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18 Orthogonal PolynomialsApplications

§18.39 Applications in the Physical Sciences

Contents
  1. §18.39(i) Quantum Mechanics
  2. §18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom
  3. §18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences
  4. §18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods
  5. §18.39(v) Other Applications

§18.39(i) Quantum Mechanics

18.39.1 Moved to (errata.1).
18.39.2 Moved to (errata.2).
18.39.3 Moved to (errata.3).
18.39.4 Moved to (errata.4).
18.39.5 Moved to (errata.5).
18.39.6 Moved to (errata.6).

An Introductory Remark

The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. This material is employed below with little additional discussion. While non-normalizable continuum, or scattering, states are mentioned, with appropriate references in what follows, focus is on the L2 eigenfunctions corresponding to the point, or discrete, spectrum, and representing bound rather than scattering states, these former being expressed in terms of OP’s or EOP’s.

Introduction and One-Dimensional (1D) Systems

The fundamental quantum Schrödinger operator, also called the Hamiltonian, , is a second order differential operator of the form

18.39.7 =22m2x2+V(x),
x(a,b) (),

where x is a spatial coordinate, m the mass of the particle with potential energy V(x), =h/(2π) is the reduced Planck’s constant, and (a,b) a finite or infinite interval.

Here the term 22m2x2 represents the quantum kinetic energy of a single particle of mass m, and V(x) its potential energy. As in classical dynamics this sum is the total energy of the one particle system. The properties of V(x) determine whether the spectrum, this being the set of eigenvalues of , is discrete, continuous, or mixed, see §1.18. Below we consider two potentials with analytically known eigenfunctions and eigenvalues where the spectrum is entirely point, or discrete, with all eigenfunctions being L2 and forming a complete set. Also presented are the analytic solutions for the L2, bound state, eigenfunctions and eigenvalues of the Morse oscillator which also has analytically known non-normalizable continuum eigenfunctions, thus providing an example of a mixed spectrum.

However, in the remainder of this section will will assume that the spectrum is discrete, and that the eigenfunctions of form a discrete, normed, and complete basis for a Hilbert space.

These eigenfunctions are the orthonormal eigenfunctions of the time-independent Schrödinger equation

18.39.8 ψn(x)=(22md2dx2+V(x))ψn(x)=ϵnψn(x),
n=0,1,2,,

which in one dimensional systems are typically non-degenerate, namely there is only a single eigenfunction corresponding to each ϵn, n0. The solutions of (18.39.8) are subject to boundary conditions at a and b. The ϵn are the observable energies of the system, and an increasing function of n. ϵ0 is referred to as the ground state, all others, n=1,2,, in order of increasing energy being excited states. These eigenfunctions are quantum wave-functions whose absolute values squared give the probability density of finding the single particle at hand at position x in the nth eigenstate, namely that probability is P(xx+Δx) = |ψn(x)|2Δ(x), Δ(x) being a localized interval on the x-axis.

also controls time evolution of the wave function Ψ(x,t) via the time-dependent Schrödinger equation,

18.39.9 iΨ(x,t)t=Ψ(x,t),

where V(x) is assumed to be independent of time.

The solutions (18.39.8) are called the stationary states as separation of variables in (18.39.9) yields solutions of form

18.39.10 Ψ(x,t)=exp(iϵnt/)ψn(x),

in which case the probability density is time-independent, as |Ψ(x,t)|2=Ψ(x,t)Ψ(x,t)¯=|ψn(x)|2.

If Ψ(x,t=0)=χ(x) is an arbitrary unit normalized function in the domain of then, by self-adjointness,

18.39.11 χ(x) =n=0cnψn(x),
cn =χ,ψn,

and

18.39.12 Ψ(x,t)=n=0cnexp(iϵnt/)ψn(x),

which is the quantum superposition principle. Such a superposition yields continuous time evolution of the probability density |Ψ(x,t)|2.

For further details about the Schrödinger equation, including applications in physics and chemistry, see Gottfried and Yan (2004) and Pauling and Wilson (1985), respectively, among many others.

