Inverse scattering and spectral problems for PDEs

Valery Serov, Markus Harju, Georgios Fotopoulos, Urpo Kyllönen, Teemu Tyni

We study the partial recovery of a potential from limited scattering data. More precisely, let q be a potential appearing in the Schrödinger equation in ℝⁿ (we are also planning to consider more general partial differential operators, in particular, the Schrödinger operator with magnetic potential). Our main interest is to locate the points of discontinuity of q from limited data. It is well known that the full (or general) scattering data A(k,θ,θ′)for k ϵ R, θ′,θ ϵ Sn-1, where A is the scattering amplitude, is overdetermined. Therefore, we turn our attention to backscattering, fixed angle and fixed energy data. The first two of this datum are formally well-determined in any dimension whereas the latter only in dimension two. We find them also suitable for practical applications.

The method used for such a recovery is called the Born approximation. In this approach the unknown potential 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 potential is smoother than the potential itself - maybe even continuous. This means that the main singularities (such as jumps across a domain) can be recovered from qB, the Born approximation. One would also like to obtain this theoretical result locally for widest possible class of potentials, i.e. q ϵ Lploc with p as small as possible.

Our current work is devoted to the generalization of the Born approximation to the nonlinear Schrödinger operator in dimensions one and two. In the linear setting we consider the two-dimensional case with backscattering and fixed angle data. The goal is to improve the best known results of Ola, Päivärinta and Serov (Comm. PDE. 26, 2001, no. 3-4, 697-715) and Ruiz and Vargas (Comm. PDE. 30, 2005, no. 1-3, 67-96) from Sobolev space Hscomp to Lebesgue space Lploc. We are also devising numerical implementations of the theoretical results.

Concerning the inverse spectral problems we study first the spectral properties of elliptic partial differential operators with singular coefficients in bounded domains in Rn. We will also study the generalization of the n-dimensional Borg-Levinson theorem (see Päivärinta and Serov, Adv. in Appl. Math. 29, 2002, no. 4, 509-520) for such operators, that is, we study the uniqueness theorem by the spectral data - the Dirichlet eigenvalues and normal derivatives of the eigenfunctions at the boundary.

Last updated: 19.10.2018