Dissociation of quarkonium in a strong electric field
The probability of tunnel decay of quarkonium (bond state of a heavy quark and a heavy antiquark) into free quarks in a strong electric field is estimated.
hep-ph↗
arXiv subjects
A. M. Ishkhanyan
Publications and source records attributed to A. M. Ishkhanyan.
Hypergeometric expansions of the general Heun function governed by two-term recurrence relations
We show that there exist infinitely many particular choices of parameters for which the three-term recurrence relations governing the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions become two-term. In these cases the coefficients are explicitly written in terms of the gamma functions.
math.CA↗
A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
We present a conditionally integrable potential, belonging to the bi-confluent Heun class, for which the Schrödinger equation is solved in terms of the confluent hypergeometric functions. The potential involves an attractive inverse square root term with arbitrary strength and a repulsive centrifugal barrier core with the strength fixed to a constant. This is a potential well defined on the half-axis. Each of the fundamental solutions composing the general solution of the Schrödinger equation is written as an irreducible linear combination, with non-constant coefficients, of two confluent hypergeometric functions. We present the explicit solution in terms of the non-integer order Hermite functions of scaled and shifted argument and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and approximate that by a highly accurate transcendental equation involving trigonometric functions. Finally, we construct an accurate approximation for the bound-state energy levels.
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A new exactly integrable hypergeometric potential for the Schrödinger equation
We introduce a new exactly integrable potential for the Schrödinger 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.
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Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
We introduce an exactly integrable singular potential for which the solution of the one-dimensional stationary Schrödinger equation is written through irreducible linear combinations of the Gauss hypergeometric functions. The potential, which belongs to a general Heun family, is a short-range one that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. We derive the exact spectrum equation for the energy and discuss the bound states supported by the potential.
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A conditionally integrable Schrödinger potential of a bi-confluent Heun class
We present a bi-confluent Heun potential for the Schrödinger equation involving inverse fractional powers and a repulsive centrifugal-barrier term the strength of which is fixed to a constant. This is an infinite potential well defined on the positive half-axis. Each of the fundamental solutions for this conditionally integrable potential is written as an irreducible linear combination of two Hermite functions of a shifted and scaled argument. We present the general solution of the problem, derive the exact energy spectrum equation and construct a highly accurate approximation for the bound-state energy levels.
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Confluent hypergeometric expansions of the solutions of the double-confluent Heun equation
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. The latter relation is reduced to a three-term one only in the case when the double-confluent Heun equation degenerates to the confluent hypergeometric equation. The conditions for obtaining finite sum solutions via termination of the expansions are discussed. The possibility of constructing expansions of different structure using certain equations related to the double-confluent Heun equation is discussed. An example of such expansion derived using the equation obeyed by a function involving the derivative of a solution of the double-confluent Heun equation is presented. In this way, expansions governed by three- or more term recurrence relations for expansion coefficients can be constructed. An expansion with coefficients obeying a seven-term recurrence relation is presented. This relation is reduced to a five-term one if the additional singularity of the equation obeyed by the considered auxiliary function coincides with a singularity of the double-confluent Heun equation.
math.CA↗
The third five-parametric hypergeometric quantum-mechanical potential
We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and Pöschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schrödinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.
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Schrödinger potentials solvable in terms of the general Heun functions
We show that there exist 35 choices for the coordinate transformation each leading to a potential for which the stationary Schrödinger equation is exactly solvable in terms of the general Heun functions. Because of the symmetry of the Heun equation with respect to the transposition of its singularities only eleven of these potentials are independent. Four of these independent potentials are always explicitly written in terms of elementary functions, one potential is given through the Jacobi elliptic sn-function, and the others are in general defined parametrically. Nine of the independent potentials possess exactly or conditionally integrable hypergeometric sub-potentials for which each of the fundamental solutions of the Schrödinger equation is written through a single hypergeometric function. Many of the potentials possess sub-potentials for which the general solution is written through fundamental solutions each of which is a linear combination of two or more Gauss hypergeometric functions. We present an example of such a potential which is a conditionally integrable generalization of the third exactly solvable Gauss hypergeometric potential.
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A periodic level-crossing two-state model of a general Heun class
We present a specific constant-amplitude periodic level-crossing model of the semi-classical quantum time-dependent two-state problem that belongs to a general Heun class of field configurations. The exact analytic solution for the probability amplitude, generally written for this class in terms of the general Heun functions, in this specific case admits series expansion in terms of the incomplete Beta functions. Terminating this series results in an infinite hierarchy of finite-sum closed-form solutions each standing for a particular two-state model, which generally is only conditionally integrable in the sense that for these field configurations the amplitude and phase modulation functions are not varied independently. However, there exists at least one exception when the model is unconditionally integrable, that is the Rabi frequency and the detuning of the driving optical field are controlled independently. This is a constant-amplitude periodic level-crossing model, for which the detuning in a limit becomes a Dirac delta-comb configuration with variable frequency of the level-crossings. We derive the exact solution for this model, determine the Floquet exponents and study the population dynamics in the system for various regions of the input parameters.
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Solutions of the bi-confluent Heun equation in terms of the Hermite functions
We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.
quant-ph↗
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.
math-ph↗
Semiclassical rates for tunnel ionization from power-law potentials induced by a constant or low-frequency electric field
The probabilities of tunnel ionization of particles confined by one-dimensional power-law and logarithmic potentials are calculated for constant and low-frequency electric fields.
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Maslov index for power-law potentials
The Maslov index in the semiclassical Bohr-Sommerfeld quantization rule is calculated for one-dimensional power-law potentials. The result for the inverse square root potential is compared with the recently reported exact solution. The case of a spherically symmetric power-law potential is also considered.
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Generation and removal of apparent singularities in linear ordinary differential equations with polynomial coefficients
We discuss several examples of generating apparent singular points as a result of differentiating particular homogeneous linear ordinary differential equations with polynomial coefficients and formulate two general conjectures on the generation and removal of apparent singularities in arbitrary Fuchsian differential equations with polynomial coefficients. We consider a model problem in polymer physics.
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A conditionally exactly solvable generalization of the inverse square root potential
We present a conditionally exactly solvable singular potential for the one-dimensional Schrödinger equation which involves the exactly solvable inverse square root potential. Each of the two fundamental solutions that compose the general solution of the problem is given by a linear combination with non-constant coefficients of two confluent hypergeometric functions. Discussing the bound-state wave functions vanishing both at infinity and in the origin, we derive the exact equation for the energy spectrum which is written using two Hermite functions of non-integer order. In specific auxiliary variables this equation becomes a mathematical equation that does not refer to a specific physical context discussed. In the two-dimensional space of these auxiliary variables the roots of this equation draw a countable infinite set of open curves with hyperbolic asymptotes. We present an analytic description of these curves by a transcendental algebraic equation for the involved variables. The intersections of the curves thus constructed with a certain cubic curve provide a highly accurate description of the energy spectrum.
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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 presented
math-ph↗
A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schrödinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one it supports only a finite number of bound states.
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