8,120 research outputs found
Complex periodic potentials with real band spectra
This paper demonstrates that complex PT-symmetric periodic potentials possess
real band spectra. However, there are significant qualitative differences in
the band structure for these potentials when compared with conventional real
periodic potentials. For example, while the potentials V(x)=i\sin^{2N+1}(x),
(N=0, 1, 2, ...), have infinitely many gaps, at the band edges there are
periodic wave functions but no antiperiodic wave functions. Numerical analysis
and higher-order WKB techniques are used to establish these results.Comment: 8 pages, 7 figures, LaTe
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
Spectral zeta functions of a 1D Schr\"odinger problem
We study the spectral zeta functions associated to the radial Schr\"odinger
problem with potential V(x)=x^{2M}+alpha x^{M-1}+(lambda^2-1/4)/x^2. Using the
quantum Wronskian equation, we provide results such as closed-form evaluations
for some of the second zeta functions i.e. the sum over the inverse eigenvalues
squared. Also we discuss how our results can be used to derive relationships
and identities involving special functions, using a particular 5F_4
hypergeometric series as an example. Our work is then extended to a class of
related PT-symmetric eigenvalue problems. Using the fused quantum Wronskian we
give a simple method for calculating the related spectral zeta functions. This
method has a number of applications including the use of the ODE/IM
correspondence to compute the (vacuum) nonlocal integrals of motion G_n which
appear in an associated integrable quantum field theory.Comment: 15 pages, version
Semiclassical analysis of a complex quartic Hamiltonian
It is necessary to calculate the C operator for the non-Hermitian
PT-symmetric Hamiltonian H=\half p^2+\half\mu^2x^2-\lambda x^4 in order to
demonstrate that H defines a consistent unitary theory of quantum mechanics.
However, the C operator cannot be obtained by using perturbative methods.
Including a small imaginary cubic term gives the Hamiltonian H=\half p^2+\half
\mu^2x^2+igx^3-\lambda x^4, whose C operator can be obtained perturbatively. In
the semiclassical limit all terms in the perturbation series can be calculated
in closed form and the perturbation series can be summed exactly. The result is
a closed-form expression for C having a nontrivial dependence on the dynamical
variables x and p and on the parameter \lambda.Comment: 4 page
New Quasi-Exactly Solvable Sextic Polynomial Potentials
A Hamiltonian is said to be quasi-exactly solvable (QES) if some of the
energy levels and the corresponding eigenfunctions can be calculated exactly
and in closed form. An entirely new class of QES Hamiltonians having sextic
polynomial potentials is constructed. These new Hamiltonians are different from
the sextic QES Hamiltonians in the literature because their eigenfunctions obey
PT-symmetric rather than Hermitian boundary conditions. These new Hamiltonians
present a novel problem that is not encountered when the Hamiltonian is
Hermitian: It is necessary to distinguish between the parametric region of
unbroken PT symmetry, in which all of the eigenvalues are real, and the region
of broken PT symmetry, in which some of the eigenvalues are complex. The
precise location of the boundary between these two regions is determined
numerically using extrapolation techniques and analytically using WKB analysis
Solution of the Skyrme HF+BCS equation on a 3D mesh. II. A new version of the Ev8 code
We describe a new version of the EV8 code that solves the nuclear
Skyrme-Hartree-Fock+BCS problem using a 3-dimensional cartesian mesh. Several
new features have been implemented with respect to the earlier version
published in 2005. In particular, the numerical accuracy has been improved for
a given mesh size by (i) implementing a new solver to determine the Coulomb
potential for protons (ii) implementing a more precise method to calculate the
derivatives on a mesh that had already been implemented earlier in our
beyond-mean-field codes. The code has been made very flexible to enable the use
of a large variety of Skyrme energy density functionals that have been
introduced in the last years. Finally, the treatment of the constraints that
can be introduced in the mean-field equations has been improved. The code Ev8
is today the tool of choice to study the variation of the energy of a nucleus
from its ground state to very elongated or triaxial deformations with a
well-controlled accuracy.Comment: 24 pages, 3 figure
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