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The Ramanujan Journal - Infinitely many cases for which two independent fundamental solutions of the biconfluent Heun equation can each be presented as an irreducible linear combination of two... 相似文献
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Theoretical and Mathematical Physics - We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two... 相似文献
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We consider the quantum above-barrier reflection of a particle by the one-dimensional Rosen-Morse potential well, for the nonlinear Schrödinger equation (the Gross-Pitaevskii equation) with a small nonlinearity. The most interesting case is realized in resonances when the reflection coefficient is exactly equal to zero for the linear Schrödinger equation. Then the reflection is determined by only a small nonlinear term in the Gross-Pitaevskii equation. The simple analytic expression is obtained for the reflection coefficient produced only by the nonlinearity. The analytic condition is found for the common action of the potential well and the nonlinearity to produce the zero reflection coefficient. The reflection coefficient is also derived analytically in the vicinity of a resonance shifted by the nonlinearity. 相似文献
4.
A. M. Ishkhanyan 《Theoretical and Mathematical Physics》2016,188(1):980-993
We show that if the potential is proportional to an energy-independent continuous parameter, then there exist 15 choices for the coordinate transformation that provide energy-independent potentials whose shape is independent of that parameter and for which the one-dimensional stationary Schrödinger equation is solvable in terms of the confluent Heun functions. All these potentials are also energy-independent and are determined by seven parameters. Because the confluent Heun equation is symmetric under transposition of its regular singularities, only nine of these potentials are independent. Five of the independent potentials are different generalizations of either a hypergeometric or a confluent hypergeometric classical potential, one potential as special cases includes potentials of two hypergeometric types (the Morse confluent hypergeometric and the Eckart hypergeometric potentials), and the remaining three potentials include five-parameter conditionally integrable confluent hypergeometric potentials. Not one of the confluent Heun potentials, generally speaking, can be transformed into any other by a parameter choice. 相似文献
5.
R. Sokhoyan H. Azizbekyan C. Leroy A. Ishkhanyan 《Journal of Experimental and Theoretical Physics》2011,112(4):543-550
We discuss the strong-coupling regime of the nonlinear Landau-Zener problem occurring at coherent photo- and magneto-association of ultracold atoms. We apply a variational approach to an exact third-order nonlinear differential equation for the molecular state probability and construct an accurate approximation describing the time dynamics of the coupled atom-molecule system. The resultant solution improves the accuracy of the previous approximation [22]. The obtained results reveal a remarkable observation that in the strong-coupling limit, the resonance crossing is mostly governed by the nonlinearity, while the coherent atom-molecule oscillations occurring soon after crossing the resonance are principally of a linear nature. This observation is supposedly general for all nonlinear quantum systems having the same generic quadratic nonlinearity, due to the basic attributes of the resonance crossing processes in such systems. The constructed approximation turns out to have a larger applicability range than it was initially expected, covering the whole moderate-coupling regime for which the proposed solution accurately describes ail the main characteristics of the system evolution except the amplitude of the coherent atom-molecule oscillation, which is rather overestimated. 相似文献
6.
Avrutskiy V. I. Ishkhanyan A. M. Krainov V. P. 《Theoretical and Mathematical Physics》2021,208(3):1201-1206
Theoretical and Mathematical Physics - The fractional derivative method is used to take wave dispersion into account in the wave equation when describing the propagation of gravitational soliton... 相似文献
7.
Resonant reflection of a Bose-Einstein condensate by a double delta-function barrier has been considered analytically using the Gross-Pitaevskii approximation for nonlinearity. The reflection coefficient has been derived taking into account a weak nonlinearity of the Schrödinger equation produced by the interaction between cold alkali atoms. Nonlinear term is given in the limit of asymptotically weak interaction and zero temperature. The one-dimensional potential is approximated by two repulsive delta-function barriers. The analytical solution was obtained for the reflection coefficient using a multiple-scale analysis in order to remove secular terms. The most interesting case corresponds to the condensate energies for which reflection is absent without nonlinear term. Thus, reflection is determined only by the nonlinearity. The reflection coefficient is derived in the first order on the nonlinearity parameter. 相似文献
8.
R. S. Sokhoyan H. H. Azizbekyan C. Leroy A. M. Ishkhanyan 《Journal of Contemporary Physics (Armenian Academy of Sciences)》2009,44(6):272-277
We study the nonlinear mean-field dynamics of molecule formation at coherent photo- and magneto-association of an atomic Bose-Einstein
condensate for the case when the external field configuration is defined by the quasi-linear level crossing Demkov-Kunike
model, characterized by a bell-shaped pulse and finite variation of the detuning. We present a general approach to construct
an approximation describing the temporal dynamics of the molecule formation in the weak interaction regime and apply the developed
method to the nonlinear Demkov-Kunike problem. The presented approximation, written as a scaled solution to the linear problem
associated to the nonlinear one we treat, contains fitting parameters which are determined through a variational procedure.
Assuming that the parameters involved in the solution of the linear problem are not modified, we suggest an analytical expression
for the scaling parameter. 相似文献
9.
We have obtained the uncertainty relations for arbitrary states of the hydrogen atom. It is shown that the minimal value of the uncertainty relation (ΔprΔr → ?2) is attained for the circular Rydberg states. 相似文献
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