EVENT DETAILS AND ABSTRACT

Title: Scattering theory in even dimensions: resonances and the scattering matrix
Speaker: Peter Hislop
Speaker Info: University of Kentucky
Brief Description:
Special Note:
Abstract:

We discuss various aspects of black-box scattering in even dimensions $d \geq 2$ obtained in joint work with T. J. Christiansen. Two basic examples are Schrödinger operators with compactly supported real-valued or complex-valued bounded potentials and the Laplacian exterior to bounded domains with self-adjoint boundary conditions. The resolvent of the black-box operator continues meromorphically to the infinitely-sheeted Riemann surface of the logarithm. For the case of Schrödinger operators, we prove that generically the counting function for resolvent resonances on each nonphysical sheet has order of growth equal to the dimension d. We discuss the S-matrix, its symmetries, and its scattering poles that appear in its meromorphic continuation. We prove a formula relating the multiplicity of the scattering poles to the multiplicity of the resolvent resonances. We prove the absence of purely imaginary resonances on each nonphysical sheet, unlike the odd dimensional case. As an example, we prove an asymptotic expansion for the resonance counting function for the Dirichlet Laplacian exterior to the sphere in even dimensions.