516 research outputs found
Nonlinearity Management in Higher Dimensions
In the present short communication, we revisit nonlinearity management of the
time-periodic nonlinear Schrodinger equation and the related averaging
procedure. We prove that the averaged nonlinear Schrodinger equation does not
support the blow-up of solutions in higher dimensions, independently of the
strength in the nonlinearity coefficient variance. This conclusion agrees with
earlier works in the case of strong nonlinearity management but contradicts
those in the case of weak nonlinearity management. The apparent discrepancy is
explained by the divergence of the averaging procedure in the limit of weak
nonlinearity management.Comment: 9 pages, 1 figure
Solutions to Maxwell's Equations using Spheroidal Coordinates
Analytical solutions to the wave equation in spheroidal coordinates in the
short wavelength limit are considered. The asymptotic solutions for the radial
function are significantly simplified, allowing scalar spheroidal wave
functions to be defined in a form which is directly reminiscent of the
Laguerre-Gaussian solutions to the paraxial wave equation in optics.
Expressions for the Cartesian derivatives of the scalar spheroidal wave
functions are derived, leading to a new set of vector solutions to Maxwell's
equations. The results are an ideal starting point for calculations of
corrections to the paraxial approximation
Solvable two-dimensional time-dependent non-Hermitian quantum systems with infinite dimensional Hilbert space in the broken PT-regime
We provide exact analytical solutions for a two-dimensional explicitly time-dependent non-Hermitian quantum system. While the time-independent variant of the model studied is in the broken PT-symmetric phase for the entire range of the model parameters, and has therefore a partially complex energy eigenspectrum, its time-dependent version has real energy expectation values at all times. In our solution procedure we compare the two equivalent approaches of directly solving the time-dependent Dyson equation with one employing the Lewis–Riesenfeld method of invariants. We conclude that the latter approach simplifies the solution procedure due to the fact that the invariants of the non-Hermitian and Hermitian system are related to each other in a pseudo-Hermitian fashion, which in turn does not hold for their corresponding time-dependent Hamiltonians. Thus constructing invariants and subsequently using the pseudo-Hermiticity relation between them allows to compute the Dyson map and to solve the Dyson equation indirectly. In this way one can bypass to solve nonlinear differential equations, such as the dissipative Ermakov–Pinney equation emerging in our and many other systems
Lie systems and integrability conditions for t-dependent frequency harmonic oscillators
Time-dependent frequency harmonic oscillators (TDFHO's) are studied through
the theory of Lie systems. We show that they are related to a certain kind of
equations in the Lie group SL(2,R). Some integrability conditions appear as
conditions to be able to transform such equations into simpler ones in a very
specific way. As a particular application of our results we find t-dependent
constants of the motion for certain one-dimensional TDFHO's. Our approach
provides an unifying framework which allows us to apply our developments to all
Lie systems associated with equations in SL(2,R) and to generalise our methods
to study any Lie system
Ermakov-Lewis angles for one-parameter supersymmetric families of Newtonian free damping modes
We apply the Ermakov-Lewis procedure to the one-parameter damped modes
\tilde{y} recently introduced by Rosu and Reyes, which are related to the
common Newtonian free damping modes y by the general Riccati solution [H.C.
Rosu and M. Reyes, Phys. Rev. E 57, 4850 (1998), physics/9707019]. In
particular, we calculate and plot the angle quantities of this approach that
can help to distinguish these modes from the common y modesComment: 6 pages, twocolumn, 18 figs embedded, only first 9 publishe
Generalizing the autonomous Kepler Ermakov system in a Riemannian space
We generalize the two dimensional autonomous Hamiltonian Kepler Ermakov
dynamical system to three dimensions using the sl(2,R) invariance of Noether
symmetries and determine all three dimensional autonomous Hamiltonian Kepler
Ermakov dynamical systems which are Liouville integrable via Noether
symmetries. Subsequently we generalize the autonomous Kepler Ermakov system in
a Riemannian space which admits a gradient homothetic vector by the
requirements (a) that it admits a first integral (the Riemannian Ermakov
invariant) and (b) it has sl(2,R) invariance. We consider both the
non-Hamiltonian and the Hamiltonian systems. In each case we compute the
Riemannian Ermakov invariant and the equations defining the dynamical system.
