Is there a reasonable notion of spectral theorem on a pre-Hilbert space? 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. 
 A: 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.
A: There is a very nice setup which is suitable for precise mathematical understanding of quantum mechanics including the “delta-function-like eigenvectors”: that of a rigged Hilbert space. It is a Hilbert space $H$ together with a fixed dense continuous inclusion of a locally convex (often assumed nuclear) topological vector space $\Phi\hookrightarrow H$. An example to think of is the inclusion of the Schwartz space $\mathcal S(\mathbb R^n)$ into $L^2(\mathbb R^n)$.
And indeed, there is a very satisfactory spectral theory of selfadjoint operators on rigged Hilbert spaces which gives, for instance, the precise meaning to the statement “the delta functions $\delta_x$, $x\in[0,1]$ form a complete system of generalised eigenvectors for the operator of multiplication by $x$ on $L^2([0,1],\mathrm{Leb})$”.
A systematic treatment can be found in the classical source
I. M. Gelfand and N. J. Vilenkin. Generalized Functions, vol. 4: Some Applications of Harmonic Analysis. Rigged Hilbert Spaces. Academic Press, New York, 1964.
Applications to some classical problems of quantum mechanics can be found in the Ph.D. thesis R. de la Madrid, Quantum Mechanics in Rigged Hilbert Space Language (2001). 
