On nodal and generalized singular structures of Laplacian eigenfunctions and applications to inverse scattering problems

Xinlin Cao, Huaian Diao*, Hongyu LIU, Jun Zou

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)

Abstract

In this paper, we present some novel and intriguing findings on the geometric structures of Laplacian eigenfunctions and their deep relationship to the quantitative behaviours of the eigenfunctions in two dimensions. We introduce a new notion of generalized singular lines of the Laplacian eigenfunctions, and carefully study these singular lines and the nodal lines. The studies reveal that the intersecting angle between two of those lines is closely related to the vanishing order of the eigenfunction at the intersecting point. We establish an accurate and comprehensive quantitative characterisation of the relationship. Roughly speaking, the vanishing order is generically infinite if the intersecting angle is irrational, and the vanishing order is finite if the intersecting angle is rational. In fact, in the latter case, the vanishing order is the degree of the rationality. The theoretical findings are original and of significant interest in spectral theory. Moreover, they are applied directly to some physical problems of great importance, including the inverse obstacle scattering problem and the inverse diffraction grating problem. It is shown in a certain polygonal setup that one can recover the support of the unknown scatterer as well as the surface impedance parameter by finitely many far-field patterns. Indeed, at most two far-field patterns are sufficient for some important applications. Unique identifiability by finitely many far-field patterns remains to be a highly challenging fundamental mathematical problem in the inverse scattering theory.

Original languageEnglish
Pages (from-to)116-161
Number of pages46
JournalJournal des Mathematiques Pures et Appliquees
Volume143
DOIs
Publication statusPublished - Nov 2020

Scopus Subject Areas

  • Mathematics(all)
  • Applied Mathematics

User-Defined Keywords

  • A single far-field pattern
  • Impedance obstacle
  • Laplacian eigenfunctions
  • Nodal and generalised singular lines

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