Non-Hermitian quantum mechanics of non-diagonalizable Hamiltonians: puzzles with self-orthogonal states

We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases. The ”self-orthogonality ” phenomenon is clarified in terms of a correct spectral decomposition and it is shown that ”self-orthogonal ” states never jeopardize decomposition of unity and thereby quantum averages of observables. The example of a complex potential leading to one Jordan cell in the Hamiltonian is constructed and its origin from level coalescence is elucidated. Some puzzles with zero-binorm bound states in continuous spectrum are unraveled with the help of a correct decomposition of unity.