Recovery of interior eigenvalues from reduced near field data

Abstract

We consider inverse obstacle and transmission scattering problems where the source of the incident waves is located on a smooth closed surface that is a boundary of a domain located outside of the obstacle/inhomogeneity of the media. The domain can be arbitrarily small but fixed.The scattered waves are measured on the same surface. An effective procedure is suggested for recovery of interior eigenvalues by these data.

Uniqueness in potential scattering with reduced data

“…Define operator L : Remark. An outline of this proof can be found in [9]. Note also the integral kernels of operators L, L * are infinitely smooth, and the arguments below prove that their ranges are dense in any Sobolev space H s , s ≥ 0, not only in L 2 .…”

“…Thus F S ϕ is not the scattered wave produced by sources on S with the density ϕ. However, u sc | S = F S ϕ can be obtained (and measured) as a scattered field on S produced by some waves emitted from S. Namely, the following lemma holds (see [9]). Lemma 2.2.…”

“…They show the uniqueness for potentials with some restrictions on angular derivatives. In [9] we considered the scattering problem (1) when the incident waves were emitted from surface S and the receivers are also distributed over the same surface S, i.e., the following data are available:…”

Борис Руфимович Вайнберг (К Восьмидесятилетию Со Дня Рождения)

“…Б. Р. Вайнберг опубликовал серию работ с Е. Л. Лакштановым [39], [41], [46] о внутренних трансмиссионных собственных значениях (объект, возникающий в рассеянии на препятствиях). В частности, они получили новый закон Вейля, в котором собственные значения подсчитываются со знаком плюс или минус в зависимости от направления вращения соответствующего собственного значения матрицы рассеяния.…”