Once you start feeling that self-orthogonality really is a puzzle
in non-Hermitian quantum mechanics, you have to clarify it. This
is the key purpose of this paper. A careful and sufficiently
rigorous account and summary are presented of the related
mathematics of linear differential operators, with particular
attention paid to the completion of the set of eigenstates using
associated functions. Particularly remarkable observations are
made concerning some subtleties emerging for bound states
embedded in the continuous part of the spectrum. A few nice
illustrative examples are constructed within the framework of the
so called supersymmetric quantum mechanics.



MR2252711 Sokolov, A. V.; Andrianov, A. A.; Cannata, F.
Non-Hermitian quantum mechanics of non-diagonalizable
Hamiltonians: puzzles with self-orthogonal states. J. Phys. A 39
(2006), no. 32, 10207--10227. 81Q05