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SUMMARY:Transmission Eigenvalues and Non-Scattering  Inhomogeneities - Fio
 ralba Cakoni (Rutgers\, The State University of New Jersey)
DTSTART:20230524T084500Z
DTEND:20230524T093000Z
UID:TALK198889@talks.cam.ac.uk
DESCRIPTION:A perplexing question in scattering theory is whether there &n
 bsp\;are incoming time harmonic waves\, at particular frequencies\, that a
 re not scattered by a given inhomogeneity\, in other words the inhomogenei
 ty is invisible to probing by such waves. &nbsp\;We refer to wave numbers\
 , that correspond to frequencies for which there exists a non-scattering i
 ncoming wave\, as non-scattering. This question is inherently related to t
 he solution of inverse scattering problem for inhomogeneous media. &nbsp\;
 The attempt to provide an answer to this question has led to the so-called
  transmission eigenvalue problem with the wave number as the eigen-paramet
 er. This is &nbsp\;non-selfadjoint eigenvalue problem with challenging mat
 hematical structure. The non-scattering wave numbers form a subset of real
  transmission eigenvalues. &nbsp\;A positive answer to the existence of no
 n-scattering wave numbers was already known for spherical inhomogeneities 
 and a negative answer &nbsp\;was &nbsp\;given for inhomogeneities with cor
 ners. Up to very recently little was known about non-scattering inhomogene
 ities that are neither spherical symmetric nor having support that contain
 s a corner. In this presentation we discuss &nbsp\;some new results for ge
 neral inhomogeneities including anisotropic. We &nbsp\;present the state o
 f the art on the spectral theory for the transmission eigenvalue problem. 
 Then we examine necessary conditions for a transmission eigenvalue to be n
 on-scattering wave number. These conditions are formulated in terms of the
  regularity of the boundary and refractive index of the inhomogeneity\, us
 ing free boundary methods. \nThis presentation is based on joint works wit
 h Michael Vogelius and Jingni Xiao.\n&nbsp\;\n&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
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