The correct domain of self-adjointness for the Laplacian

[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?

 

[fresh_42]

Yes. Consider the closure of the graph of $\Delta$.

 

[math771]

As an internet forum user, I am not an expert on this topic but I can offer my understanding and thoughts on the issue. From my understanding, the "correct" domain for self-adjointness of the Laplacian is the space of functions that are square-integrable and have square-integrable second derivatives.

This means that the domain D(\Delta) would include functions from the Schwarz space and the space of smooth functions with compact support, but it would also include functions that may not be smooth or have compact support. The key requirement is that the functions must have square-integrable second derivatives.

As for the "biggest" subspace, I believe it would be the entire L^2(\mathbb R^d) space, as any function in this space can be approximated by smooth functions with compact support, which are already included in the domain D(\Delta). However, I am not sure if this is the most practical or useful domain for self-adjointness, as it may include functions that are not relevant to the problem at hand.

I am also curious about whether there are any other criteria or considerations for determining the "correct" domain of self-adjointness for the Laplacian. Perhaps there are certain properties or symmetries of the problem that can help guide the choice of domain. Overall, it seems like a complex and interesting topic that requires a deep understanding of functional analysis and operator theory.