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

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 ${\mathcal H}(\Omega) = \left{u \in L^2(\Omega),|,\Delta u = 0 ; \mbox{in};\Omega\right}$,$\Omega \subset {\mathbb R}^d$,$d \geq 2$,isaboundeddomainwith$\partial \Omega \in C^\infty$,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 $T_V \in S_{p, {\rm w}}$,theweakcounterpartof$S_p$\mathrm{.}Next,weassumethat$\Omega$istheunitballin${\mathbb R}^d$,and$V = \overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $\mathrm{\partial }\mathrm{\Omega }$ or $V$ is compactly supported in $\mathrm{\Omega }$. Further, we consider general $\mathrm{\Omega }$ and $V\ge 0$ which is regular in $\mathrm{\Omega }$, and admits a power-like decay of rate $\gamma >0$ at $\partial \Omega$,andweshowthatinthiscase$T_V$ 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 $L^2(\mathrm{\Omega })$; it is known that $\mathrm{K}\mathrm{e}\mathrm{r}K=\mathcal{H}(\mathrm{\Omega })$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V\in C(\bar{\mathrm{\Omega }};\mathbb{R})$, and show that ${\sigma }_{\mathrm{e}\mathrm{s}\mathrm{s}}(K+V)=V(\mathrm{\partial }\mathrm{\Omega })$. Assuming that $V\ge 0$ and $V\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }=0$, 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 $T_V$.