T. J. Christiansen

Resonances for steplike potentials: forward and inverse results

We consider resonances associated to the one dimensional Schrodinger operator -d 2 d x 2 + V(x), where V(x) = V + if x > x M and V(x) = V- if x < -x M , with V + ≠ V_. We obtain asymptotics of the resonance-counting function for several regions. Moreover, we show that in several situations, the resonances, V + , and V- determine V uniquely up to translation.

Some upper bounds on the number of resonances for manifolds with infinite cylindrical ends

Abstract. We prove some sharp upper bounds on the number of resonances associated with the Laplacian, or Laplacian plus potential, on a manifold with infinite cylindrical ends.

Weyl asymptotics for the Laplacian on asymptotically Euclidean spaces

We prove a trace-type formula for the Laplacian on an asymptotically Euclidean space and use this to obtain Weyl asymptotics of the scattering phase. Other results include an explicit calculation of the leading order singularity of the scattering matrix and results on the behavior of the resolvent and scattering matrix near 0.

Several Complex Variables and the Distribution of Resonances in Potential Scattering

We study resonances associated to Schrödinger operators with compactly supported potentials on ℝd, d≥3, odd. We consider potentials depending holomorphically on a parameter For certain such families, for all z except those in a pluripolar set, the associated resonance–counting function has order of growth d.

Several complex variables and the order of growth of the resonance counting function in Euclidean scattering

We study four classes of compactly supported perturbations of the Laplacian on Rd, d ≥ 3 odd. They are a fairly general class of black box perturbations, a class of second order, self-adjoint elliptic differential operators, Laplacians associated to metric perturbations, and the Dirichlet Laplacian on the exterior of a star-shaped obstacle. In each case, we show that generically the resonance counting function has maximal order of growth.

Harmonic functions of polynomial growth on certain complete manifolds

Given a complete manifold with a particular structure at infinity, we give the dimension of the space of harmonic functions with prescribed polynomial growth.

Probabilistic Weyl Laws for Quantized Tori

For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false” eigenvalues created by pseudospectral effects.

SCATTERING THEORY FOR PERTURBED STRATIFIED MEDIA

We study the operatorH = -c2x,y)Μx,y)∇ · Μ-1(x,y)∇, wherec andΜ are perturbations of functionsc0(y) andΜ0(y) which depend only on the one-dimensional variabley. In particular, we study the spatial asymptotics of limε↺0(H - (λ +iε)2)-1 applied to functions which have compact support or are otherwise well-behaved at infinity and relate the scattering matrix to the asymptotics of the generalized eigenfunctions. We then prove a trace formula for the operatorH in terms of the scattering phase, and, in a very special situation, use the trace formula to find spectral asymptotics forH.

Lower bounds for resonance counting functions for obstacle scattering in even dimensions

In even dimensional Euclidean scattering, the resonances lie on the logarithmic cover of the complex plane. This paper studies resonances for obstacle scattering in

Inverse Problems for Obstacles in a Waveguide

{\Bbb R}^d