Singularities of the scattering kernel and inverse scattering problems

In his classical work "Can one hear the shape of a drum ?'', (1966) Kac posed the problem to recover the shape of a strictly convex compact domain $D \subset \R^2$ from the spectrum $\{\lambda_j^2\}_{j = 1}^{\infty}$ of the Laplacian $-\Delta$ with Dirichlet boundary condition on $\partial D$. The distribution $\sum_{j = 1}^{\infty} \cos (\lambda_j t)$ is well defined and its singularities are included in the set of the periods of periodic orbits in $D$. In this talk we study unbounded domains $\Omega = \R^d \setminus K$ and the problem is to prove that the form of the obstacle $K$ is uniquely determined by some scattering data. Consider the scattering kernel $s(t, \theta, \omega)$ which is the Fourier transform of the scattering amplitude related to the scattering operator for $K$. First, we prove the Poisson relation which says that the singularities of scattering kernel with respect to $t$ are included in the set of sojourn times of generalised rays incoming with direction $\omega$ and outgoing with direction $\theta$. Second, we establish that for almost all directions $(\omega, \theta)$ the Poisson relation becomes an equality. Thus the sojourn times are observables and they can be considered as scattering data. We will discuss different inverse scattering problems related to the set of sojourn times. In particular, for a large class of obstacles the knowledge of the sojourn times for almost all directions determines uniquely the form of the obstacle. The results are obtained in joint works with L. Stoyanov.