pith. sign in

Spectral properties of the Dirichlet-to-Neumann map for the Helmholtz equation

· 2026 · math.SP · arXiv 2604.11526

2 Pith papers cite this work. Polarity classification is still indexing.

2 Pith papers citing it
open full Pith review browse 2 citing papers arXiv PDF
abstract

The study of the Dirichlet-to-Neumann map and the associated Steklov problem for the Laplace equation has been a central topic in spectral geometry over the past decade. In this survey, we consider a more general framework in which the Laplace equation is replaced by the Helmholtz equation. We examine how the properties of the Dirichlet-to-Neumann eigenvalues and eigenfunctions depend on the parameter in the Helmholtz equation and describe new phenomena arising when this parameter is nonzero, as opposed to the Laplace case. In particular, we present various eigenvalue inequalities, analyse spectral asymptotics in different regimes, and investigate nodal domains and other features of eigenfunctions. We also discuss applications where the Helmholtz parameter plays an essential role, as well as challenges encountered in the numerical computation of the Dirichlet-to-Neumann spectrum.

fields

math.SP 2

years

2026 2

verdicts

UNVERDICTED 2

clear filters

representative citing papers

Eigenvalue optimization via a first-variation formula

math.SP · 2026-06-30 · unverdicted · novelty 7.0

Derives Clarke subdifferential and first-variation formula for the kth eigenvalue on self-adjoint operators (valid at essential spectrum edge) and applies it to characterize optimal weights in weighted Laplace/Steklov problems.

citing papers explorer

Showing 2 of 2 citing papers after filters.

  • Eigenvalue optimization via a first-variation formula math.SP · 2026-06-30 · unverdicted · none · ref 9 · internal anchor

    Derives Clarke subdifferential and first-variation formula for the kth eigenvalue on self-adjoint operators (valid at essential spectrum edge) and applies it to characterize optimal weights in weighted Laplace/Steklov problems.

  • Geometric bounds for Steklov and weighted Neumann eigenvalues on Euclidean domains math.SP · 2026-04-03 · unverdicted · none · ref 5 · internal anchor

    Sharp upper bounds are obtained for the first two nonzero Steklov eigenvalues in dimensions d >= 7 under volume-boundary normalization, derived from optimal weighted Neumann characterizations, plus strict bounds for higher eigenvalues on planar simply connected domains.