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

Abstract

We consider harmonic Toeplitz operators T V = P V : H(Ω) → H(Ω) where P : L 2 (Ω) → H(Ω) is the orthogonal projection onto H(Ω) = u ∈ L 2 (Ω) | ∆u = 0 in Ω , Ω ⊂ R d , 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 T V is in the pth Schatten-von Neumann class S p , by simple sufficient conditions which imply T V ∈ S p,w , the weak counterpart of S p . Next, we consider symbols V ≥ 0 which have a regular power-like decay of rate γ > 0 at ∂Ω, and we show that T V is unitarily equivalent to a pseudo-differential operator of order −γ, self-adjoint in L 2 (∂Ω). Utilizing this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for T V , and establish a sharp remainder estimate. Further, we assume that Ω is the unit ball in R d , and V = V is compactly supported in Ω, and investigate the eigenvalue asymptotics of the Toeplitz operator T V . Finally, we introduce the Krein Laplacian K, self-adjoint in L 2 (Ω), perturb it by a multiplier V ∈ C(Ω; R), and show that σ ess (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 T V .

Spectral asymptotics for Toeplitz operators and an application to banded matrices

“…They obtain asymptotics of eigenvalues very similar to (1.2). The method of proof of [3] is quite different from ours and consists in a reduction to a pseudodifferential operator on the boundary. The same method can probably be applied to give an alternative proof of Theorem 1.1.…”

“…Another closely related recent result is [3], where the authors consider harmonic Toeplitz operators in a bounded domain in R d with symbols that have a power decay near the boundary. They obtain asymptotics of eigenvalues very similar to (1.2).…”

Boundary Value Problems, Weyl Functions, and Differential Operators

“…However, the problem of lower boundedness was also discussed with different methods in [349,355,362]. The Kreȋn-von Neumann extension which is of special interest in this context was investigated in, e.g., [14,68,69,70,71,93,181,356,362,578,579]. For coupling methods for elliptic differential operators based on boundary triplet techniques in the spirit of Section 8.6 we refer to the recent paper [88], where also an abstract version of the third Green identity was proved.…”