Eigenproblems of large powers of the Laplacian in bounded domains
A. Ramani, Basil Grammaticos, Yves Pomeau · Journal of Physics A Mathematical and Theoretical · 2007
We present a method for computing the spectrum of large powers of the Laplacian in a bounded domain restricting ourselves to the one- and three-dimensional cases. Since it does not seem possible to obtain information on the eigenvalues directly from the transcendental equation that gives the spectrum, we introduce a Wallis-inspired method. We obtain the expansion of the eigenfunction and the eigenvalues in power series where the inverse of the power at which the Laplacian is raised plays the role of the small parameter. We compare these eigenvalues to those obtained through a simple variational approach and remark that the latter offers an excellent approximation to the exact result.