In his book The Principles of Quantum Mechanics, Dirac states:
"We call a real dynamical variable whose eigenstates form a complete set an observable."
To Dirac, any observable has a complete set of eigenstates. This philosophy is taken seriously in the physicist approach of quantum mechanics. Usually one starts learning quantum mechanics by solving the time-independent Schrödinger equation, in which one looks for eigenstates of the Hamiltonian operator. Since the Hamiltonian operator is an observable, their eigenstates are supposed to form a complete set and every state of that Hilbert space could be expressed in terms of these eigenstates.
From the mathematical point of view, we know that this approach has serious limitations. If we consider $\mathscr{H} = L^{2}(\mathbb{R}^{n})$ for instance, we know that some self-adjoint operators have no eigenvectors at all. However, in some cases we can find a complete orthonormal set of eigenvectors of the Hamiltonian operator (e.g. the harmonic oscillator).
I would like to know what is/are some theorems which can be used to justify the physicist approach, at least for a large class of models of interest. For example, a large class of models deals with Hamiltonians which are of the form $H = -\Delta + V$ on $L^{2}(\mathbb{R}^{n})$, where $-\Delta$ is the Laplace operator and $V$ some external potential function. In my experience, most books on mathematical aspects of quantum mechanics put a lot of effort discussing conditions on $V$ for which $H$ becomes self-adjoint. But I have seen barely no discussion (at least explicitly) about what conditions would make $H$ self-adjoint and have a complete orthonormal set of eigenstates.
Are there any results in this direction? Or one should analyse case by case (and thus there is no point of discussing it in depth on quantum mechanics math book)? I would like something general enough so the basic examples (harmonic oscillator, square well etc) would follow from it. Are there such results? Reference suggestions are also welcome!
