14 research outputs found
Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
We examine the expansions of the solutions of the general Heun equation in
terms of the Gauss hypergeometric functions. We present several expansions
using functions, the forms of which differ from those applied before. In
general, the coefficients of the expansions obey three-term recurrence
relations. However, there exist certain choices of the parameters for which the
recurrence relations become two-term. The coefficients of the expansions are
then explicitly expressed in terms of the gamma functions. Discussing the
termination of the presented series, we show that the finite-sum solutions of
the general Heun equation in terms of generally irreducible hypergeometric
functions have a representation through a single generalized hypergeometric
function. Consequently, the power-series expansion of the Heun function for any
such case is governed by a two-term recurrence relation
Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions
We derive 35 five-parametric classes of the quantum time-dependent two-state
models solvable in terms of the general Heun functions. Each of the classes is
defined by a pair of generating functions the first of which is referred to as
the amplitude- and the second one as the detuning-modulation function. The
classes suggest numerous families of specific field configurations with
different physical properties generated by appropriate choices of the
transformation of the independent variable, real or complex. There are many
families of models with constant detuning or constant amplitude, numerous
classes of chirped pulses of controllable amplitude and/or detuning, families
of models with double or multiple (periodic) crossings, periodic amplitude
modulation field configurations, etc. We present several families of
constant-detuning field configurations the members of which are symmetric or
asymmetric two-peak finite-area pulses with controllable distance between the
peaks and controllable amplitude of each of the peaks. We show that the edge
shapes, the distance between the peaks as well as the amplitude of the peaks
are controlled almost independently, by different parameters. We identify the
parameters controlling each of the mentioned features and discuss other basic
properties of pulse shapes. We show that the pulse edges may become step-wise
functions and determine the positions of the limiting vertical-wall edges. We
show that the pulse width is controlled by only two of the involved parameters.
For some values of these parameters the pulse width diverges and for some other
values the pulses become infinitely narrow. We show that the effect of the two
mentioned parameters is almost similar, that is, both parameters are able to
independently produce pulses of almost the same shape and width
Expansions of the solutions of the general Heun equation in terms of the incomplete Beta functions
Applying the approach based on the equation for the derivative, we construct
several expansions of the solutions of the general Heun equation in terms of
the incomplete Beta functions. Several expansions in terms of the Appell
generalized hypergeometric functions of two variables of the fist kind are also
presented. The constructed expansions are applicable for arbitrary sets of the
involved parameters. The coefficients of the expansions obey four-, five- or
six-term recurrence relations. However, there exist several sets of the
parameters for which the recurrence relations involve fewer terms, not
necessarily successive. The conditions for deriving finite-sum solutions via
termination of the series are discussed.Comment: arXiv admin note: text overlap with arXiv:1505.0217
Complete-return spectrum for a generalized Rosen-Zener two-state term-crossing model
The general semiclassical time-dependent two-state problem is considered for
a specific field configuration referred to as the generalized Rosen-Zener
model. This is a rich family of pulse amplitude- and phase-modulation functions
describing both non-crossing and term-crossing models with one or two crossing
points. The model includes the original constant-detuning non-crossing
Rosen-Zener model as a particular case. We show that the governing system of
equations is reduced to a confluent Heun equation. When inspecting the
conditions for returning the system to the initial state at the end of the
interaction with the field, we reformulate the problem as an eigenvalue problem
for the peak Rabi frequency and apply the Rayleigh-Schr\"odinger perturbation
theory. Further, we develop a generalized approach for finding the higher-order
approximations, which is applicable for the whole variation region of the
involved input parameters of the system. We examine the general surface in the
3D space of input parameters, which defines the position of the n-th order
return-resonance, and show that the section of the general surface is
accurately approximated by an ellipse. We find a highly accurate analytic
description through the zeros of a Kummer confluent hypergeometric function.
From the point of view of the generality, the analytical description of
mentioned curve for the whole variation range of all involved parameters is the
main result of the present paper