1D Quantum Systems with Analytically Known Stationary States

Here are three examples of solutions for (18.39.8) for explicit choices of V(x) and with the ψn(x) corresponding to the discrete spectrum. All are written in the same form as the product of three factors: the square root of a weight function w(x), the corresponding OP or EOP, and constant factors ensuring unit normalization. Brief mention of non-unit normalized solutions in the case of mixed spectra appear, but as these solutions are not OP’s details appear elsewhere, as referenced.

argument a) The Harmonic Oscillator

18.39.13 V(x)=kx2/2

defines the potential for a symmetric restoring force kx for displacements from x=0. Then ω=2πν=k/m is the circular frequency of oscillation (with ν the ordinary frequency), independent of the amplitude of the oscillations. By Table 18.3.1#12 the normalized stationary states and corresponding eigenvalues are

18.39.14 ψn(x) =(chn)1/2w1/2(cx)Hn(cx),
c =(mk)1/41/2,
n=0,1,,
18.39.15 ϵn =k/m(n+12).

Here the Hn(x) are Hermite polynomials, w(x)=ex2, and hn=2nn!π. With the normalization factor (chn)1/2 the ψn are orthonormal in L2(,dx). The spectrum is entirely discrete as in §1.18(v).

argument b) The Morse Oscillator

18.39.16 V(x)=Deα(xxe)(eα(xxe)2)

allows anharmonic, or amplitude dependent, frequencies of oscillation about xe, and also escape of the particle to x=+ with dissociation energy D. The orthonormal stationary states and corresponding eigenvalues are then of the form

18.39.17 ψn(x) =α(2λ2n1)n!Γ(2λn)ϕn(α(xxe);λ),
λ =2mDα,
n=0,1,,N=λ32,

and the corresponding eigenvalues are

18.39.18 ϵn=(α)22m(λn12)2.

The functions ϕn are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). The finite system of functions ψn is orthonormal in L2(,dx), see (18.34.7_3). The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by (αγ)2/(2m) (γ0) with corresponding eigenfunctions eα(xxe)/2Wλ,iγ(2λeα(xxe)) expressed in terms of Whittaker functions (13.14.3). The corresponding eigenfunction transform is a generalization of the Kontorovich–Lebedev transform §10.43(v), see Faraut (1982, §IV).

c) A Rational SUSY Potential argument

The Schrödinger equation with potential

18.39.19 V(x)=12x2+32+8(4x22)(4x2+2)2,

and =k=m=1, has eigenfunctions

18.39.20 ψ^n+3(x)=w(x)H^n+3(x),
n=3,0,1,2,,

and eigenvalues n+3, with n as above, with w(x) the weight function of (18.36.10), and H^n+3(x) a type III Hermite EOP defined by (18.36.8) and (18.36.9). There is no need for a normalization constant here, as appropriate constants already appear in §18.36(vi). The spectrum is entirely discrete as in §1.18(v).

An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with L2 eigenfunctions vanishing at the end points, in this case ± see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. Namely the kth eigenfunction, listed in order of increasing eigenvalues, starting at k=0, has precisely k nodes, as real zeros of wave-functions away from boundaries are often referred to. This seems odd at first glance as H^n+3(x) is a polynomial of order n+3 for n=0,1,2,, seemingly suggesting that for n=0, this being the first excited state, i.e., k=1, there might be 3 nodes, rather than Sturm’s 1. This is illustrated in Figure 18.39.1 where the first and fourth excited state eigenfunctions of the Schrödinger operator with the rationally extended harmonic potential, of (18.39.19), are shown, and compared with the first and fourth excited states of the harmonic oscillator eigenfunctions of (18.39.14) of paragraph a), above. Both satisfy Sturm’s theorem. Thus the two missing quantum numbers corresponding to EOP’s of order 1 and 2 of the type III Hermite EOP’s are offset in the node counts by the fact that all excited state eigenfunctions also have two missing real zeros. Kuijlaars and Milson (2015, §1) refer to these, in this case complex zeros, as exceptional, as opposed to regular, zeros of the EOP’s, these latter belonging to the (real) orthogonality integration range.

See accompanying text
Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for =k=m=1, of (18.39.13), in ψ1(x), ψ4(x) and those of the rational potential of (18.39.19), in ψ^3(x), ψ^6(x). Both sets satisfy the Sturm oscillation theorem. Magnify

Other Analytically Solved Schrödinger Equations

These are overviewed in §18.38(iii), and §18.36(vi), and typically involve OP’s or EOP’s.

