Searcharxiv⌕ Search

arXiv subjects

A. M. Ishkhanyan

Publications and source records attributed to A. M. Ishkhanyan.

At least 19 recordsLinked to original sources
arXiv2021

A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations

We review the series solutions of the general and single-confluent Heun equations in terms of powers, ordinary-hypergeometric and confluent-hypergeometric functions. The conditions under which the expansions reduce to finite sums as well as the cases when the coefficients of power-series expansions obey two-term recurrence relations are discussed. Infinitely many cases for which a solution of a general or single-confluent Heun equation is written through a single generalized hypergeometric function are indicated. It is shown that the parameters of this generalized hypergeometric function are the roots of a certain polynomial equation, which we explicitly present for any given order.

math.CA↗
arXiv2020

Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation

We apply the Darboux transformation to construct new exactly-solvable cases of the two-dimensional massless Dirac equation for potential classes of Lambert-W and inverse exponential type. Both of these classes originate from the Heun equation. Conditions are devised for transformed potentials to be real-valued, and to be in terms of elementary functions.

quant-ph↗
arXiv2020

Exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$

We present the exact solution of the 1D Dirac equation for the inverse-square-root potential $1/\sqrt{x}$ for several configurations of vector, pseudo-scalar and scalar fields. Each fundamental solution of the problem can be written as an irreducible linear combination of two Hermite functions of a scaled and shifted argument. We derive the exact equations for bound-state energy eigenvalues and construct accurate approximations for the energy spectrum.

quant-ph↗
arXiv2020

Inverse square root level-crossing quantum two-state model

We introduce a new unconditionally solvable level-crossing two-state model given by a constant-amplitude optical field configuration for which the detuning is an inverse-square-root function of time. This is a member of one of the five families of bi-confluent Heun models. We prove that this is the only non-classical exactly solvable field configuration among the bi-confluent Heun classes, solvable in terms of finite sums of the Hermite functions. The general solution of the two-state problem for this model is written in terms of four Hermite functions of a shifted and scaled argument (each of the two fundamental solutions presents an irreducible combination of two Hermite functions). We present the general solution, rewrite it in terms of more familiar physical quantities and analyze the time dynamics of a quantum system subject to excitation by a laser field of this configuration.

quant-ph↗
arXiv2020

Exactly-solvable quantum systems in terms of Lambert-W functions

We construct a variety of new exactly-solvable quantum systems, the potentials of which are given in terms of Lambert-W functions. In particular, we generate Schrödinger models with energy-dependent potentials, conventional Schrödinger models using the supersymmetry formalism, and two-dimensional Dirac systems. In addition, we derive Wronskian integral formulas for Lambert-W functions.

quant-ph↗
arXiv2020

Comment on the paper "Third-harmonic generation investigated by a short-range bottomless exponential potential well" by M. Hu, K. Guo, Q. Yu, Z. Zhang [Superlattices and Microstructures, 122 (2018) 538-547]

We have discovered several severe errors in the recent paper M. Hu, K. Guo, Q. Yu, Z. Zhang [Superlattices and Microstructures, 122 (2018) 538-547]. Specifically, we demonstrate that both the solution of the Schrödinger equation and the bound-state wave functions used in the paper are incorrect.

quant-ph↗
arXiv2020

Generalized hypergeometric solutions of the Heun equation

We present infinitely many solutions of the general Heun equation in terms of generalized hypergeometric functions. Each solution assumes that two restrictions are imposed on the involved parameters: a characteristic exponent of one of the singularities should be a non-zero integer, and the accessory parameter must satisfy a polynomial equation.

