Three-term Recursion Relation Research Articles

Three-body Coulomb problem in the dipole approximation: and states

In the asymmetric doubly excited states of the two-electron atom one electron is excited much more than the other. The characteristic separation of the strongly excited electron from the atomic nucleus significantly exceeds that for the weakly excited electron . The electron - electron interaction is approximated by the first non-trivial (dipole) term of its expansion over . The same approach is easily extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. The dynamics of three-body states with and symmetry is analysed in quantum, semiclassical and classical approximations in comparison with the S and states considered previously. Mathematically, the problem is reduced to two coupled three-term recursion relations which were not studied previously. The exact and approximate (semiclassical) degeneracies of the levels with different values of the total orbital momentum L are revealed, and the energies of the `planetary atom' states are estimated.

On a generalized oscillator system: Interbasis expansions

This article deals with a nonrelativistic quantum mechanical study of a dynamical system which generalizes the isotropic harmonic oscillator system in three dimensions. The Schrodinger equation for this generalized oscillator system is separable in spherical, cylindrical, and spheroidal (prolate and oblate) coordinates. The quantum mechanical spectrum of this system is worked out in some details. The problem of interbasis expansions of the wavefunctions is completely solved. The coefficients for the expansion of the cylindrical basis in terms of the spherical basis, and vice-versa, are found to be analytic continuations (to real values of their arguments) of Clebsch-Gordan coefficients for the group SU(2). The interbasis expansion coefficients for the prolate and oblate spheroidal bases in terms of the spherical or the cylindrical bases are shown to satisfy three-term recursion relations. Finally, a connection between the generalized oscillator system (projected on the z-line) and the Morse system (in one dimension) is discussed.

Quasi-exactly solvable potentials on the line and orthogonal polynomials

In this paper we show that a quasi-exactly solvable (normalizable or periodic) one-dimensional Hamiltonian satisfying very mild conditions defines a family of weakly orthogonal polynomials which obey a three-term recursion relation. In particular, we prove that (normalizable) exactly solvable one-dimensional systems are characterized by the fact that their associated polynomials satisfy a two-term recursion relation. We study the properties of the family of weakly orthogonal polynomials defined by an arbitrary one-dimensional quasi-exactly solvable Hamiltonian, showing in particular that its associated Stieltjes measure is supported on a finite set. From this we deduce that the corresponding moment problem is determined, and that the kth moment grows like the kth power of a constant as k tends to infinity. We also show that the moments satisfy a constant coefficient linear difference equation, and that this property actually characterizes weakly orthogonal polynomial systems.

Off-diagonal matrix elements < nl | r k | nl' > for hydrogen-like states: an exact correspondence relationship in terms of orthogonal polynomials and the WKB approximation

A closed formula for and interpretation of exact quantum mechanical off-diagonal matrix elements of powers of r is given in terms of orthogonal polynomials. The compact formulae for r-k-2 and rk-1 are in exact correspondence to the semiclassical (diagonal) expectation values and are most suitable for the practical evaluation of the matrix elements. The expressions for < nl | r-k-2 | l'n > and < nl | rk-1 | l'n > consist only of the minimum of n-l or k+1 terms. An off-diagonal generalization of the well known diagonal inversion formula is derived and the matrix elements can thus be evaluated for arbitrary (complex) powers k. Also, an analytic continuation to an arbitrary effective n* is possible. It is thus established that, complementary to the O(2,1) approach by Armstrong (1970, 1971), orthogonality properties for purely radial quantities can be obtained without the introduction of a fictitious angular variable. Using the analytical results obtained, certain deficiencies of the semiclassical Coulomb (or WKB) approximation are discussed. The normalization error involved in the WKB approximation is identified analytically and an improved version of the semiclassical limit, a correspondence limit, is derived. Furthermore, it is established that, like the exact results, the WKB matrix elements obey a three-term recursion relation and an inversion formula. For low k some explicit formulae for exact matrix elements are tabulated.

Arbitrary mode number boundary-layer theory for nonideal toroidal Alfvén modes

The theory of toroidicity-induced Alfvén eigenmodes (TAE) and kinetic TAE (KTAE) is generalized to arbitrary mode numbers for a large aspect ratio low-beta circular tokamak. The interaction between nearest neighbors is described by a three-term recursion relation that combines elements from an outer region, described by the ideal magnetohydrodynamic equations of a cylinder, and an inner region, which includes the toroidicity and the nonideal effects of finite ion Larmor radius, electron inertia, and collisions. By the use of quadratic forms, it is proven that the roots of the recursion relation are stable and it is shown how perturbation theory can be applied to include frequency shifts due to other kinetic effects. Analytic forms are derived which display the competition between the resistive and radiative damping, where the radiation is carried by kinetic Alfvén waves. When the nonideal parameter is small, the KTAE modes appear in pairs. When this parameter is large, previously found scaling for the single gap case is reproduced analytically.

