I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be complicated but could there still be some reasonable version of the spectral theorem? Why or why not? Some elaboration and perhaps even some references would be appreciated. 



Is there a reasonable notion of spectral theorem on a pre-Hilbert space?
Here is a simple example that shows that the idea of spectral theory on pre-Hilbert spaces in the sense you are asking is hopeless. Consider the pre-Hilbert space consisting of the restrictions of all complex polynomials to $[0,1]$, as a dense subspace of $L^2[0,1]$. Then let $A$ be the operator of multiplication by $x$. The spectral projections of this operator are characteristic functions; none of them except $0$ and $1$ are polynomials.


