The eigenvalues of the Laplacian on domains with small slits (2008)
We introduce a small slit into a planar domain and study the resulting effect upon the eigenvalues of the Laplacian. In particular, we show that as the length of the slit tends to zero, each...
Determinants of Laplacians and Isopolar Metrics on Surfaces of In Area (2007)
David Borthwick, Chris Judge, A. Perry
Abstract. 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...
David Borthwick, Chris Judge, Peter Perry
Abstract. A determinant of the Laplacian is dened for convex co-compact Riemann surfaces, i.e., Riemann surfaces of innite area obtained as quotients of the Poincare upper half-plane by a nitely...
Generic spectral simplicity of polygons (2007)
We study the Laplace operator with Dirichlet or Neumann boundary condition on polygons in the Euclidean plane. We prove that almost every simply connected polygon with at least four vertices has...
Determinants of Laplacians and isopolar metrics on surfaces of infinite area (2003)
Borthwick, David, Judge, Chris, Perry, Peter A.
We construct a determinant of the Laplacian for infinite-area surfaces that are hyperbolic near $\infty$ and without cusps. In the case of a convex cocompact hyperbolic metric, the determinant can be...
Tracking eigenvalues to the frontier of moduli space II: Limits for eigenvalue branches (2001)
We prove the existence of limits of real-analytic Laplace eigenvalue branches for real-analytic families of metrics that degenerate along a compact hypersurface.
Tracking eigenvalues to the frontier of moduli space I: Convergence and spectral accumulation (2001)
We study the limiting behavior of eigenfunctions/eigenvalues of the Laplacian of a family of Riemannian metrics that degenerates on a hypersurface. Our results generalize earlier work concerning the...