This paper (the third or fourth one in a series) describes and
develops an idea of a possible future broadening of the scope of the
standard quantum mechanics via a certain specific complex extension
of the underlying phase space. The paper investigates the first steps
and possibilities of such an extension by a thorough analysis of a
few elementary examples. Technically, an onset of this study may be
sought in an implicit critique of the standard procedure where one
starts from the principle of correspondence and, having chosen a
suitable generator of the time evolution of the system (i.e., a
``phenomenological" or ``realistic" Hamiltonian operator), one
considers (and solves) the differential Schroedinger equation,
usually with some great technical difficulties. In the paper, this
procedure is simply reversed. In the spirit of the so called
quasi-exact Schr\"{o}dinger equations (QESE), one postulates here
just the knowledge of an elementary form of some particular (say,
ground) state. The reason is that the re-construction of the
Hamiltonian is then virtually trivial [interested reader may find a
lot of details in the monograph ``Quasi-exactly solvable models in
quantum mechanics" (IOP Publishing, Bristol, 1994) by A. Ushveridze].
The authors employ this technical advantage in the new setting and
offer a long list of the most elementary Hamiltonian-wavefunction
pairs with the latter QESE property. A further study in this
direction is necessary/promised. Here, several hints supporting the
possible applicability of the new scheme are mentioned, with a
particular emphasis on multiple parallels of this project with
several recent studies of some non-Hermitian Hamiltonians with real
spectra as performed within the so called PT symmetric quantum
mechanics.



CNO: 1946969 Kaushal, R. S. ; Parthasarathi . Quantum mechanics of
complex Hamiltonian systems in one dimension. J. Phys. A 35 (2002),
no. 41, 8743--8761.