10,222 research outputs found
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
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
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
Guidance, flight mechanics and trajectory optimization. Volume 12 - Relative motion, guidance equations for terminal rendezvous
Equations of relative motion and guidance for orbital transfer and docking maneuvers in spacecraft rendezvou
Semiclassical Analysis of Quasi-Exact Solvability
Higher-order WKB methods are used to investigate the border between the
solvable and insolvable portions of the spectrum of quasi-exactly solvable
quantum-mechanical potentials. The analysis reveals scaling and factorization
properties that are central to quasi-exact solvability. These two properties
define a new class of semiclassically quasi-exactly solvable potentials.Comment: 12 pages, ReVTe
Exact PT-Symmetry Is Equivalent to Hermiticity
We show that a quantum system possessing an exact antilinear symmetry, in
particular PT-symmetry, is equivalent to a quantum system having a Hermitian
Hamiltonian. We construct the unitary operator relating an arbitrary
non-Hermitian Hamiltonian with exact PT-symmetry to a Hermitian Hamiltonian. We
apply our general results to PT-symmetry in finite-dimensions and give the
explicit form of the above-mentioned unitary operator and Hermitian Hamiltonian
in two dimensions. Our findings lead to the conjecture that non-Hermitian
CPT-symmetric field theories are equivalent to certain nonlocal Hermitian field
theories.Comment: Few typos have been corrected and a reference update
Vacuum Stability of the wrong sign Scalar Field Theory
We apply the effective potential method to study the vacuum stability of the
bounded from above (unstable) quantum field potential. The
stability ( and the mass renormalization
( conditions force the effective
potential of this theory to be bounded from below (stable). Since bounded from
below potentials are always associated with localized wave functions, the
algorithm we use replaces the boundary condition applied to the wave functions
in the complex contour method by two stability conditions on the effective
potential obtained. To test the validity of our calculations, we show that our
variational predictions can reproduce exactly the results in the literature for
the -symmetric theory. We then extend the applications
of the algorithm to the unstudied stability problem of the bounded from above
scalar field theory where classical analysis prohibits the
existence of a stable spectrum. Concerning this, we calculated the effective
potential up to first order in the couplings in space-time dimensions. We
find that a Hermitian effective theory is instable while a non-Hermitian but
-symmetric effective theory characterized by a pure imaginary
vacuum condensate is stable (bounded from below) which is against the classical
predictions of the instability of the theory. We assert that the work presented
here represents the first calculations that advocates the stability of the
scalar potential.Comment: 21pages, 12 figures. In this version, we updated the text and added
some figure
Interactions of Hermitian and non-Hermitian Hamiltonians
The coupling of non-Hermitian PT-symmetric Hamiltonians to standard Hermitian
Hamiltonians, each of which individually has a real energy spectrum, is
explored by means of a number of soluble models. It is found that in all cases
the energy remains real for small values of the coupling constant, but becomes
complex if the coupling becomes stronger than some critical value. For a
quadratic non-Hermitian PT-symmetric Hamiltonian coupled to an arbitrary real
Hermitian PT-symmetric Hamiltonian, the reality of the ground-state energy for
small enough coupling constant is established up to second order in
perturbation theory.Comment: 9 pages, 0 figure
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