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

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 ${\mathcal H}(Ω) = \left\{u \in L^2(Ω)\,|\,Δu = 0 \; \mbox{in}\;Ω\right\}$, $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$, 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 $T_V\in S_{p,\mathrm{w}}$, 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 $V\ge 0$ 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 $K\ge 0$, self-adjoint in $L^2(\Omega )$; it is known that $\mathrm{Ker}K=\mathcal{H}(\Omega )$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V\in C(\overline{\Omega };\mathbb{R})$, and show that ${\sigma }_{\mathrm{ess}}(K+V)=V(\partial \Omega )$. Assuming that $V\ge 0$ 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 $T_V$.

arXiv comment22 pages, Sections 3 and 4 rearranged, Subsection 2.4 added, Section 4 shortened, and the result of Corollary 5.1 improved