§18.39(ii) A 3D Separable Quantum System, the Hydrogen Atom

Analogous to (18.39.7) the 3D Schrödinger operator is

18.39.21 =22m2+V(𝐱),
𝐱=(x,y,z)3,

where 2 is the Laplacian (1.5.17). Now use spherical coordinates (1.5.16) with r instead of ρ, and assume the potential V to be radial. Then write V(r) instead of V(𝐱). By (1.5.17) the first term in (18.39.21), which is the quantum kinetic energy operator Te, can be written in spherical coordinates r,θ,ϕ as

18.39.22 Te22m2=22m1r2ddrr2ddr+L22mr2,

where L2 is the (squared) angular momentum operator (14.30.12). The eigenfunctions of L2 are the spherical harmonics Yl,ml(θ,ϕ) with eigenvalues 2l(l+1), each with degeneracy 2l+1 as ml=l,l+1,,l. See (14.30.11).

Analogous to (18.39.8) the 3D time-independent Schrödinger equation with potential V(r) is

18.39.23 (Te+V(r))Ψ(r,θ,ϕ)=EΨ(r,θ,ϕ).

Since the operators Te+V(r) and L2 commute and have simultaneous eigenfunctions (see §1.3(iv)), the wave function Ψ(r,θ,ϕ) separates as

18.39.24 Ψn,l,ml(r,θ,ϕ)=Rn,l(r)Yl,ml(θ,ϕ).

Orthogonality and normalization of eigenfunctions of this form is respect to the measure r2drsinθdθdϕ. Substitution of (18.39.24) into (18.39.23) then gives the ordinary differential equation for the radial wave function Rn,l(r),

18.39.25 (22mr2ddrr2ddr+2l(l+1)2mr2+V(r))Rn,l(r)=ϵn,lRn,l(r),
n=0,1,2,;l=0,1,2,.

The eigenvalues and radial wave functions are independent of ml and they both do depend on l due to the presence of the ‘fictitious’ centrifugal potential 2l(l+1)/(2mr2), which is a result of the choice of co-ordinate system, and not the physical potential energy of interaction V(r). argument

An alternative, and often used, form of (18.39.25) is that for the spherical radial function ψn,l(r)=rRn,l(r),

18.39.26 (22md2dr2+2l(l+1)2mr2+V(r))ψn,l=ϵn,lψn,l(r),

where the orthogonality measure is now dr, r[0,).

The Quantum Coulomb Problem

In the case of a single electron, charge e and mass me, interacting with a fixed (infinite mass) nucleus of charge +Ze at the co-ordinate origin, with the choice of SI units, V(r)=Ze2/(4πϵ0r). This is Coulomb’s Law. The spectrum is mixed, as in §1.18(viii), the positive energy, non-L2, scattering states are the subject of Chapter 33.

The non-relativistic Schrödinger equation describing a single, bound (negative energy) electron, in an L2 eigenstate of energy E is:

18.39.27 (22me2Ze24πϵ0r)Ψ(r,θ,ϕ)=EΨ(r,θ,ϕ).

In what follows the radial and spherical radial eigenfunctions corresponding to (18.39.27) are found in four different notations, with identical eigenvalues, all of which appear in the current and past mathematical and theoretical physics and chemistry literatures, regarding this central problem. Explicit normalization is given for the second, third, and fourth of these, paragraphs c) and d), below. All results are presented in Hartree atomic units, or a.u., i.e., =me=e2=4πϵ0=1, Mohr and Taylor (2005, Table XXX, p. 71), where the relationship of a.u. to SI units is spelled out.

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

The functions ψp,l(r) satisfy the equation,

18.39.28 (12d2dr2+l(l+1)2r2Zr)ψp,l=ϵp,lψp,l(r),

with an infinite set of orthonormal L2 eigenfunctions

18.39.29 ψp,l(r)=Zp!(p+2l+1)!eρp/2ρpl+1p+l+1Lp(2l+1)(ρp),
p=0,1,2,; l=0,1,2,,

p here being the order of the Laguerre polynomial, Lp(2l+1) of Table 18.8.1, line 11, and l the angular momentum quantum number, and where

18.39.30 ρp=2Zrp+l+1,

with eigenvalues

18.39.31 ϵp,l=Z22(p+l+1)2.

See Titchmarsh (1962a, pp. 99–100).

