11 research outputs found
Effects of complex parameters on classical trajectories of Hamiltonian systems
Anderson have shown that for complex energies, the classical
trajectories of quartic potentials are closed and periodic only
on a discrete set of eigencurves. Moreover, recently it was revealed that, when
time is complex certain real hermitian
systems possess close periodic trajectories only for a discrete set of values
of . On the other hand it is generally true that even for real
energies, classical trajectories of non - symmetric Hamiltonians
with complex parameters are mostly non-periodic and open. In this paper we show
that for given real energy, the classical trajectories of
quartic Hamiltonians , (where is real, is
complex and ) are closed and periodic only for a discrete set of
parameter curves in the complex -plane. It was further found that given
complex parameter , the classical trajectories are periodic for a discrete
set of real energies (i.e. classical energy get discretized or quantized by
imposing the condition that trajectories are periodic and closed). Moreover, we
show that for real and positive energies (continuous), the classical
trajectories of Hamiltonian are periodic when for
and .Comment: 9 pages, 2 tables, 6 figure
Explicit energy expansion for general odd degree polynomial potentials
In this paper we derive an almost explicit analytic formula for asymptotic
eigenenergy expansion of arbitrary odd degree polynomial potentials of the form
where s are real or complex for
. The formula can be used to find semiclassical analytic
expressions for eigenenergies up to any order very efficiently. Each term of
the expansion is given explicitly as a multinomial of the parameters and of the potential. Unlike in the even
degree polynomial case, the highest order term in the potential is pure
imaginary and hence the system is non-Hermitian. Therefore all the integrations
have been carried out along a contour enclosing two complex turning points
which lies within a wedge in the complex plane. With the help of some examples
we demonstrate the accuracy of the method for both real and complex
eigenspectra.Comment: 10 page
Non-Hermitian systems of Euclidean Lie algebraic type with real energy spectra
We study several classes of non-Hermitian Hamiltonian systems, which can be expressed in terms of bilinear combinations of Euclidean Lie algebraic generators. The classes are distinguished by different versions of antilinear (PT)-symmetries exhibiting various types of qualitative behaviour. On the basis of explicitly computed non-perturbative Dyson maps we construct metric operators, isospectral Hermitian counterparts for which we solve the corresponding time-independent Schr\"{o}dinger equation for specific choices of the coupling constants. In these cases general analytical expressions for the solutions are obtained in the form of Mathieu functions, which we analyze numerically to obtain the corresponding energy spectra. We identify regions in the parameter space for which the corresponding spectra are entirely real and also domains where the PT symmetry is spontaneously broken and sometimes also regained at exceptional points. In some cases it is shown explicitly how the threshold region from real to complex spectra is characterized by the breakdown of the Dyson maps or the metric operator. We establish the explicit relationship to models currently under investigation in the context of beam dynamics in optical lattices
Spontaneous PT-Symmetry Breaking for Systems of Noncommutative Euclidean Lie Algebraic Type
We propose a noncommutative version of the Euclidean Lie algebra E 2. Several types of non-Hermitian Hamiltonian systems expressed in terms of generic combinations of the generators of this algebra are investigated. Using the breakdown of the explicitly constructed Dyson maps as a criterium, we identify the domains in the parameter space in which the Hamiltonians have real energy spectra and determine the exceptional points signifying the crossover into the different types of spontaneously broken PT-symmetric regions with pairs of complex conjugate eigenvalues. We find exceptional points which remain invariant under the deformation as well as exceptional points becoming dependent on the deformation parameter of the algebra
New solitary wave solutions and stability analysis for the generalized (3+1)-dimensional nonlinear wave equation in liquid with gas bubbles
In this study, solitary wave solutions for the generalized (3+1)-dimensional nonlinear wave equation (NLWE) are extracted using a new generalized exponential rational function method (GERFM). Numerous nonlinear behaviors in liquids with gas bubbles are described by this equation. The suggested method is used to derive the various kinds of new accurate soliton solutions for the equation. Also, the physical interpretations of some obtained solutions are represented. In addition, a modulational instability analysis framework is used to look at the system’s stability