Work

Low energy resolvent expansions in dimension two

Year: 2025

Type: article

Abstract: The behavior of the resolvent at low energies has implications for many kinds of asymptotics, including for the scattering matrix and phase, for the Dirichlet-to-Neumann map, and for wave evolution. In this paper we present a robust method, based in part on resolvent identity arguments following Vodev and boundary pairing arguments following Melrose, for deriving such expansions, and implement it in detail for compactly supported perturbations of the Laplacian on R 2 \mathbb {R}^2 . We obtain precise results for general self-adjoint black-box perturbations, in the sense of Sjöstrand–Zworski, and also for some non-self-adjoint ones. The most important terms are the most singular ones, and we compute them in detail, relating them to spaces of zero eigenvalues and resonances.

Source: Communications of the American Mathematical Society

DOI: 10.1090/cams/46

Authors: T. Christiansen, K. Datchev

Institutions: University of Missouri, Purdue University West Lafayette

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: National Science Foundation, Simons Foundation

Awards: Geometric Scattering Theory, and Propagation and Decay of Waves, G7737820138