Teemu Tyni, Valery Serov. Scattering problems for perturbations of the multidimensional biharmonic operator. Inverse Problems & Imaging, 2018, 12 (1) : 205-227. doi: 10.3934/ipi.2018008

### Scattering problems for perturbations of the multidimensional biharmonic operator

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Author: Tyni, Teemu1; Serov, Valery1
Organizations: 1Department of Mathematical Sciences, FI-90014 University of Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.4 MB)
Language: English
Published: American Institute of Mathematical Sciences, 2018
Publish Date: 2019-04-04
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# Abstract

Some scattering problems for the multidimensional biharmonic operator are studied. The operator is perturbed by first and zero order perturbations, which maybe complex-valued and singular. We show that the solutions to direct scattering problem satisfy a Lippmann-Schwinger equation, and that this integral equation has a unique solution in the weighted Sobolev space $$H_{-δ}^2$$. The main result of this paper is the proof of Saito’s formula, which can be used to prove a uniqueness theorem for the inverse scattering problem. The proof of Saito’s formula is based on norm estimates for the resolvent of the direct operator in $$H_{-δ}^1$$.

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Series: Inverse problems and imaging
ISSN: 1930-8337
ISSN-E: 1930-8345
ISSN-L: 1930-8337
Volume: 12
Issue: 1
Pages: 205 - 227
DOI: 10.3934/ipi.2018008