Affiliations: [a] Institut de Mathématiques de Bordeaux, UMR 5251 du CNRS, Université de Bordeaux, 351 cours de la Libération, 33405 Talence cedex, France. E-mail: [email protected] | [b] Departamento de Matemáticas, Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Santiago de Chile, Chile. E-mail: [email protected]
Abstract: We consider harmonic Toeplitz operators TV=PV:H(Ω)→H(Ω) where P:L2(Ω)→H(Ω) is the orthogonal projection onto H(Ω)={u∈L2(Ω)∣Δu=0 in Ω}, Ω⊂Rd, d⩾2, is a bounded domain with boundary ∂Ω∈C∞, and V:Ω→C is an appropriate multiplier. First, we complement the known criteria which guarantee that TV is in the pth Schatten–von Neumann class Sp, by simple sufficient conditions which imply TV∈Sp,w, the weak counterpart of Sp. Next, we consider symbols V⩾0 which have a regular power-like decay of rate γ>0 at ∂Ω, and we show that TV is unitarily equivalent to a classical pseudo-differential operator of order −γ, self-adjoint in L2(∂Ω). Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for TV, and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in Rd, and V=V‾ is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator TV. Finally, we introduce the Krein Laplacian K, self-adjoint in L2(Ω), perturb it by a multiplier V∈C(Ω‾;R), and show that σess(K+V)=V(∂Ω). Assuming that V⩾0 and V|∂Ω=0, we study the asymptotic distribution of the discrete spectrum of K±V near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator TV.
