Spectral properties of harmonic Toeplitz operators and applications to the perturbed Krein Laplacian
Vincent Bruneau, Georgi D. Raikov · Asymptotic Analysis · 2018
We consider harmonic Toeplitz operators [Formula: see text] where [Formula: see text] is the orthogonal projection onto [Formula: see text], [Formula: see text], [Formula: see text], is a bounded domain with boundary [Formula: see text], and [Formula: see text] is an appropriate multiplier. First, we complement the known criteria which guarantee that [Formula: see text] is in the pth Schatten–von Neumann class [Formula: see text], by simple sufficient conditions which imply [Formula: see text], the weak counterpart of [Formula: see text]. Next, we consider symbols [Formula: see text] which have a regular power-like decay of rate [Formula: see text] at [Formula: see text], and we show that [Formula: see text] is unitarily equivalent to a classical pseudo-differential operator of order [Formula: see text], self-adjoint in [Formula: see text]. Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for [Formula: see text], and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in [Formula: see text], and [Formula: see text] is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator [Formula: see text]. Finally, we introduce the Krein Laplacian K, self-adjoint in [Formula: see text], perturb it by a multiplier [Formula: see text], and show that [Formula: see text]. Assuming that [Formula: see text] and [Formula: see text], we study the asymptotic distribution of the discrete spectrum of [Formula: see text] near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator [Formula: see text].