# American Institute of Mathematical Sciences

January  2018, 12(1): 205-227. doi: 10.3934/ipi.2018008

## Scattering problems for perturbations of the multidimensional biharmonic operator

 Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

* Corresponding author: Teemu Tyni

Received  February 2017 Revised  August 2017 Published  December 2017

Fund Project: This work was supported by the Academy of Finland (application number 250215, Finnish Programme for Centres of Excellence in Research 2012-2017)

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 $H_{-δ}^1$.

Citation: 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
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