DE019820271

Znojil, Miloslav $\cal{PT}$ symmetric models in more dimensions and
solvable square-well versions of their angular Schr"odinger
equations.

J. Phys. A, Math. Gen. 36, No.28, 7825-7838 (2003).

[ISSN 0305-4470]

http://www.iop.org/Journals/ja

The exactly solvable square well potential is fairly popular as a
model of bound states of a single particle in one dimension. It also
finds its use when you move beyond the framework of the standard
quantum mechanics and weaken the usual Hermiticity of the observables
to their mere PT symmetry (= parity-pseudo-Hermiticity), with the
basic formulae to be found in my older study of the PT symmetric
square well in Physics Letters A 285 (2001) on pp. 7 - 10. In the
present continuation of this study, more particles (A=3) or more
dimensions $D > 1$ are considered in the respective spirit of the
Calogero's and Smorodinsky-Winternitz' models. The PT-symmetrization
and square-well modeling are tentatively used in their angular
Schroedinger equations. Obviously, the parity-violating terms lose
their traditional physical sink/source meaning and acquire a new
importance as interactions which mediate the tunneling between
(originally, completely separated) Weyl chambers. This modifies the
underlying physics connected, usually, with a freedom of the choice
of the statistics of the system which must be replaced by a
non-traditional, Floquetian mathematical re-interpretation of the
statistics. In the paper in question, this is illustrated by a few
tabulated sets of the numerically evaluated energy levels which
clearly obey the expected generalized Floquetian classification. In
the simplest double-well example, four different series of the energy
levels are seen to emerge. The study is waiting for its further
continuations (paying attention, e.g., to the mechanisms responsible
for the PT-symmetry breaking) as well as for its better integration
into the main stream of the development of the subject as summarized
recently, during the 1st International Workshop ``Pseudo-Hermitian
Hamiltonians in Quantum Physics", with proceedings to appear, early
in the year 2004, in the January issue of Czechoslovak J. Physics,
vol. 54.


Review Submission Reviewernumber: 9689

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Primary classification: 81Q05 Closed and approximate solutions to the
Schroedinger, Dirac, Klein-Gordon and other quantum-mechanical
equations Secondary classification: 81-08 Computational methods 34L16
Numerical approximation of eigenvalues and of other parts of the
spectrum 34M15 Algebraic aspects differential-algebraic,
hypertranscendence, group-theoretical 34M60 Singular perturbation
problems in the complex domain complex WKB, turning points, steepest
descent

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Review text:

The exactly solvable square well potential is fairly popular as a
model of bound states of a single particle in one dimension. It also
finds its use when you move beyond the framework of the standard
quantum mechanics and weaken the usual Hermiticity of the observables
to their mere PT symmetry (= parity-pseudo-Hermiticity), with the
basic formulae to be found in my older study of the PT symmetric
square well in Physics Letters A 285 (2001) on pp. 7 - 10. In the
present continuation of this study, more particles (A=3) or more
dimensions $D > 1$ are considered in the respective spirit of the
Calogero's and Smorodinsky-Winternitz' models. The PT-symmetrization
and square-well modeling are tentatively used in their angular
Schroedinger equations. Obviously, the parity-violating terms lose
their traditional physical sink/source meaning and acquire a new
importance as interactions which mediate the tunneling between
(originally, completely separated) Weyl chambers. This modifies the
underlying physics connected, usually, with a freedom of the choice
of the statistics of the system which must be replaced by a
non-traditional, Floquetian mathematical re-interpretation of the
statistics. In the paper in question, this is illustrated by a few
tabulated sets of the numerically evaluated energy levels which
clearly obey the expected generalized Floquetian classification. In
the simplest double-well example, four different series of the energy
levels are seen to emerge. The study is waiting for its further
continuations (paying attention, e.g., to the mechanisms responsible
for the PT-symmetry breaking) as well as for its better integration
into the main stream of the development of the subject as summarized
recently, during the 1st International Workshop ``Pseudo-Hermitian
Hamiltonians in Quantum Physics", with proceedings to appear, early
in the year 2004, in the January issue of Czechoslovak J. Physics,
vol. 54.