Dipole-dipole interaction of two excited hydrogen atoms.

The interaction of two excited hydrogen atoms at large separations R is governed by the dipole-dipole term (\ensuremath{\simeq}${\mathit{R}}^{\mathrm{\ensuremath{-}}3}$). The potential-curve asymptotes are obtained by diagonalizing the dipole-dipole-interaction operator in the subspace of the degenerate unperturbed states. The symmetry properties are used for the reduction of the matrix size. In some special cases the eigenvalues are found analytically. For the interaction of highly excited atoms an approximate semiclassical quantization procedure is developed. The problem is reformulated via the three-dimensional three-term recursion relations. The discrete (index) variables are converted into continuous variables and approximate (adiabatic-type) separation of variables is achieved. This makes the problem effectively one dimensional and provides a qualitative interpretation of the spectrum structure. The simplest explicit expression for the spectrum is obtained by using the harmonic approximation in the index space. It applies for the extreme eigenvalues.

Discrete semiclassical methods in the theory of Rydberg atoms in external fields

The properties of the WKB method in the discrete representation are reviewed. The method provides the eigenvalues and the eigenvectors of the three-term recursion relations or, which is the same thing, the tridiagonal band matrices. Applications of the method to the splitting of the Rydberg atom levels in the external electric and magnetic fields are considered. Analytical treatment is given to the problem of the oscillator strength distribution in the quadratic Zeeman and the Stark-Zeeman spectra. In the case of the nonhydrogenic Rydberg atoms, the effect of the core on the pattern of the splitting is studied. Certain alternative applications of the discrete WKB method are considered in brief (the quasienergy spectra of nonlinear oscillators in resonant fields, rotational molecular spectra, calculation of infinite continued fractions).

Mode structure and continuum damping of high-n toroidal Alfvén eigenmodes*

An asymptotic theory is described for calculating the mode structure and continuum damping of short-wavelength toroidal Alfvén eigenmodes (TAE). The formalism somewhat resembles the treatment used for describing low-frequency toroidal modes with singular structure at a rational surface, where an inner solution, which for the TAE mode has toroidal coupling, is matched to an outer toroidally uncoupled solution. A three-term recursion relation among coupled poloidal harmonic amplitudes is obtained, whose solution gives the structure of the global wave function and the complex eigenfrequency, including continuum damping. Both analytic and numerical solutions are presented. The magnitude of the damping is essential for determining the thresholds for instability driven by the spatial gradients of energetic particles (e.g., neutral-beam-injected ions or fusion-product alpha particles) contained in a tokamak plasma.

Three-body Coulomb problem in the dipole approximation.

The doubly excited states of the two-electron atom (or ion) are considered in the case when one electron is excited much more than the other. The characteristic separation of the strongly excited electron from the atomic nucleus significantly exceeds that for the weakly excited electron. The electron-electron interaction can be approximated by the dipole term in its multipole expansion. The inner electron velocity is assumed to be much larger than that of the outer electron, which permits assigning a fixed principal quantum number for the inner electron. The same approach is extended to the general Coulomb three-body problem with the particles of arbitrary mass and charge. For the three-body states with symmetry S and ${\mathit{P}}^{\mathit{e}}$, the quantum problem reduces to the three-term recursion relations, which means that the system is effectively one dimensional. The slow evolution of the classical particle trajectories under the influence of interaction is described. It is shown that, depending on the parameters of the system, two types of librations or rotation can be realized. The semiclassical quantization rules (analogous to the well-known Bohr-Sommerfeld rules) are deduced. For S and ${\mathit{P}}^{\mathit{e}}$ states they differ only by the substitution of a half-integer quantum number by an integer. Particularly simple results are obtained in the harmonic approximation, which is valid in the vicinity of the effective-potential curve extreme.

Generalized polyspheroidal periodic functions and the quantum inverse scattering method

Bounded and π-periodic eigenfunctions of the operator d2/dv2+[2(μ−ν +(μ+ν+1)cos 2v)/sin 2v] (d/dv)+8ξ cos 2v+4γ2 sin2 2v are studied. These solutions are new special functions, particular cases of which will be the spheroidal, the Coulomb spheroidal, and the hyperspheroidal harmonics. The algebraic technique due to the quantum inverse scattering method is applied to obtain the three-term recursion relations for the coefficients of eigenfunctions expansion at the Jacobi polynomials series and hence the suitable equation (the continued fraction is equal to zero) for eigenvalues. The special functions considered generalize the Truskova polyspheroidal periodic functions (γ=0).

Recursion Relations for Solutions to the Schrödinger Equation

We will consider the general eigenfunction for a second-order differential operator in one variable. For many well-known elementary functions, we can also find a three-term recursion relation in the eigenvalue parameter. For practical computation, this is a very desirable property. Examples include Legendre functions, Bessel functions, etc. In [Math. Z., 29 (1929), pp. 730–736], Bochner showed that the only polynomial solutions to this problem were the well-known ones. This paper will look for solutions that may not be polynomials. It will be shown that, unfortunately, for the simple recursion relation considered here, no really new examples exist.

