# The correct domain of self-adjointness for the Laplacian

1. Oct 9, 2012

### AxiomOfChoice

The "correct" domain of self-adjointness for the Laplacian

Consider the Hilbert space $L^2(\mathbb R^d)$, and consider the Laplacian operator $\Delta$ on this space. We want to find a domain, $D(\Delta) \subset L^2(\mathbb R^d)$, such that this guy is a self-adjoint operator. We have been talking about this in class recently, and I know that the Schwarz space and the space of smooth functions with compact support are both cores for $\Delta$. But can one easily describe the "biggest" subspace of $L^2(\mathbb R^d)$ on which we can define $\Delta$ such that it's self-adjoint there?