## Inverse scattering and spectral problems for PDEs

#### Valery Serov, Andreas Hauptmann, Valter Pohjola, Teemu Tyni, Markus Harju, Georgios Fotopoulos, Urpo Kyllönen

We study the partial recovery of unknown coefficients of partial differential operators from limited scattering data. Our main interest is to locate the points of discontinuity of the unknown from limited data. Mainly our data is connected to the asymptotical behaviour of the solution to a partial differential equation for large argument. This data is known as the scattering amplitude (or sometimes the farfield pattern). It is well known that the full (or general) scattering data A(k,θ,θ′)for k ϵ R, θ′,θ ϵ S^{n-1}, where A is the scattering amplitude, quarantees the unique reconstruction of the unknown. This is one reason why it is interesting to study backscattering, fixed angle and fixed energy data where uniqueness can fail. We find these datum also suitable for practical applications.

The method used for such a recovery is called the Born approximation. In this approach the unknown is essentially the inverse Fourier transform of the scattering amplitude A. The main idea of the technique is to prove that the difference between the Born approximation and the true unknown is smoother than the unknown itself - maybe even continuous. This means that the main singularities (such as jumps across a domain) can be recovered from the Born approximation. One would also like to obtain this theoretical result locally for widest possible class of unknowns, for instance from L^{p}_{loc}.

Our current work is devoted to the generalization of the Born approximation to the magnetic Schrödinger operator in two and three dimensions. In the linear setting we consider the two-dimensional case with backscattering and fixed angle data. We are also devising numerical implementations of the theoretical results.

We also study scattering problems for perturbations of the biharmonic operator **Δ**^{2}. In this case the operator can have higher order perturbations, such as perturbations of the gradient. Given the full scattering data A(k,θ,θ′) for θ′,θ ϵ S^{n-1} and high enough k ϵ R we have proved the so-called Saito's formula, which yields a uniqueness theorem for the inverse problem of recovering the perturbations. We have also studied backscattering data in two and three dimensional cases and concluded that this data is sufficient for the recovery of singularities in the coefficients. Numerical results support these theorems.

We study (direct and inverse) spectral problems for elliptic differential operators of higher order. The famous Borg-Levinson theorem is proved for elliptic differential operators of order two and four.

Yet another research topic is the study of so called monotonicity methods, and in particular the use of monotonicity methods for shape reconstruction. A shape reconstruction method can be thought of as an procedure that reconstructs the shape of an unknown scatterer from scattering data. This type of problem is of direct interest in applications. This line of research focuses mainly on the use of monotonicity methods in the context of Helmholtz type equations.

Investigation of nonlinear equations appearing in nonlinear optics is also one of our interests (see the list of publications).

Last updated: 12.2.2019