Orthogonality, with measure dr for r[0,), for fixed l

18.39.32 0ψp,l(r)ψq,l(r)dr=δp,q,

noting that the ψp,l(r) are real, follows from the fact that the Schrödinger operator of (18.39.28) is self-adjoint, or from the direct derivation of Dunkl (2003). This is not the orthogonality of Table 18.8.1, as the co-ordinate arguments depend, independently on p and q. This indicates that the Laguerre polynomials appearing in (18.39.29) are not classical OP’s, and in fact, even though infinite in number for fixed l, do not form a complete set. Namely for fixed l the infinite set labeled by p describe only the L2 bound states for that single l, omitting the continuum briefly mentioned below, and which is the subject of Chapter 33, and so an unusual example of the mixed spectra of §1.18(viii).

b) The Bohr Quantum Number

The fact that both the eigenvalues of (18.39.31) and the scaling of the r co-ordinate in the eigenfunctions, (18.39.30), depend on the sum p+l+1 leads to the substitution

18.39.33 n=p+l+1,

where n is now the Bohr Principle Quantum Number.

Physical scientists use the n of Bohr as, to 0th and 1st order, it describes the structure and organization of the Periodic Table of the Chemical Elements of which the Hydrogen atom is only the first.

This follows from the fact that the eigenvalues (18.39.31), ϵn,l=ϵn=Z2/(2n2) depend only on the single quantum number n, the Bohr Principal quantum number, n=1,2,, and depend explicitly neither on l nor ml. Thus the overall degeneracy of the solutions (18.39.29) (the number of independent eigenfunctions corresponding to a single eigenvalue (18.39.31) for all values of l and ml) consistent with each n is n2, which controls the lengths of the rows in the Periodic Table. Interactions between electrons, in many electron atoms, breaks this degeneracy as a function of l, but n still dominates.

c) Spherical Radial Coulomb Wave Functions

The solution, (18.39.29), of the spherical radial equation (18.39.28), now expressed in terms of the Bohr quantum number n, is

18.39.34 ψn,l(r)=1nZ(nl1)!(n+l)!eρn/2ρnl+1Lnl1(2l+1)(ρn),
n=1,2,,l=0,1,n1,
18.39.35 ρn=2Zrn,ϵn=Z22n2,

thus recapitulating, for Z=1, line 11 of Table 18.8.1, now shown with explicit normalization for the measure dr. This is also the normalization and notation of Chapter 33 for Z=1 , and the notation of Weinberg (2013, Chapter 2). The radial Coulomb wave functions Rn,l(r), solutions of

18.39.36 (12r2ddrr2ddr+l(l+1)2r2Zr)Rn,l(r)=ϵnRn,l(r),
n=1,2,,l=0,1,,n1,

with ρn and ϵn being those of (18.39.35), are then

18.39.37 Rn,l(r)=2n2Z3(nl1)!(n+l)!eρn/2ρnlLnl1(2l+1)(ρn),

normalized with measure r2dr, r[0,).

d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

The same solutions as in paragraph c), above, appear frequently in the literature in terms of associated Laguerre polynomials, which are referred to here as associated Coulomb–Laguerre polynomials to avoid confusion with the more recent meaning of ‘associated’ of §18.30. The associated Coulomb–Laguerre polynomials are defined as

18.39.38 𝐋pm(ρ)=dmdρm𝐋p0(ρ),

where

18.39.39 𝐋p0(ρ)=eρdpdρp(ρpeρ),

see Bethe and Salpeter (1957, p. 13), Pauling and Wilson (1985, pp. 130, 131); and noting that this differs from the Rodrigues formula of (18.5.5) for the Laguerre OP’s, in the omission of an n! in the denominator.

From (18.9.23) and (18.5.5) with Table 18.5.1, line 8:

18.39.40 𝐋p+mm(ρ)=(1)m(p+m)!Lp(m)(ρ),

and thus replacing p by nl1 as in Table 18.8.1, line 11, or as in (18.39.33),

18.39.41 Lnl1(2l+1)(ρn)=𝐋n+l2l+1(ρn)/(n+l)!,

(where the minus sign is often omitted, as it arises as an arbitrary phase when taking the square root of the real, positive, norm of the wave function), allowing equation (18.39.37) to be rewritten in terms of the associated Coulomb–Laguerre polynomials 𝐋n+l2l+1(ρn).

18.39.42 Rn,l(r)=2n2Z3(nl1)!((n+l)!)3eρn/2ρnl𝐋n+l2l+1(ρn).

Derivations of (18.39.42) appear in Bethe and Salpeter (1957, pp. 12–20), and Pauling and Wilson (1985, Chapter V and Appendix VII), where the derivations are based on (18.39.36), and is also the notation of Piela (2014, §4.7), typifying the common use of the associated Coulomb–Laguerre polynomials in theoretical quantum chemistry. In all of these references these OP’s are simply referred to as the associated Laguerre OP’s.

