Fundamental properties of Hamiltonian operators of Schrödinger type

The fundamental quality required of operators representing physical quantities in quantum mechanics is that they be hypermaximalQ-) or self-adjoint(2) in the strict sense employed in the theory of Hubert space, which is equivalent to saying that the eigenvalue problem is completely solvable for them, that is, that there exists a complete set {discrete or continuous) of eigenfunctions.