Not only Hermitian or symmetric operators-matrices have real eigenvalues, what does QM

have to say about these cases.?

It turns out that these do not have orthogonal eigenvectors, so what to do?

So you invent a metric operator, or metric, or weight function for which the eigenvectors

are orthogonal with respect to this metric S.

Probability density is then psistar S psi, and the conservation of matter is met.

You start redifining everything , the wave function, the new Hamiltonian, so everything looks normal again.

An alternative view is a Sturm - Liouville problem

Everything looks fine again, is this valid QM? What is the new physics?