Spectrum of the Laplacian for free space versus spectrum of Laplacian with boundary conditions?

Question

In this question the spectrum of the Laplacian in free space is defined as $]-\infty, 0]$. So it seems the spectrum can be determined without regard to boundary conditions? If instead we consider the Laplacian with Dirichlet (or Neumann) boundary conditions, would you expect this to effect the spectrum? If so, why would boundary conditions affect the spectrum?

Answer 1

The spectrum of an operator, in this case an unbounded operator, greatly depends on the domain of definition and the surrounding Banach or Hilbert space. For example on the space $C^2[0,1]$ of twice differentiable functions with the usual Banach norm, the Laplacian is a bounded operator, and hence its spectrum is bounded.

On the other hand, if we take the $X=\{ f \in C^2[0,1]\bigm| f(0) = 0\}$, then the spectrum is just $\{0\}$.

I suggest you start to read about unbounded operators on Banach and Hilbertspaces, for example in the book by Weidmann http://www.springer.com/de/book/9781461260295