I am slightly confused with regards to the way one obtains self - adjoint differential operators in spectral theory. The aspect that I'd like to understand better is the following:
Suppose we are interested in the spectral analysis of the Laplacian (say). Then, to obtain a bounded (i.e. continuous) operator we define it to act between Sobolev spaces, ok. Let us assume that we are considering the $p = 2$ case here to obtain a Hilbert space.
On the other hand, in order to turn it into a self - adjoint operator we need to ensure that it is symmetric relative to the inner product. Now, which inner product is taken here -- the one that I obtain via the Sobolev space consideration, or the standard $L^2$ - inner product?