There is a question that puzzles me, so may be someone here has an answer. Assume we have a symmetric operator $A$ that is defined on a space $D$ that is dense in $L^2$, so $A:D\rightarrow L^2$, and $A$ is unbounded when one uses the usual $L^2$ inner product. For example, the momentum operator defined on the first Sobolev space. Assume we know the spectrum of $A$ in this case and it consists of the entire real line. Now, we change the inner product on $D$ such that $D$ is becomes a Hilbert space and $A$ becomes bounded. Therefore, also the spectrum of $A$ should now be bounded. How can one understand this fact and relate the two spectra to each other through the two scalar products?