We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.
We develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum.
We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable 2-graded Hilbert spaces of super-holomorphic functions. The quantized supermanifold arises as the *-algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck′s constant tends to zero.
We construct a determinant of the Laplacian for infinite-area surfaces that are hyperbolic near ∞ and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order 2 with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near ∞ case, the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order 2 with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.
For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(rn+1) upper bound on the resonance counting function, correcting a mistake in the existing literature. In the case of a compactly supported perturbation of a hyperbolic manifold, we establish a Poisson formula expressing the regularized wave trace as a sum over scattering resonances. This leads to an rn+1 lower bound on the counting function for scattering poles.
The purpose of this paper is to apply the framework of non- commutative differential geometry to quantum deformations of a class of Kahler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute the explicit formula for the Chern character on generators of the Toeplitz C^* -algebra.
For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [20] and Müller [23] to groups which are not necessarily cofinite.