David Borthwick

Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

For hyperbolic Riemann surfaces of finite geometry, we study Selbergs 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

Scattering Poles for Asymptotically Hyperbolic Manifolds

\SL(2,{\mathbb R})

Super Toeplitz operators and non-perturbative deformation quantization of supermanifolds

is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [20] and Muller [23] to groups which are not necessarily cofinite

UPPER AND LOWER BOUNDS ON RESONANCES FOR MANIFOLDS HYPERBOLIC NEAR INFINITY

For a class of manifolds X that includes quotients of real hyperbolic (n + 1)-dimensional space by a convex co-compact discrete group, we show that the resonances of the meromorphically continued resolvent kernel for the Laplacian on X coincide, with multiplicities, with the poles of the meromorphically continued scattering operator for X. In order to carry out the proof, we use Shmuel Agmons perturbation theory of resonances to show that both resolvent resonances and scattering poles are simple for generic potential perturbations.

Determinants of Laplacians and isopolar metrics on surfaces of infinite area

The purpose of this paper is to construct non-perturbative deformation quantizations of the algebras of smooth functions on Poisson supermanifolds. For the examplesU1¦1 andCm¦n, algebras of super Toeplitz operators are defined with respect to certain Hilbert spaces of superholomorphic functions. Generators and relations for these algebras are given. The algebras can be thought of as algebras of “quantized functions,” and deformation conditions are proven which demonstrate the recovery of the super Poisson structures in a semi-classical limit.

Distribution of Resonances for Hyperbolic Surfaces

For a conformally compact manifold that is hyperbolic near infinity and of dimension n + 1, we complete the proof of the optimal O(r n+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 r n+1 lower bound on the counting function for scattering poles.

Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity 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 two 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.

Supersymmetry and Fredholm modules over quantized spaces

We study the distribution of resonances for geometrically finite hyperbolic surfaces of infinite area by counting resonances numerically. The resonances are computed as zeros of the Selberg zeta function, using an algorithm for computation of the zeta function for Schottky groups. Our particular focus is on three aspects of the resonance distribution that have attracted attention recently: the fractal Weyl law, the spectral gap, and the concentration of decay rates.

The Pfaffian line bundle

Given a holomorphic iterated function scheme with a finite symmetry group

Sharp upper bounds on resonances for perturbations of hyperbolic space

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