We address an issue in the proof of Lemma 4.1 in [J. Spectr. Theory 15 (2025), no. 4, 1593–1613].
We characterize locally finite metric trees with finitely many ends of infinite volume for which the Neumann Laplacian has purely discrete spectrum. In particular, for infinite comb graphs this criterion admits a transparent formulation by means of heights and distances between neighboring teeth.
We characterize spectra of Schrödinger operators with small quasi-periodic analytic potentials in terms of their comb domains, and study action variables motivated by the KdV integrable system.
We study the Laplace operator in an ellipse with either Dirichlet or Neumann boundary. We prove that except for a countable set of eccentricities, the spectrum is simple.
We study a generalized Steklov problem involving a rough potential on the boundary. We establish sharp L^{p} estimates for the Steklov eigenfunctions on compact manifolds with boundary, controlled by their L^{2} norms on the boundary. We first establish sharp boundary estimates by heat kernel bounds and resolvent estimates for the Dirichlet-to-Neumann operator with a rough potential. And then we combine harmonic extension with the Littlewood–Paley decomposition to obtain sharp interior estimates. These results are new even when there is no potential. As an application, we prove the eigenfunctions are C^{1} if the potential is Lipschitz and refine the previous results by Wang and Zhu (2015) on the lower bound of the size of the boundary nodal sets. A key tool is the commutator estimate for first-order pseudo-differential operators by Calderón (1965), and Coifman and Meyer (1978).
