Spectral Properties of Harmonic Toeplitz Operators and Applications to the Perturbed Krein Laplacian

TL;DR

This paper studies the spectral properties of harmonic Toeplitz operators and their applications to the perturbed Krein Laplacian, providing new criteria for Schatten class membership, eigenvalue asymptotics, and spectral distribution analysis.

Contribution

It introduces new conditions for Toeplitz operators to belong to weak Schatten classes, analyzes eigenvalue asymptotics under decay conditions, and links these operators to the spectral behavior of the Krein Laplacian under perturbations.

Findings

Criteria for $T_V$ in $S_{p,w}$ classes.

Eigenvalue asymptotics for radially symmetric $V$.

Spectral distribution of $K o K extpm V$.

Abstract

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(\Omega) \to {\mathcal H}(\Omega)$ where $P: L^2(\Omega) \to {\mathcal H}(\Omega)$ is the orthogonal projection onto ${\mathcal H}(\Omega) = \left\{u \in L^2(\Omega)\,|\,\Delta u = 0 \; \mbox{in}\;\Omega\right\}$, $\Omega \subset {\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\partial \Omega \in C^\infty$, and $V: \Omega \to {\mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V \in S_{p, {\rm w}}$, the weak counterpart of $S_p$. Next, we assume that $\Omega$ is the unit ball in ${\mathbb R}^d$, and $V = \overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $\partial \Omega$ or $V$ is compactly supported in $\Omega$. Further, we consider general $\Omega$ and $V \geq 0$ which is regular in $\Omega$, and admits a power-like decay of rate $\gamma > 0$ at $\partial \Omega$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-\gamma$, self-adjoint in $L^2(\partial \Omega)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K \geq 0$, self-adjoint in $L^2(\Omega)$; it is known that ${\rm Ker}\,K = {\mathcal H}(\Omega)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V \in C(\overline{\Omega};{\mathbb R})$, and show that $\sigma_{\rm ess}(K+V) = V(\partial \Omega)$. Assuming that $V \geq 0$ and $V{|\partial \Omega} = 0$, we study the asymptotic distribution of the eigenvalues of $K \pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.