We consider the scattering theory and the related inverse problem on a non-compact Riemannian manifold (or orbifold) $M$ having the following structure : $M= K \cup M_1 \cup M_{N+N′}$ where $K$ is compact, $M_i$ is diffeomorphic to $(1;\infty)\times M_i$ with $M_i$ being a compact $n-1$ dimensional manifold (or orbifold) endowed with the metric $h_{M_i}$. On each end $M_i$, the metric of $M$ is assumed to behave like $ds^2 \sim (dr)^2 + \rho_i(r)^2 h_{M_i}$ when $r \to \infty$ and $\rho_i(r)$ has the form either $A exp (c_0 r + \beta r^\alpha)$ for $0 < \alpha < 1$, or $Ar^\beta$. Moreover, for 1 <= i <= N, we assume that c0 >= 0 and if c0 = 0, then $\beta > 0$, and for $N + 1 <= i <= N + N′$, we assume $c0 \leq 0$ and if $c0 = 0$, then $\beta < 0$. This means that the ends $M_i$ have regular infinities (i.e. with infinite volume) for $1 \leq i \leq N$, while they have cusps for $N +1 \leq i \leq N +N′$.
Many important examples from physics or mathematics, especially hyperbolic manifolds appearing in number theory fall into this category. We shall solve the Helmholtz equation $(-\Delta_g - \lambda )u = 0$, $\Delta_g$ being the Laplace operator on M, and construct a family of generalized eigenfunctions of $-\Delta_g$ which makes it possible to introduce a Fourier transformation on M. By observing the asymptotic behavior of generalized eigenfunctions at infinity, we introduce the S-matrix, which is a counter part of Heisenberg’s S-matrix in quantum mechanics, and then solve the inverse scattering problem, i.e. the recovery of the manifold M from one component of the S-matrix (for all energies). One can also consider the inverse scattering from cusp by introducing the generalized S-matrix.
This is a joint work with Y. Kurylev and M. Lassas.