Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

Abstract

Abstract. We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface M = Γ\H 2 associated with a Fuchsian group of the 1st kind Γ containing parabolic elements. M is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.

Inverse scattering at fixed energy on asymptotically hyperbolic Liouville surfaces

“…The Liouville surfaces S we are considering in this paper thus possess two asymptotically hyperbolic ends at {x = 0} and {x = A} as well as this remarkable property of separability for the wave equation mentioned above. We refer to [3,11,12,13] for a detailed presentation of general asymptotically hyperbolic surfaces (and higher dimensional asymptotically hyperbolic manifolds) with no hypothesis of (hidden) symmetries or integrability of the geodesic flow. We start defining the main object from which we shall study some inverse problems at fixed energy for such asymptotically hyperbolic Liouville surfaces.…”

“…where λ = 0 is a fixed energy. Indeed, it is known ( [12]) that the essential spectrum of −△ g is [ 1 4 , +∞) and thus, we shift the bottom of the essential spectrum to 0 by our choice of wave equation. It is also known that the operator −△ g − 1 4 has no eigenvalues embedded into the essential spectrum [0, +∞) (see [4,11,12]).…”

“…in order to separate these two ends. It is shown in [12] that the solutions of the shifted stationary equation (1.7) are unique when imposing some radiation conditions at infinities. Precisely, we define some Besov spaces that encode these radiation conditions at infinities as follows.…”

Inverse Scattering at Fixed Energy on Three-Dimensional Asymptotically Hyperbolic Stäckel Manifolds

“…We recall here the construction of the scattering operator given in [40,41] for asymptotically hyperbolic manifolds. This construction has been used in [22] in the case of asymptotically hyperbolic Liouville surfaces.…”

“…It is known that the operator −∆ g − 1 has no eigenvalues embedded into the essential spectrum [0, +∞) (see [15,40,41]). It is shown in [41] that the solutions of the shifted stationary equation…”