- #1
AxiomOfChoice
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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?
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?