One of the best, exceptionally accessible introduction into the
problems with non-self-adjointness in quantum mechanics. Presented
in the context of their ``numerical signatures", via two examples.
The first one is pedagogical and methodical. It is solvable,
contrasting the operators of kinetic energy and momentum and
delivering the main message: In the finite-matrix simulations
exemplified here by the Runge-Kutta-type N-point discretization, the
property of non-self-adjointness of the continuous-limit operators
may be often deduced from a pseudoconvergence of the eigenvalues or,
even better, from the divergence of the eigenvectors with increasing
N. The second, physics-oriented illustrative example deals with the
one-dimensional Klein–Gordon Coulombic states. It shows that
numerically, one can succeed here in the approximate detection and
localization of the critical-charge-boundary change of status.
Cleverly, the majority of technical details - forming one third of
the text and developed in explicit detail - is stored in appendices.

MR2823450 Ruf, M.; M\"{u}ller, C.; Grobe, R. Numerical signatures of
non-self-adjointness in quantum Hamiltonians. J. Phys. A 44 (2011),
no. 34, 345205, 18 pp. 81Q12