61 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
A new exactly integrable hypergeometric potential for the Schr\"odinger equation
We introduce a new exactly integrable potential for the Schr\"odinger
equation for which the solution of the problem may be expressed in terms of the
Gauss hypergeometric functions. This is a potential step with variable height
and steepness. We present the general solution of the problem, discuss the
transmission of a quantum particle above the barrier, and derive explicit
expressions for the reflection and transmission coefficients
Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
Starting from equations obeyed by functions involving the first or the second
derivatives of the biconfluent Heun function, we construct two expansions of
the solutions of the biconfluent Heun equation in terms of incomplete Beta
functions. The first series applies single Beta functions as expansion
functions, while the second one involves a combination of two Beta functions.
The coefficients of expansions obey four- and five-term recurrence relations,
respectively. It is shown that the proposed technique is potent to produce
series solutions in terms of other special functions. Two examples of such
expansions in terms of the incomplete Gamma functions are presente
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
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