[Paper Review] Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain

Alexander Pushnitski, Grigori Rozenblum|ArXiv.org|Jul 29, 2007

Spectral Theory in Mathematical Physics10 references4 citations

TL;DR

This paper investigates the spectral clustering of eigenvalues for the Schrödinger operator with a constant magnetic field in the exterior of a compact domain on the plane. It establishes that eigenvalues accumulate to Landau levels at a rate governed by the logarithmic capacity of the obstacle, providing precise asymptotic estimates for the eigenvalue gaps using operator-theoretic and potential-theoretic methods.

ABSTRACT

We consider the Schrodinger operator with a constant magnetic field in the exterior of a compact domain on the plane. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We discuss the rate of accumulation of eigenvalues in a fixed cluster.

Motivation & Objective

• To analyze the rate of eigenvalue accumulation in spectral clusters around Landau levels for the Schrödinger operator with a constant magnetic field in the exterior of a compact domain.
• To characterize the asymptotic distribution of eigenvalues in a fixed cluster using the logarithmic capacity of the obstacle.
• To extend spectral results from the whole plane to exterior domains by incorporating geometric and potential-theoretic invariants.
• To establish sharp upper and lower bounds on the rate of convergence of eigenvalues to the Landau levels using operator-theoretic techniques.
• To connect spectral behavior to the geometric capacity of the obstacle, particularly through the use of inner capacity and polynomial convex hulls.

Proposed method

• The authors use the magnetic Schrödinger operator defined on the exterior of a compact set $K \subset \mathbb{R}^2$, with Dirichlet boundary conditions on $\partial K$.
• They analyze the spectrum of $X(K^c)$, showing that eigenvalues accumulate only from above to the Landau levels $\Lambda_q = (2q+1)B$.
• The rate of accumulation is quantified via the spectral characteristics $\Delta_q(K) = \limsup_{n\to\infty} [n!(\lambda_n^q - \Lambda_q)]^{1/n}$ and $\delta_q(K) = \liminf_{n\to\infty} [n!(\lambda_n^q - \Lambda_q)]^{1/n}$.
• The key tool is the logarithmic capacity $\operatorname{Cap}(K)$, with the main result showing $\Delta_q(K)$ and $\delta_q(K)$ are bounded in terms of $\operatorname{Cap}(K)$ and $\operatorname{Cap}_-(K)$.
• Operator-theoretic constructions from previous works on perturbed Landau Hamiltonians are adapted, particularly involving the resolvent difference $X_0^{-1} - R(K^c)$.
• The proof uses a partition of unity and estimates on the $H^1_A$-norm, combined with compactness arguments and the use of the $\mathfrak{Q}^*$ operator to control the growth of eigenvalues.

Experimental results

Research questions

• RQ1 How does the geometry of the obstacle $K$ influence the rate of eigenvalue accumulation in spectral clusters of the Landau Hamiltonian in the exterior domain?
• RQ2 What is the precise asymptotic behavior of the eigenvalue gaps $\lambda_n^q - \Lambda_q$ as $n \to \infty$ for a fixed Landau level $\Lambda_q$?
• RQ3 Can the logarithmic capacity of the obstacle be used to characterize the spectral clustering rate in terms of $\Delta_q(K)$ and $\delta_q(K)$?
• RQ4 How do the spectral characteristics $\Delta_q(K)$ and $\delta_q(K)$ relate to the geometric properties of $K$, such as its inner capacity and polynomial convex hull?
• RQ5 What role does the operator-theoretic structure of the resolvent difference $X_0^{-1} - R(K^c)$ play in deriving bounds on eigenvalue accumulation?

Key findings

• The eigenvalues $\lambda_n^q$ of the exterior Landau Hamiltonian accumulate to the Landau level $\Lambda_q$ from above, with $\lambda_n^q \to \Lambda_q$ as $n \to \infty$.
• The rate of accumulation is quantified by $\Delta_q(K) = \limsup_{n\to\infty} [n!(\lambda_n^q - \Lambda_q)]^{1/n} \leq \frac{1}{2}B(\operatorname{Cap}(K))^2$.
• A lower bound is established: $\delta_q(K) \geq \frac{1}{2}B(\operatorname{Cap}_-(K))^2$, where $\operatorname{Cap}_-(K)$ is the inner capacity of $K$.
• For a smooth obstacle $K$, the bounds are sharp in the sense that $\Delta_q(K)$ and $\delta_q(K)$ are determined by the logarithmic capacity of $K$.
• The results show that the spectral clustering rate is directly governed by the logarithmic capacity, linking spectral theory to potential theory.
• The asymptotic behavior $\lambda_n^q - \Lambda_q \sim \frac{c^n}{n!}$ is confirmed, with $c$ related to the capacity of the obstacle, providing a precise exponential-type decay rate for the eigenvalue gaps.