The spectrum of large powers of the Laplacian in bounded domains

TL;DR

This paper investigates the spectrum of the Nth power of the Laplacian operator in bounded domains, providing exact results, numerical methods, and discussing extensions to non-integer N and three-dimensional cases.

Contribution

It offers the first exact spectrum results for large powers of the Laplacian and introduces a numerical approach applicable for any N, including non-integer values.

Findings

Exact spectrum in 1D for large N

Numerical method valid for all N

Extensions to non-integer N and 3D cases

Abstract

We present exact results for the spectrum of the Nth power of the Laplacian in a bounded domain. We begin with the one dimensional case and show that the whole spectrum can be obtained in the limit of large N. We also show that it is a useful numerical approach valid for any N. Finally, we discuss implications of this work and present its possible extensions for non integer N and for 3D Laplacian problems.