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### Open Access Journals

IPI

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

IPI

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

IPI

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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