9 research outputs found
How Do Energies Complexify?
Some particular properties of the parametric dependence of eigenvalues with emphasis on their complexification are discussed. The non-diagonalisability of PT-symmetric matrix Hamiltonians in exceptional points is compared with level-crossing prohibition of Hermitian systems. For non-matrix Hamiltonians, the different way of complexification between Klein-Gordon and Dirac Hamiltonians is demonstrated.
Comment on `Solution of the Dirac equation for the Woods-Saxon potential with spin and pseudospin symmetry' [J. Y. Guo and Z-Q. Sheng, Phys. Lett. A 338 (2005) 90]
Out of the four bound-state solutions presented in loc. cit., only one (viz.,
the spin-symmetric one, in the low-mass regime) is shown compatible with the
physical boundary conditions. We clarify the problem, correct the method and
offer another, "missing" (viz., pseudospin-symmetric) new solution with certain
counterintuitive "repulsion-generated" property.Comment: 6 p
Quantum Big Bang without fine-tuning in a toy-model
The question of possible physics before Big Bang (or after Big Crunch) is
addressed via a schematic non-covariant simulation of the loss of observability
of the Universe. Our model is drastically simplified by the reduction of its
degrees of freedom to the mere finite number. The Hilbert space of states is
then allowed time-dependent and singular at the critical time . This
option circumvents several traditional theoretical difficulties in a way
illustrated via solvable examples. In particular, the unitary evolution of our
toy-model quantum Universe is shown interruptible, without any fine-tuning, at
the instant of its bang or collapse .Comment: 20 pp., 1 fig., invited talk for the conference "10th Workshop on
Quantization, Dualities and Integrable Systems" (April 22 - 24, 2011),
http://qdis.emu.edu.tr/index.htm
Level Crossings in Complex Two-Dimensional Potentials
Two-dimensional PT-symmetric quantum-mechanical systems with the complex
cubic potential V_{12}=x^2+y^2+igxy^2 and the complex Henon-Heiles potential
V_{HH}=x^2+y^2+ig(xy^2-x^3/3) are investigated. Using numerical and
perturbative methods, energy spectra are obtained to high levels. Although both
potentials respect the PT symmetry, the complex energy eigenvalues appear when
level crossing happens between same parity eigenstates.Comment: 9 pages, 4 figures. Submitted as a conference proceeding of PHHQP
Closed formula for the metric in the Hilbert space of a PT-symmetric model
We introduce a very simple, exactly solvable PT-symmetric non-Hermitian model
with real spectrum, and derive a closed formula for the metric operator which
relates the problem to a Hermitian one.Comment: LaTeX, 13 pages; to appear in Journal of Physics
Quadratic pseudosupersymmetry in two-level systems
Using the intertwining relation we construct a pseudosuperpartner for a
(non-Hermitian) Dirac-like Hamiltonian describing a two-level system
interacting in the rotating wave approximation with the electric component of
an electromagnetic field. The two pseudosuperpartners and pseudosupersymmetry
generators close a quadratic pseudosuperalgebra. A class of time dependent
electric fields for which the equation of motion for a two level system placed
in this field can be solved exactly is obtained. New interesting phenomenon is
observed. There exists such a time-dependent detuning of the field frequency
from the resonance value that the probability to populate the excited level
ceases to oscillate and becomes a monotonically growing function of time
tending to 3/4. It is shown that near this fixed excitation regime the
probability exhibits two kinds of oscillations. The oscillations with a small
amplitude and a frequency close to the Rabi frequency (fast oscillations) take
place at the background of the ones with a big amplitude and a small frequency
(slow oscillations). During the period of slow oscillations the minimal value
of the probability to populate the excited level may exceed 1/2 suggesting for
an ensemble of such two-level atoms the possibility to acquire the inverse
population and exhibit lasing properties.Comment: 5 figure