12,198 research outputs found
Semiclassical Calculation of the C Operator in PT-Symmetric Quantum Mechanics
To determine the Hilbert space and inner product for a quantum theory defined
by a non-Hermitian -symmetric Hamiltonian , it is necessary to
construct a new time-independent observable operator called . It has
recently been shown that for the {\it cubic} -symmetric
Hamiltonian one can obtain as a
perturbation expansion in powers of . This paper considers the more
difficult case of noncubic Hamiltonians of the form
(). For these Hamiltonians it is shown how to calculate
by using nonperturbative semiclassical methods.Comment: 11 pages, 1 figur
WKB Analysis of PT-Symmetric Sturm-Liouville problems
Most studies of PT-symmetric quantum-mechanical Hamiltonians have considered
the Schroedinger eigenvalue problem on an infinite domain. This paper examines
the consequences of imposing the boundary conditions on a finite domain. As is
the case with regular Hermitian Sturm-Liouville problems, the eigenvalues of
the PT-symmetric Sturm-Liouville problem grow like for large .
However, the novelty is that a PT eigenvalue problem on a finite domain
typically exhibits a sequence of critical points at which pairs of eigenvalues
cease to be real and become complex conjugates of one another. For the
potentials considered here this sequence of critical points is associated with
a turning point on the imaginary axis in the complex plane. WKB analysis is
used to calculate the asymptotic behaviors of the real eigenvalues and the
locations of the critical points. The method turns out to be surprisingly
accurate even at low energies.Comment: 11 pages, 8 figure
Systematics of quadrupolar correlation energies
We calculate correlation energies associated with the quadrupolar shape
degrees of freedom with a view to improving the self-consistent mean-field
theory of nuclear binding energies. The Generator Coordinate Method is employed
using mean-field wave functions and the Skyrme SLy4 interaction. Systematic
results are presented for 605 even-even nuclei of known binding energies, going
from mass A=16 up to the heaviest known. The correlation energies range from
0.5 to 6.0 MeV in magnitude and are rather smooth except for large variations
at magic numbers and in light nuclei. Inclusion of these correlation energies
in the calculated binding energy is found to improve two deficiencies of the
Skyrme mean field theory. The pure mean field theory has an exaggerated shell
effect at neutron magic numbers and addition of the correlation energies reduce
it. The correlations also explain the phenomenon of mutually enhanced magicity,
an interaction between neutron and proton shell effects that is not explicable
in mean field theory.Comment: 4 pages with 3 embedded figure
Green Functions for the Wrong-Sign Quartic
It has been shown that the Schwinger-Dyson equations for non-Hermitian
theories implicitly include the Hilbert-space metric. Approximate Green
functions for such theories may thus be obtained, without having to evaluate
the metric explicitly, by truncation of the equations. Such a calculation has
recently been carried out for various -symmetric theories, in both quantum
mechanics and quantum field theory, including the wrong-sign quartic
oscillator. For this particular theory the metric is known in closed form,
making possible an independent check of these approximate results. We do so by
numerically evaluating the ground-state wave-function for the equivalent
Hermitian Hamiltonian and using this wave-function, in conjunction with the
metric operator, to calculate the one- and two-point Green functions. We find
that the Green functions evaluated by lowest-order truncation of the
Schwinger-Dyson equations are already accurate at the (6-8)% level. This
provides a strong justification for the method and a motivation for its
extension to higher order and to higher dimensions, where the calculation of
the metric is extremely difficult
Interpretation of satellite images of the Republic of Niger
Interpretations of LANDSAT pictures were carried out for an area located in the west of the Niger Republic in the geological, hydrogeological and pedological sectors. Checking of the extent of vegetation and use of the soils and effects of desertification for the purpose of yearly map making was carried out. The proposed control of land use may be optimized by the direct reception of LANDSAT data by the receiving station planned for Ouagadougou. Since that station will not be operating before 1983, the establishment of a mobile reception station in the Republic of Niger to enable the installation of the required control system is advised
PT-symmetric sextic potentials
The family of complex PT-symmetric sextic potentials is studied to show that
for various cases the system is essentially quasi-solvable and possesses real,
discrete energy eigenvalues. For a particular choice of parameters, we find
that under supersymmetric transformations the underlying potential picks up a
reflectionless part.Comment: 8 pages, LaTeX with amssym, no figure
Quantum tunneling as a classical anomaly
Classical mechanics is a singular theory in that real-energy classical
particles can never enter classically forbidden regions. However, if one
regulates classical mechanics by allowing the energy E of a particle to be
complex, the particle exhibits quantum-like behavior: Complex-energy classical
particles can travel between classically allowed regions separated by potential
barriers. When Im(E) -> 0, the classical tunneling probabilities persist.
Hence, one can interpret quantum tunneling as an anomaly. A numerical
comparison of complex classical tunneling probabilities with quantum tunneling
probabilities leads to the conjecture that as ReE increases, complex classical
tunneling probabilities approach the corresponding quantum probabilities. Thus,
this work attempts to generalize the Bohr correspondence principle from
classically allowed to classically forbidden regions.Comment: 12 pages, 7 figure
Harmonic oscillator well with a screened Coulombic core is quasi-exactly solvable
In the quantization scheme which weakens the hermiticity of a Hamiltonian to
its mere PT invariance the superposition V(x) = x^2+ Ze^2/x of the harmonic and
Coulomb potentials is defined at the purely imaginary effective charges
(Ze^2=if) and regularized by a purely imaginary shift of x. This model is
quasi-exactly solvable: We show that at each excited, (N+1)-st
harmonic-oscillator energy E=2N+3 there exists not only the well known harmonic
oscillator bound state (at the vanishing charge f=0) but also a normalizable
(N+1)-plet of the further elementary Sturmian eigenstates \psi_n(x) at
eigencharges f=f_n > 0, n = 0, 1, ..., N. Beyond the first few smallest
multiplicities N we recommend their perturbative construction.Comment: 13 pages, Latex file, to appear in J. Phys. A: Math. Ge
Quantum counterpart of spontaneously broken classical PT symmetry
The classical trajectories of a particle governed by the PT-symmetric
Hamiltonian () have been studied in
depth. It is known that almost all trajectories that begin at a classical
turning point oscillate periodically between this turning point and the
corresponding PT-symmetric turning point. It is also known that there are
regions in for which the periods of these orbits vary rapidly as
functions of and that in these regions there are isolated values of
for which the classical trajectories exhibit spontaneously broken PT
symmetry. The current paper examines the corresponding quantum-mechanical
systems. The eigenvalues of these quantum systems exhibit characteristic
behaviors that are correlated with those of the associated classical system.Comment: 11 pages, 7 figure
Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model
We study a two-dimensional exactly solvable non-Hermitian non-symmetric
quantum model with real spectrum, which is not amenable to separation of
variables, by supersymmetrical methods. Here we focus attention on the property
of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To
our knowledge this is the first time that pseudo-Hermiticity is realized
explicitly for a nontrivial two-dimensional case. It is shown that the
Hamiltonian of the model is not diagonalizable.Comment: 14 page
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