Spectral measure of Laplacean operators in Paley-Wiener space

Abstract: We are interested in computing the spectral measure of Laplacean operators in Paley-Wiener space, the Hilbert space of all square integrable functions having Fourier transforms supported in a compact set $K$, the closure of an open bounded set in $\R^N$. I is well-known that every differential operator is bounded in this space. Among others, we will prove that the spectrum of Laplace operator is the set $${-|x|^2: x\in K}.$$