AxiomOfChoice
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The "correct" domain of self-adjointness for the Laplacian
Consider the Hilbert space [itex]L^2(\mathbb R^d)[/itex], and consider the Laplacian operator [itex]\Delta[/itex] on this space. We want to find a domain, [itex]D(\Delta) \subset L^2(\mathbb R^d)[/itex], 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 [itex]\Delta[/itex]. But can one easily describe the "biggest" subspace of [itex]L^2(\mathbb R^d)[/itex] on which we can define [itex]\Delta[/itex] such that it's self-adjoint there?
Consider the Hilbert space [itex]L^2(\mathbb R^d)[/itex], and consider the Laplacian operator [itex]\Delta[/itex] on this space. We want to find a domain, [itex]D(\Delta) \subset L^2(\mathbb R^d)[/itex], 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 [itex]\Delta[/itex]. But can one easily describe the "biggest" subspace of [itex]L^2(\mathbb R^d)[/itex] on which we can define [itex]\Delta[/itex] such that it's self-adjoint there?