The Relativistic Quantum Coulomb Problem

Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with Z protons, involve Laguerre polynomials of fractional index. A relativistic treatment becoming necessary as Z becomes large as corrections to the non-relativistic Schrödinger picture are of approximate order (αZ)2(Z/137)2, α being the dimensionless fine structure constant e2/(4πε0c), where c is the speed of light. See Martinez-y-Romero (2000).

The Quantum Coulomb Problem: Scattering States

The positive energy (scattering) eigenfunctions for the above Coulomb problem, with potential V(r)=Ze2/r are discussed in Chapter 33 for each integer l. These, taken together with the infinite sets of bound states for each l, form complete sets. As the scattering eigenfunctions of Chapter 33, are not OP’s, their further discussion is deferred to §18.39(iv), where discretized representations of these scattering states are introduced, Laguerre and Pollaczek OP’s then playing a key role.

For the potential V(r)=+Ze2/r, corresponding to interaction of particles with like charges, there are no bound states, the continuum scattering states form a complete set for each l, as discussed in Chapter 33, and their discretized versions in §18.39(iv).

§18.39(iii) Non Classical Weight Functions of Utility in DVR Method in the Physical Sciences

Shizgal (2015) gives a broad overview of techniques and applications of spectral and pseudo-spectral methods to problems arising in theoretical chemistry, chemical kinetics, transport theory, and astrophysics.

The discrete variable representations (DVR) analysis is simplest when based on the classical OP’s with their analytically known recursion coefficients (Table 3.5.17_5), or those non-classical OP’s which have analytically known recursion coefficients, making stable computation of the xi and wi, from the J-matrix as in §3.5(vi), straightforward. For many applications the natural weight functions are non-classical, and thus the OP’s and the determination of the Gaussian quadrature points and weights represent a computational challenge. Table 18.39.1 lists typical non-classical weight functions, many related to the non-classical Freud weights of §18.32, and §32.15, all of which require numerical computation of the recursion coefficients (i.e., the J-matrix elements) as in Gautschi (1968), Golub and Welsch (1969), Gordon (1968). A major difficulty in such calculations, loss of precision, is addressed in Gautschi (2009) where use of variable precision arithmetic is discussed and employed. Shizgal (2015, Chapter 2), contains a broad-ranged discussion of methods and applications for these, and other, non-classical weight functions. Further details of such calculations are contained in the papers cited.

Table 18.39.1: Typical Non-Classical Weight Functions Of Use In DVR Applicationsa
Name of OP System w(x) [a,b] Notation Applications
Freud-Bimodal exp((x4/4x2/2)/α) (,) Bn(x) Fokker–Planck DVRb,c
Quartic Freud exp(x4/4zx2) (,) pn(x) §32.15 and application refs. therein: Quantum Gravity and Graph Theory Combinatorics
Half-Freud Druvesteyn exp(x4) [0,) Dn(x) Electron Transport in Plasmasd
Half-Freud Gaussian exp((xx0)2) [0,) Gn(x) Fokker–Planck DVRe
Half-Freud Maxwell xpexp(x2) [0,) Mnp(x) Kinetic Theoryf,g and Fokker–Planck DVRsh
Half-Rys exp(αx2) [0,1] Rn(x,α) Quantum Chemistry Quadraturesi,j
Multi-Exponential ln2x [0,1] MExpn(x) Quadrature Sums of Exponentialsk
Radiative Transfer exp(α/x) [0,1] Rn(x) Radiative Transferl

a) Shizgal (2015), b) Blackmore et al. (1986), c) Gammaitoni et al. (1998), d) Liboff (2003). e) Garashchuk and Light (2001), f) Gross and Ziering (1958), g) Desai and Nelkin (1966), h) Blackmore and Shizgal (1985), i) Helgaker et al. (2012), j) Rys et al. (1983), k) Gill and Chen (2003), l) Gander and Karp (2001).