Exact evaluation and recursion relations of two-center harmonic oscillator matrix elements

Using vibrational wave functions of two relatively displaced harmonic oscillators of arbitrary frequencies, Franck–Condon overlap integrals and matrix elements of xl, exp(−2cx), and exp(−cx2) (x is the internuclear separation) are obtained. Useful three-term, four-term, and five-term recursion relations among these matrix elements are derived. It is shown that all of the relevant matrix elements can be obtained from a mere knowledge of the lowest two Franck–Condon overlap integrals. Results are illustrated by computation of Franck–Condon factors for the A 1∑+u–X 1∑+g and the B 1Πu–X 1∑+g systems of 7Li2.

Elliptic basis for a circular oscillator

This paper presents a complete set of mutually commuting operators determining an elliptic basis for a quantum circular oscillator. The elliptic basis of the circular oscillator is introduced and three-term recursion relations generating it are found. The elliptic corrections to the polar and Cartesian bases of the circular oscillator are calculated.

Three-term recursion relations for hydrogen wave functions: Exact calculations and semiclassical approximations

Three-term recursion relations with respect to the angular momentum l are given for the normalized hydrogen wave functions associated with r, pr, p (r is the radial polar coordinate in configuration representation, pr is the momentum conjugated to r, p is the radial polar coordinate in momentum representation). These three-term recursion relations [Eqs. (4), (6), (16)] are found numerically stable in the order of decreasing l values, even for large quantum numbers. The three-term recursion relations in r and p are used to derive semiclassical approximations for the radial wave functions Pn,l( p) and Rn,l(r). These semiclassical approximations [Eqs. (67) and (84)] are valid even at the classical turning points and are still markedly good at small quantum numbers.

Modified moments and continued fraction coefficients for the diatomic linear chain

A simple, stable, and rapid procedure for the calculation of modified moments and continued fraction coefficients for the diatomic linear chain is presented. The Chebyshev modified moments are obtained stably from a simple three-term recursion relation. Stable determination of the continued fraction coefficients requires a more sophisticated approach, also based on three-term recursion relations. This procedure also allows stable calculation of the continued fraction coefficients for more complex model densities with singularities within the bands. Examples are given and the corresponding behavior of the continued fraction coefficients is discussed.

Transmission coefficient for one-dimensional potential barriers using continued fractions

An explicit formula is given for evaluating the transmission coefficient of one-dimensional potential barriers corresponding to localised forces. The method is based on a simple discretisation of the Schrodinger equation leading to a three-term recursion relation. A continued fraction is then introduced in a quite natural way. Examples are provided to shed some light on the efficiency of the procedure, as compared to semiclassical approximations.

Some aspects of two-point Padé approximants

Recently McCabe and Murphy have considered the two-point Padé approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.

On a general system of orthogonal q-polynomials

A general set of orthogonal q-polynomials {P m (x); m = 0, 1, 2, …, N} is introduced and characterized by its three-term recursion relation. This set unifies many of the different known systems of orthogonal q-polynomials, e.g. the Stieltjes-Wigert polynomials and their several generalizations, the Brenke-Chihara polynomials, the Al Salam-Carlitz polynomials, the Al Salam-Chihara polynomials, …. Compact expressions of the moments of the asymptotical density of zeros of this global set of q-polynomials are explicitly found in terms of the coefficients of the three-term recurrence relation. As an example the asymptotical density of zeros of the known, above-mentioned systems of orthogonal q-polynomials are calculated through its moments.

WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator

WKB method for three-term recursion relations and quasienergies of an anharmonic oscillator

A general theory of linear finite-difference equations with a few applications

Recently a new formalism has been developed giving the solutions in explicit form of multiterm linear homogeneous recursion relations with nonconstant coefficients. The basic idea was to relate this problem to the one of partitioning an interval into parts of given lengths. This idea is extensively used here to obtain the solutions of linear inhomogeneous difference equations. The resulting method has the advantage of being general: It does not rely on any special device and does not assume any special form for the recursion relation. Applications of these techniques to physical problems are presented elsewhere. Here we show how the method works for first and second order equations. The three-term Legendre polynomial recursion relation with an arbitrary inhomogeneous term is discussed in detail. The Legendre polynomials are then viewed as special cases of combinatorics functions, P (j,m;z), based on the partitions of an interval (j,m), 0⩽j⩽m, into parts of lengths 1 and 2. P (j,m;z) reduces to the Legendre polynomial, Pm(z), for j=0. It is also interesting to note that P (j,m;z) are not orthogonal in general. However, a new set of orthogonal functions can be constructed as the solutions of the inhomogeneous Legendre recursion relation with a suitable choice of the inhomogeneous term.