Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian

Abstract

We consider harmonic Toeplitz operators ${\mathit{T}}_{\mathit{V}}=\mathit{P}\mathit{V}:\mathcal{H}\mathrm{(}\mathrm{\Omega }\mathrm{)}\to \mathcal{H}\mathrm{(}\mathrm{\Omega }\mathrm{)}$ where $\mathit{P}:{\mathit{L}}^2\mathrm{(}\mathrm{\Omega }\mathrm{)}\to \mathcal{H}\mathrm{(}\mathrm{\Omega }\mathrm{)}$ is the orthogonal projection onto $\mathcal{H}\mathrm{(}\mathrm{\Omega }\mathrm{)}=\{\mathit{u}\in {\mathit{L}}^2\mathrm{(}\mathrm{\Omega }\mathrm{)}\mid \mathrm{\Delta }\mathit{u}=0\text{in }\mathrm{\Omega }\}$ , $\mathrm{\Omega }\subset {\mathbb{R}}^{\mathit{d}}$ , $\mathit{d}⩾2$ , is a bounded domain with boundary $\partial \mathrm{\Omega }\in {\mathit{C}}^{\infty }$ , and $\mathit{V}:\mathrm{\Omega }\to \mathbb{C}$ is an appropriate multiplier. First, we complement the known criteria which guarantee that ${\mathit{T}}_{\mathit{V}}$ is in the pth Schatten–von Neumann class ${\mathit{S}}_{\mathit{p}}$ , by simple sufficient conditions which imply ${\mathit{T}}_{\mathit{V}}\in {\mathit{S}}_{\mathit{p}\mathrm{,}\mathrm{w}}$ , the weak counterpart of ${\mathit{S}}_{\mathit{p}}$ . Next, we consider symbols $\mathit{V}⩾0$ which have a regular power-like decay of rate $\mathit{\gamma }\mathrm{>}0$ at $\partial \mathrm{\Omega }$ , and we show that ${\mathit{T}}_{\mathit{V}}$ is unitarily equivalent to a classical pseudo-differential operator of order $-\mathit{\gamma }$ , self-adjoint in ${\mathit{L}}^2\mathrm{(}\partial \mathrm{\Omega }\mathrm{)}$ . Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for ${\mathit{T}}_{\mathit{V}}$ , and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in ${\mathbb{R}}^{\mathit{d}}$ , and $\mathit{V}=\bar{\mathit{V}}$ is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator ${\mathit{T}}_{\mathit{V}}$ . Finally, we introduce the Krein Laplacian K, self-adjoint in ${\mathit{L}}^2\mathrm{(}\mathrm{\Omega }\mathrm{)}$ , perturb it by a multiplier $\mathit{V}\in \mathit{C}\mathrm{(}\bar{\mathrm{\Omega }};\mathbb{R}\mathrm{)}$ , and show that ${\mathit{\sigma }}_{\mathrm{ess}}\mathrm{(}\mathit{K}+\mathit{V}\mathrm{)}=\mathit{V}\mathrm{(}\partial \mathrm{\Omega }\mathrm{)}$ . Assuming that $\mathit{V}⩾0$ and ${\mathit{V}}_{|\partial \mathrm{\Omega }}=0$ , we study the asymptotic distribution of the discrete spectrum of $\mathit{K}\pm \mathit{V}$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator ${\mathit{T}}_{\mathit{V}}$ .