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

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 $.

keywords: Saito's formula biharmonic operator scattering problem inverse problem perturbation
Recovery of jumps and singularities in the multidimensional Schrodinger operator from limited data
Lassi Päivärinta Valery Serov
Inverse Problems & Imaging 2007, 1(3): 525-535 doi: 10.3934/ipi.2007.1.525
The inverse scattering problem for multidimensional Schrödinger operator is studied. More exactly we prove a new formula for the first nonlinear term to estimate more accurately this term. This estimate allows to conclude that all singularities and jumps of the unknown potential can be recovered from the Born approximation. Especially, we show for the potentials in $L^p$ for certain values of $p$ that the approximation agrees with the true potential up to the continuous function.% Text of abstract
keywords: Born approximation. Inverse problems Schrödinger operator
Inverse scattering problem for quasi-linear perturbation of the biharmonic operator on the line
Teemu Tyni Valery Serov
Inverse Problems & Imaging 2019, 13(1): 159-175 doi: 10.3934/ipi.2019009

We consider an inverse scattering problem of recovering the unknown coefficients of quasi-linearly perturbed biharmonic operator on the line. These unknown complex-valued coefficients are assumed to satisfy some regularity conditions on their nonlinearity, but they can be discontinuous or singular in their space variable. We prove that the inverse Born approximation can be used to recover some essential information about the unknown coefficients from the knowledge of the reflection coefficient. This information is the jump discontinuities and the local singularities of the coefficients.

keywords: Inverse scattering biharmonic operator nonlinear perturbation Born approximation reconstruction of singularities
Inverse fixed angle scattering and backscattering for a nonlinear Schrödinger equation in 2D
Georgios Fotopoulos Markus Harju Valery Serov
Inverse Problems & Imaging 2013, 7(1): 183-197 doi: 10.3934/ipi.2013.7.183
We investigate two inverse scattering problems for the nonlinear Schrödinger equation $$ -\Delta u(x) + h(x,|u(x)|)u(x) = k^{2}u(x), \quad x \in \mathbb{R}^2, $$ where $h$ is a very general and possibly singular combination of potentials. The method of Born approximation is applied for the recovery of local singularities and jumps from fixed angle scattering and backscattering data.
keywords: Schrödinger operator nonlinearity Born approximation. inverse problem backscattering fixed angle

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