We apply the results in General Relativity and determine the autonomous
Hamiltonian Riemannian Kepler Ermakov system in the spatially flat Friedman
Robertson Walker spacetime. We consider a locally rotational symmetric (LRS)
spacetime of class A and discuss two cosmological models. The first
cosmological model consists of a scalar field with exponential potential and a
perfect fluid with a stiff equation of state. The second cosmological model is
the f(R) modified gravity model of {\Lambda}_{bc}CDM. It is shown that in both
applications the gravitational field equations reduce to those of the
generalized autonomous Riemannian Kepler Ermakov dynamical system which is
Liouville integrable via Noether integrals.Comment: Reference [25] update, 21 page
Unified Treatment of Heterodyne Detection: the Shapiro-Wagner and Caves Frameworks
A comparative study is performed on two heterodyne systems of photon
detectors expressed in terms of a signal annihilation operator and an image
band creation operator called Shapiro-Wagner and Caves' frame, respectively.
This approach is based on the introduction of a convenient operator
which allows a unified formulation of both cases. For the Shapiro-Wagner
scheme, where , quantum phase and amplitude
are exactly defined in the context of relative number state (RNS)
representation, while a procedure is devised to handle suitably and in a
consistent way Caves' framework, characterized by , within the approximate simultaneous measurements of
noncommuting variables. In such a case RNS phase and amplitude make sense only
approximately.Comment: 25 pages. Just very minor editorial cosmetic change
On Close Relationship between Classical Time-Dependent Harmonic Oscillator and Non-Relativistic Quantum Mechanics in One Dimension
In this paper, I present a mapping between representation of some quantum
phenomena in one dimension and behavior of a classical time-dependent harmonic
oscillator. For the first time, it is demonstrated that quantum tunneling can
be described in terms of classical physics without invoking violations of the
energy conservation law at any time instance. A formula is presented that
generates a wide class of potential barrier shapes with the desirable
reflection (transmission) coefficient and transmission phase shift along with
the corresponding exact solutions of the time-independent Schr\"odinger's
equation. These results, with support from numerical simulations, strongly
suggest that two uncoupled classical harmonic oscillators, which initially have
a 90\degree relative phase shift and then are simultaneously disturbed by the
same parametric perturbation of a finite duration, manifest behavior which can
be mapped to that of a single quantum particle, with classical 'range
relations' analogous to the uncertainty relations of quantum physics.Comment: 20 pages, 8 figures, 1 table, final versio
Superposition rules for higher-order systems and their applications
Superposition rules form a class of functions that describe general solutions
of systems of first-order ordinary differential equations in terms of generic
families of particular solutions and certain constants. In this work we extend
this notion and other related ones to systems of higher-order differential
equations and analyse their properties. Several results concerning the
existence of various types of superposition rules for higher-order systems are
proved and illustrated with examples extracted from the physics and mathematics
literature. In particular, two new superposition rules for second- and
third-order Kummer--Schwarz equations are derived.Comment: (v2) 33 pages, some typos corrected, added some references and minor
commentarie
Milne quantization for non-Hermitian systems
We generalize the Milne quantization condition to non-Hermitian systems. In the general case the underlying nonlinear Ermakov–Milne–Pinney equation needs to be replaced by a nonlinear integral differential equation. However, when the system is PT-symmetric or/and quasi/pseudo-Hermitian the equations simplify and one may employ the original energy integral to determine its quantization. We illustrate the working of the general framework with the Swanson model and two explicit examples for pairs of supersymmetric Hamiltonians. In one case both partner Hamiltonians are Hermitian and in the other a Hermitian Hamiltonian is paired by a Darboux transformation to a non-Hermitian one
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