math.CA↗
arXiv2020

Hermite function solutions of the Schrödinger equation for the sextic oscillator

We examine the conditions under which the solution of the radial stationary Schrödinger equation for the sextic anharmonic oscillator can be expanded in terms of Hermite functions. We find that this is possible for an infinite hierarchy of potentials discriminated by the parameter setting the strength of the centrifugal barrier. The $N$'th member of the hierarchy involves $N$ solutions for $N$ generally different values of the energy. For a particular member of the hierarchy, there exist infinitely many bound states with square integrable wave functions, written in terms of the Hermite functions, which vanish at the origin and at infinity. These bound states correspond to distinct values of the parameter setting the strength of the harmonic term. We also investigate connection with the polynomial solutions of the sextic oscillator obtained from the formalism of quasi-exactly solvable potentials.

quant-ph↗
arXiv2019

Confluent hypergeometric expansions of the confluent Heun function governed by two-term recurrence relations

We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansion coefficients are explicitly calculated in terms of the Euler gamma functions.

math.CA↗
arXiv2019

Generalized-hypergeometric solutions of the biconfluent Heun equation

We examine the power-series solutions and the series solutions in terms of the Hermite functions for the biconfluent Heun equation. Infinitely many cases for which a solution of the biconfluent equation is presented as an irreducible linear combination of four generalized hypergeometric functions, that in general do not reduce to polynomials, are identified.

math.CA↗
arXiv2019

A Schrödinger potential involving $x^\frac{2}{3}$ and centrifugal-barrier terms conditionally integrable in terms of the confluent hypergeometric functions

The solution of the one-dimensional Schrödinger equation for a potential involving an attractive $x^\frac{2}{3}$ and a repulsive centrifugal-barrier $\sim x^{-2}$ terms is presented in terms of the non-integer-order Hermite functions. The potential belongs to one of the five bi-confluent Heun families. This is a conditionally integrable potential in that the strength of the centrifugal-barrier term is fixed. The general solution of the problem is composed using fundamental solutions each of which presents an irreducible linear combination of two Hermite functions of a scaled and shifted argument. The potential presents an infinitely extended confining well defined on the positive semi-axis and sustains infinitely many bound states.

quant-ph↗
arXiv2019

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the $δ=0$ choice the quartic potential term is cancelled, leading to the {\it reduced} sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the quasi-exactly solvable formalism. Possible generalizations of the method are also presented.

quant-ph↗
arXiv2019

WKB-approach for the 1D hydrogen atom

Taking into account results of WKB-approximation, we derive exact quantum energies and wave functions of even and odd states in the one-dimensional Coulomb potential

quant-ph↗
arXiv2019

Bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle

The variety of bi-confluent Heun potentials for a stationary relativistic wave equation for a spinless particle is presented. The physical potentials and energy spectrum of this wave equation are related to those for a corresponding Schrödinger equation in the sense that all the potentials derived for the latter equation are also applicable for the wave equation under consideration. We show that in contrast to the Schrödinger equation the characteristic spatial length of the potential imposes a restriction on the energy spectrum that directly reflects the uncertainty principle. Studying the inverse-square-root bi-confluent Heun potential, it is shown that the uncertainty principle limits, from below, the principal quantum number for the bound states, i.e., physically feasible states have an infimum cut so that the ground state adopts a higher quantum number as compared to the Schrödinger case.

quant-ph↗
arXiv2018

Low-frequency electromagnetic radiation of hydrogen molecular gas: effect of the ortho-para conversion

We calculate the rate of the spontaneous magnetic dipole radiation transitions from ortho- to para-states of the hydrogen molecule at the room temperature with the radiation wavelengths about 0.01 - 0.1 cm. Exponentially small differences between the rotational energies and heat capacities of parahydrogen and orthohydrogen molecules and also diatomic molecules consisting of different hydrogen atoms for high temperatures are derived.

physics.atom-ph↗
arXiv2018

Generalized confluent hypergeometric solutions of the Heun confluent equation

We show that the Heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions. For each of these solutions a characteristic exponent of a regular singularity of the Heun confluent equation is a non-zero integer and the accessory parameter obeys a polynomial equation. Each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the Kummer confluent hypergeometric functions or the Bessel functions.

math-ph↗
arXiv2018

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.

math.CA↗