ V $. The result extends spectral asymptotics to non-compactly supported, expanding potentials.

ABSTRACT

Under a perturbation by a decaying electric potential, the Landau Hamiltonian acquires some discrete eigenvalues between the Landau levels. We study the perturbation by an "expanding" electric potential $V(t^{-1}x)$, $t>0$, and derive a quasi-classical formula for the counting function of the discrete spectrum as $t o \infty$.

Motivation & Objective

• To analyze the distribution of discrete eigenvalues of the Landau Hamiltonian under perturbation by an expanding electric potential $ V(t^{-1}x) $ as $ t \to \infty $.
• To derive a quasi-classical formula for the counting function of eigenvalues in gaps between Landau levels.
• To establish sharp upper and lower bounds for the rescaled eigenvalue count $ t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) $ in terms of the potential $ V $.
• To extend spectral asymptotics to non-smooth, $ L^1 \cap L^2 $ potentials using approximation and operator theory techniques.

Proposed method

• The analysis relies on the spectral decomposition of the Landau Hamiltonian into Landau level subspaces and the study of the Toeplitz-type operator $ T^{(t)} = P_q V^{(t)} P_q $, where $ P_q $ is the projection onto the $ q $-th Landau level.
• The eigenvalue asymptotics are analyzed via the Birman-Schwinger principle applied to the operator $ L^{(t)}(V,Z) = H_{0a}^2 + V^{(t)}H_{0a} + H_{0a}V^{(t)} + Z^{(t)} $, with $ Z $ related to $ V^2 $.
• A key step involves estimating the spectral distribution function $ \mathfrak{B}(\eta; L(V,Z)) $ using trace-class bounds and the diamagnetic inequality to control the operator norm.
• The method uses approximation of $ V $ and $ Z $ by smooth, compactly supported functions and applies a limiting argument via Lemma 2.3 to extend results to $ L^1 \cap L^2 $ potentials.
• The proof leverages the fact that the non-zero spectra of $ AB $ and $ BA $ coincide, allowing reduction to the study of $ S^{(t)} = W P_q \overline{W} $ for radial $ W $, which decomposes into one-dimensional operators.
• The final asymptotic formula is derived by relating the counting function $ N(\lambda_1,\lambda_2; H^{(t)}) $ to $ \mathscr{B}(b^2; V, V^2) $, which equals $ \mathscr{A}(\lambda_1,\lambda_2; V) $ under genericity conditions.

Experimental results

Research questions

• RQ1 How does the discrete spectrum of the Landau Hamiltonian behave under a perturbation by an expanding electric potential $ V(t^{-1}x) $ as $ t \to \infty $?
• RQ2 What is the quasi-classical asymptotic formula for the number of eigenvalues in a gap between Landau levels?
• RQ3 Under what conditions does the rescaled eigenvalue count $ t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) $ converge to a limit involving the distribution of $ V $?
• RQ4 How can spectral asymptotics be extended to non-smooth, $ L^1 \cap L^2 $ potentials using operator-theoretic methods?

Key findings

• The rescaled eigenvalue count satisfies the inequality $ \mathscr{A}(\lambda_1,\lambda_2; V) \leq \liminf_{t\to\infty} t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) \leq \limsup_{t\to\infty} t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) \leq \mathscr{A}(\lambda_1 - 0, \lambda_2 + 0; V) $, where $ \mathscr{A} $ is defined via level sets of $ V $.
• Under the condition that $ \lambda_1 - \Lambda_q $ and $ \lambda_2 - \Lambda_q $ are generic values for $ V $ for all $ q $, the limit exists and equals $ \mathscr{A}(\lambda_1, \lambda_2; V) $.
• The limit $ \lim_{t\to\infty} t^{-2}N(\lambda_1,\lambda_2; H^{(t)}) $ coincides with the canonical quasi-classical expression for the magnetic Schrödinger operator.
• The proof relies on the spectral analysis of Toeplitz-type operators $ T^{(t)} = P_q V^{(t)} P_q $, which are reduced to one-dimensional operators via radial symmetry and explicit kernel formulas.
• The method extends results to $ L^1 \cap L^2 $ potentials by approximating them with smooth, compactly supported functions and using the Birman-Schwinger principle with trace-class estimates.
• The final result is established by showing that $ \mathscr{A}(\lambda_1,\lambda_2; V) = \mathscr{B}(b^2; V, V^2) $, and that $ \mathfrak{B}(\eta; L(V,Z)) $ converges to $ \mathscr{B}(\eta; V, Z) $ under approximation.