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

The bound state L2 eigenfunctions of the radial Coulomb Schrödinger operator are discussed in §§18.39(i) and 18.39(ii), and the δ-function normalized (non-L2) in Chapter 33, where the solutions appear as Whittaker functions. These same solutions are expressed here in terms of Laguerre and Pollaczek OP’s. The technique to accomplish this follows the DVR idea, in which methods are based on finding tridiagonal representations of the co-ordinate, x. Here tridiagonal representations of simple Schrödinger operators play a similar role. The radial operator (18.39.28)

18.39.43 (Z)=12d2dr2+l(l+1)2r2+Zr,
r[0,),

is tridiagonalized in the complete L2 non-orthogonal (with measure dr, r[0,)) basis of Laguerre functions:

18.39.44 ϕn,l(sr)=(sr)l+1esr/2Ln(2l+1)(sr),
n=0,1,2,,

where s is a real, positive, scaling factor, and l a non-negative integer.

As in this subsection both positive (repulsive) and negative (attractive) Coulomb interactions are discussed, the prefactor of Z/r in (18.39.43) has been set to +1, rather than the 1 of (18.39.28) implying that Z<0 is an attractive interaction, Z>0 being repulsive.

This operator may be discretized by projecting it onto the sub-space defined by the first N members, n=0,1,2,,N1, of the complete basis of (18.39.44), the eigenfunctions, may be expressed as

18.39.45 ΨϵN(r)=n=0N1n!Γ(n+2l+2)cn(ϵ)ϕn,l(sr),

with matrix eigenvalues ϵ=ϵiN, i=1,2,,N, and the eigenvectors, 𝐜(ϵ)=(c0(ϵ),c1(ϵ),,cN1(ϵ)), are determined by the recursion relation (18.39.46) below.

Following the method of Schwartz (1961), Yamani and Reinhardt (1975), Bank and Ismail (1985), and Ismail (2009, §5.8)  have shown this is equivalent to determination of x such that cN(x)=0 in the recursion scheme

18.39.46 (n+1)cn+1=2((n+l+1+a)xa)cn(n+2l+1)cn1,

with initial data c0=1, c1=0, where

18.39.47 a=2Zs

and

18.39.48 x=x(ϵ)=8ϵs28ϵ+s2

which maps ϵ[0,) onto x[1,1]. The recursion of (18.39.46) is that for the type 2 Pollaczek polynomials of (18.35.2), with λ=l+1, a=b=2Z/s, and c=0, and terminates for x=xiN being a zero of the polynomial of order N. Thus the cN(x)=PN(l+1)(x;2Zs,2Zs) and the eigenvalues

18.39.49 ϵiN=s281+xiN1xiN

are determined by the N zeros, xiN of the Pollaczek polynomial PN(l+1)(x;2Zs,2Zs).

The Coulomb–Pollaczek Polynomials

The polynomials PN(l+1)(x;2Zs,2Zs), for both positive and negative Z, define the Coulomb–Pollaczek polynomials (CP OP’s in what follows), see Yamani and Reinhardt (1975, Appendix B, and §IV). For Z>0 these are the repulsive CP OP’s with x[1,1] corresponding to the continuous spectrum of (Z), ϵ(0,), and for Z<0 we have the attractive CP OP’s, where the spectrum is complemented by the infinite set of bound state eigenvalues for fixed l. These cases correspond to the two distinct orthogonality conditions of (18.35.6) and (18.35.6_3).

Given that a=b in both the attractive and repulsive cases, the expression for the absolutely continuous, x[1,1], part of the function of (18.35.6) may be simplified:

18.39.50 wCP(x)=(l+1+2Zs)πΓ(2l+2)e(2θ(x)π)τ(x)(4(1x2))l+12|Γ(l+1+iτ(x))|2,
θ(x)=arccos(x), τ(x)=2Zs1x1+x.

Note that violation of the Favard inequality, l+1+(2Z/s)>0, possible when Z<0, results in a zero or negative weight function. While s in the basis of (18.39.44) is simply a variational parameter, care must be taken, or the relationship between the results of the matrix variational approximation and the Pollaczek polynomials is lost, although this has no effect on the use of the variational approximations Reinhardt (2021a, b). Graphs of the weight functions of (18.39.50) are shown in Figure 18.39.2.

