This lecture reports on joint work with Christopher Judge
and David Borthwick. Determinants of Laplacians are nonlocal spectral invariants which carry important geometric information about Riemannian manifolds and have played a crucial role in spectral geometry. Osgood, Phillips, and Sarnak defined and analyzed determinants of Laplacians for compact surfaces and planar domains, and used them to partially characterize the set of all surfaces or plane domains with a given spectrum. They use a spectral zeta function, formed from the eigenvalues of the Laplacian, to define the determinant, and use heat kernel asymptotics to prove some its key properties. In this lecture I will describe how determinants of Laplacians are being used to study inverse scattering problems on noncompact Riemann surfaces and certain perturbations. The discrete data analogous to the eigenvalues of a compact surface are the scattering poles for the Laplacian, a discrete set of complex numbers which solve a nonselfadjoint eigenvalue problem. We use a Green's function approach to define the determinant and use recent work of Patterson and Perry to connect the determinant for convex cocompact Riemann surfaces with Selberg's zeta function. Perturbation theory, entire function theory, and fine estimates on the resolvent and scattering operator for twodimensional Riemann surfaces due to Guillope and Zworski enable us to elucidate the geometric information contained in the determinant. Using the determinant as an `isopolar' invariant, we are able to prove a compactness theorem for `isopolar' manifolds analogous to the one proved by Osgood, Phillips, and Sarnak for compact surfaces References:
