54 research outputs found
New 1-step extension of the Swanson oscillator and superintegrability of its two-dimensional generalization
We derive a one-step extension of the well known Swanson oscillator that
describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected
to an extended harmonic oscillator model. Our analysis is based on the use of
the techniques of supersymmetric quantum mechanics and address various
representations of the ladder operators starting from a seed solution of the
harmonic oscillator given in terms of a pseudo-Hermite polynomial. The role of
the resulting chain of Hamiltonians related via similarity transformation is
then exploited. In the second part we write down a two dimensional
generalization of the Swanson Hamiltonian and establish superintegrability of
such a system.Comment: accepted for publication in Physics Letters
Comment on "Non-Hermitian Quantum Mechanics with Minimal Length Uncertainty"
We demonstrate that the recent paper by Jana and Roy entitled ''Non-Hermitian
quantum mechanics with minimal length uncertainty'' [SIGMA 5 (2009), 083, 7
pages, arXiv:0908.1755] contains various misconceptions. We compare with an
analysis on the same topic carried out previously in our manuscript
[arXiv:0907.5354]. In particular, we show that the metric operators computed
for the deformed non-Hermitian Swanson models differs in both cases and is
inconsistent in the former
Minimal areas from q-deformed oscillator algebras
We demonstrate that dynamical noncommutative space-time will give rise to
deformed oscillator algebras. In turn, starting from some q-deformations of
these algebras in a two dimensional space for which the entire deformed Fock
space can be constructed explicitly, we derive the commutation relations for
the dynamical variables in noncommutative space-time. We compute minimal areas
resulting from these relations, i.e. finitely extended regions for which it is
impossible to resolve any substructure in form of measurable knowledge. The
size of the regions we find is determined by the noncommutative constant and
the deformation parameter q. Any object in this type of space-time structure
has to be of membrane type or in certain limits of string type.Comment: 14 pages, 1 figur
Tracking down localized modes in PT-symmetric Hamiltonians under the influence of a competing nonlinearity
The relevance of parity and time reversal (PT)-symmetric structures in
optical systems is known for sometime with the correspondence existing between
the Schrodinger equation and the paraxial equation of diffraction where the
time parameter represents the propagating distance and the refractive index
acts as the complex potential. In this paper, we systematically analyze a
normalized form of the nonlinear Schrodinger system with two new families of
PT-symmetric potentials in the presence of competing nonlinearities. We
generate a class of localized eigenmodes and carry out a linear stability
analysis on the solutions. In particular, we find an interesting feature of
bifurcation charaterized by the parameter of perturbative growth rate passing
through zero where a transition to imaginary eigenvalues occurs.Comment: 10pages, To be published in Acta Polytechnic
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