How to show that an atomic Hamiltonian spectrum is lower-bounded? I'm looking for a proof that the spectrum of an atomic (or molecular) Hamiltonian is lower-bounded.
Right now, the closest I've got is the proof in [1] that the spectrum of a second-order elliptic PDE is lower bounded when we consider solutions in $H^1$ (i.e, the Sobolev space $W^{1,2}$).
I can handle restricting to elliptic PDEs, because the time-independent Schroedinger equation is elliptic, while the time-dependent Schroedinger equation is parabolic (right?).
I can rationalize restricting solutions to $L^2$, since the solution wouldn't make sense physically if the total probability of the system being in some state wasn't finite.  Is there some similar rationalization for restricting to $H^1$?
Also, I think I read somewhere years ago that we can prove that the lower bound on a neutral (i.e, electrically balanced) atomic/molecular Hamiltonian is negative (and thus a stable ground state exists), while no such proof is known for a charged system (i.e. an ion) like CH$_3^+$.  Is this true?
Any insights would be welcome.
[1]: Evans, L., Partial Differential Equations, AMS, Graduate Studies in Mathematics 19.
 A: Starting with the last assessment, I do not know of an analysis of the atomic/molecular ion-probem with respect to semi-boundedness.
There is a direct proof of the following theorem:
Theorem: The operator $H_{\text{atom}}$ given by:
$$ H_{\text{atom}}=-\sum_i \frac{\hbar^2}{2m_i}\nabla_{r_i}^2 +
\frac{e^2}{4\pi\epsilon_0}\sum_{i<j}\frac{1}{|r_i-r_j|} - \frac{Ze^2}{4\pi\epsilon_0}\sum_{i}\frac{1}{|r_j|}, \\ D(H_\text{atom})=C_{0}^{\infty} (\mathbb R^{3n})$$
is essentially self-adjoint and semi-bounded in the Hilbert space $L^2 (\mathbb{R}^{3n})$ and $D\left(\overline{H_{\text{atom}}}\right) = W_{2}^{2} (\mathbb{R}^{3n}).$
This is found as theorem 7.3.7 on page 449 in Hans Triebel's book Higher Analysis (1992 Johann Ambrosius Barth Verlag GmbH) which is the slightly abridged English translation of the 1972 original German edition.  Comparing to my statement, Triebel uses a simplified notation and misses the sum of Laplacians for the electrons in his statement, but uses the sum of all second-order derivatives in the proof, so the proof is good, no mistake.
This theorem is formulated under the assumption that the chosen coordinates refer to a "fixed" nucleus, thus simplifying the intricacies of the relative motion.
The self-adjointness proof is essentially that of Kato (1951), while the semiboundedness result (according to Teschl, Sect. 11.2) is known by the name HVZ theorem. Teschl's approach is different to Triebel's (Teschl makes an argument from the spectrum's point of view, while Triebel uses estimates and inequalities to each potential term of the Hamiltonian).
Now ending with your beginning, yes, from a PDE theory perspective, the result of Evans is good enough, under the proviso that the 2nd order PDF of the atomic/molecular Hamiltonian's spectral equation is shown to be elliptic.
