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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 theorthogonal projection onto ${\mathcal H}(\Omega) = \left\{u \inL^2(\Omega)\,|\,\Delta u = 0 \; \mbox{in}\;\Omega\right\}$, $\Omega \subset{\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\partial \Omega \inC^\infty$, and $V: \Omega \to {\mathbb C}$ is a suitable multiplier. First, wecomplement the known criteria which guarantee that $T_V$ is in the $p$thSchatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V \inS_{p, {\rm w}}$, the weak counterpart of $S_p$. Next, we assume that $\Omega$is the unit ball in ${\mathbb R}^d$, and $V = \overline{V}$ is radiallysymmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has apower-like decay at $\partial \Omega$ or $V$ is compactly supported in$\Omega$. Further, we consider general $\Omega$ and $V \geq 0$ which is regularin $\Omega$, 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 apseudo-differential operator of order $-\gamma$, self-adjoint in $L^2(\partial\Omega)$. Using this unitary equivalence, we obtain the main asymptotic term ofthe eigenvalue counting function for the operator $T_V$. Finally, we introducethe 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 $K$ isisolated. We perturb $K$ by $V \in C(\overline{\Omega};{\mathbb R})$, and showthat $\sigma_{\rm ess}(K+V) = V(\partial \Omega)$. Assuming that $V \geq 0$ and$V{|\partial \Omega} = 0$, we study the asymptotic distribution of theeigenvalues of $K \pm V$ near the origin, and find that the effectiveHamiltonian which governs this distribution is the Toeplitz operator $T_V$.

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