Year of Publication


Degree Name

Doctor of Philosophy (PhD)

Document Type



Arts and Sciences



First Advisor

Dr. Peter Hislop


We establish the upper bounds for the difference between the first two eigenvalues of the relative and absolute eigenvalue problems. Relative and absolute boundary conditions are generalization of Dirichlet and Neumann boundary conditions on functions to differential forms respectively. The domains are taken to be a family of symmetric regions in Rn consisting of two cavities joined by a straight thin tube. Our operators are Hodge Laplacian operators acting on k-forms given by the formula Δ(k) = +δd, where d and δ are the exterior derivatives and the codifferentials respectively. A result has been established on Dirichlet case (0-forms) by Brown, Hislop, and Martinez [2]. We use the same techniques to generalize the results on exponential decay of eigenforms, singular perturbation on domains [1], and matrix representation of the Hodge Laplacian restricted to a suitable subspace [2]. From matrix representation, we are able to find exponentially small upper bounds for the difference between the first two eigenvalues.

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Mathematics Commons