See accompanying text
Figure 18.39.2: Coulomb–Pollaczek weight functions, x[1,1], (18.39.50) for s=10, l=0, and Z=±1. For Z=+1 the weight function, red curve, has an essential singularity at x=1, as all derivatives vanish as x1+; the green curve is 1xwCP(y)dy, to be compared with its histogram approximation in §18.40(ii). For Z=1 the weight function, blue curve, is non-zero at x=1, but this point is also an essential singularity as the discrete parts of the weight function of (18.39.51) accumulate as k, xk1. Magnify

In the attractive case (18.35.6_4) for the discrete parts of the weight function where with xk<1, are also simplified:

18.39.51 xk =(2Zs)2+(k+l+1)2(2Zs)2(k+l+1)2,
ρ =k+l+1+2Zsk+l+12Zs,
wxkCP =(l+1+2Zs)ρ2k1(1ρ2)2l+3(2l+2)k2(k+l+1)k!.

The weight functions for both the attractive and repulsive cases are now unit normalized, see Bank and Ismail (1985), and Ismail (2009). Note that these discrete xk, from (18.39.51), giving, for Z<0, xk1,

18.39.52 ϵk=Z22(k+l+1)2,
k=0,1,2,,

which is Bohr’s quantization for the Coulomb bound state energies (18.39.31). The Schrödinger operator essential singularity, seen in the accumulation of discrete eigenvalues for the attractive Coulomb problem, is mirrored in the accumulation of jumps in the discrete Pollaczek–Stieltjes measure as x1. As the Coulomb–Pollaczek OP’s are members of the Nevai-Blumenthal class, this is, for Z<0, a physical example of the properties of the zeros of such OP’s, and their possible accumulation at x=1, as discussed in §18.2(xi).

Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory

The Coulomb–Pollaczek polynomials provide alternate representations of the positive energy Coulomb wave functions of Chapter 33. For either sign of Z, and s chosen such that n+l+1+(2Z/s)>0, n=0,1,2,, truncation of the basis to N terms, with xiN[1,1], the discrete eigenvectors are the orthonormal L2 functions

18.39.53 Ψxi,lN(r)=Axi,ln=0N1n!Γ(n+2l+2)Pn(l+1)(xi;2Zs,2Zs)ϕn,l(sr),
xi=8ϵis28ϵi+s2.

With N the functions normalized as δ(ϵϵ) with measure dr are, formally,

18.39.54 Ψx,l(r)=Bl(x)n=0n!Γ(n+2l+2)Pn(l+1)(x;2Zs,2Zs)ϕn,l(sr),
x=8ϵs28ϵ+s2,

which corresponds to the exact results, in terms of Whittaker functions, of §§33.2 and 33.14, in the sense that projections onto the functions ϕn,l(sr)/r, the functions bi-orthogonal to ϕn,l(sr), are identical. Full expressions for both Axi,l and Bl(x) are given in Yamani and Reinhardt (1975) and it is seen that |Axi,l/Bl(xi)|2 = wiN/wCP(xi) where wiN is the Gaussian-Pollaczek quadrature weight at x=xi, and wCP(xi) is the Gaussian-Pollaczek weight function at the same quadrature abscissa. This equivalent quadrature relationship, see Heller et al. (1973), Yamani and Reinhardt (1975), allows extraction of scattering information from the finite dimensional L2 functions of (18.39.53), provided that such information involves potentials, or projections onto L2 functions, exactly expressed, or well approximated, in the finite basis of (18.39.44). The equivalent quadrature weight, wi/wCP(xi), also forms the foundation of a novel inversion of the Stieltjes–Perron moment inversion discussed in §18.40(ii).

The fact that non-L2 continuum scattering eigenstates may be expressed in terms or (infinite) sums of L2 functions allows a reformulation of scattering theory in atomic physics wherein no non-L2 functions need appear. As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. See Yamani and Fishman (1975) for L2 for expansions of both the regular and irregular spherical Bessel functions, which are the Pollaczeks with a=Z=0, and Coulomb functions for fixed l, Broad and Reinhardt (1976) for a many particle example, and the overview of Alhaidari et al. (2008). Mathematical underpinnings appear in Ismail (2009, §5.8), and Ismail and Koelink (2011).

§18.39(v) Other Applications

For applications of Legendre polynomials in fluid dynamics to study the flow around the outside of a puff of hot gas rising through the air, see Paterson (1983). For applications and an extension of the Szegő–Szász inequality (18.14.20) for Legendre polynomials (α=β=0) to obtain global bounds on the variation of the phase of an elastic scattering amplitude, see Cornille and Martin (1972, 1974). For physical applications of q-Laguerre polynomials see §17.17. For interpretations of zeros of classical OP’s as equilibrium positions of charges in electrostatic problems (assuming logarithmic interaction), see Ismail (2000a, b).