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Open Problem in Orthogonal Polynomials
Vol. 84 (2019)
REPORTS ON MATHEMATICAL PHYSICS
No. 3
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS A BDULAZIZ D. A LHAIDARI Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438, Saudi Arabia (e-mail: [email protected]) (Received May 7, 2019 — Revised June 13, 2019) Using an algebraic method for solving the wave equation in quantum mechanics, we encountered new families of orthogonal polynomials on the real line. The properties of the physical system (e.g. energy spectrum, phase shift, density of states, etc.) are obtained from the properties of these polynomials. One of these new families is composed of four-parameter polynomials describing a discrete spectrum of the corresponding quantum mechanical system. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their threeterm recursion relations and initial values. However, their other properties like weight functions, generating functions, orthogonality, Rodrigues-type formula, etc., are yet to be derived analytically. Obtaining these properties is an open problem in orthogonal polynomials. Keywords: tridiagonal representation, orthogonal polynomials, potential functions, asymptotics, recursion relation, spectrum formula.
1.
Introduction The wave function in quantum mechanics could be viewed as an element of an infinite-dimensional vector space with local unit vectors. Therefore, in one of the algebraic formulations of quantum mechanics, the wave function at an energy E, ψE (x), is written as a convergent sum over a complete set of square integrable basis functions in configuration space with coordinate x, X ψE (x) = fn (E)φn (x), (1) n
where {φn (x)} are the basis elements (local unit vectors) and {fn (E)} are the expansion coefficients (projections of the wave function along the unit vectors) as a function of energy. All physical information about the system, both structural and dynamical, are contained in these expansion coefficients. In our earlier publications [1–4], we developed the so-called “Tridiagonal Representation Approach (TRA)” — a new algebraic method for solving the wave equation (e.g. the Schr¨odinger equation or the Dirac equation). In the TRA, the basis elements are chosen such that the matrix representation of the wave operator is tridiagonal. Consequently, the resulting matrix wave equation reduces to a three-term recursion relation for {fn (E)}. If [393]
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A. D. ALHAIDARI
we write fn (E) = f0 (E)Pn (ε), where the argument ε depends on the energy and physical parameters, then P0 (ε) = 1 and Favard’s theorem [5, 6] can be used to show that {Pn (ε)} is a complete set of orthogonal polynomials satisfying the said recursion relation. We have shown that the corresponding positive definite weight function is [f0 (E)]2 (see page 4 of [1] and page 3 of [7]). This was derived using the completeness R of the basis set {φn (x)} and normalizability of the wavefunction ψE (x) giving fn (E)fm (E) dε ∼ δn,m . Substituting fn (E) = f0 (E)Pn (ε), turns this integral into [1–7] Z (2) [f0 (E)]2 Pn (ε)Pm (ε) dε ∼ δn,m .
These polynomials are associated with the continuum scattering states of the system when E is a continuous set. On the other hand, the discrete bound states are associated with the discrete version of these polynomials. We reproduced all such polynomials that correspond to well-known physical systems and found new ones as well. For example, we reproduced earlier results by Reinhardt, Heller, Yamani and Fishmann [8–10] that the scattering states of the Coulomb problem are associated with the Meixner–Pollaczek polynomials whereas the bound states are associated with one of its discrete versions; the Meixner polynomials. Moreover, we also reproduced earlier findings by Ojha [11] that the scattering states of the Morse oscillator are associated with the continuous dual Hahn polynomials, whereas the finite number of bound states is associated with their discrete version, the dual Hahn polynomials. Additionally, the continuum scattering states of the hyperbolic P¨oschl–Teller potential correspond to the Wilson polynomials, whereas the finite number of bound states is associated with the Racah polynomials, which are the discrete version of the Wilson polynomials. On the other hand, while using the TRA [12–19], we have encountered a new family of exactly solvable problems that are associated with new orthogonal polynomials that seem to have been over-looked in the traditional mathematics and physics literature. These polynomials are defined, up to now, by their three-term recursion relations and initial value P0 (ε) = 1. However, their other important properties are yet to be derived analytically. These properties include the weight functions, generating functions, asymptotics, orthogonality, Rodrigues-type formulae, etc. The purpose of this outline is to shed light on the most important properties of these remarkable families of orthogonal polynomials, bring awareness, and raise enough interest in the community of orthogonal polynomials specialists in order to study them. In physics, any special functions or orthogonal polynomials that enter in the description of a physical system are considered to be of prime significance. The orthogonal polynomials presented here are associated with nonconventional but interesting and highly significant potential functions in quantum mechanics. All attempts to compare this family of polynomials with known ones failed to produce a match. For example, we tried using the table of recurrence formulae in Chihara’s book [5] and the properties of the hypergeometric orthogonal polynomials in the
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
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book of Koekoek, Lesky and Swarttouw [20]. We also studied the chapter on Orthogonal Polynomials in the Digital Library of Mathematical Functions [21] and compared with the information available in CAOP (Computer Algebra & Orthogonal Polynomials) [22]. Members of our group tried using computer algebra systems (such as rec2ortho of Koornwinder and Swarttouw or retode of Koepf and Schmersau [23]) to identify the polynomials from their recurrence relations. Additionally, we had various private communications with experts in the field including M. E. H. Ismail, W. Van Assche, and Y. T. Li. Some colleagues and students of Van Assche studied several of these polynomials and partial results were communicated [24] and reported in [25]. Furthermore, in the latter reference [25], Van Assche treated two polynomial families submitted by us as an open problem [26] to the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA14) held at the University of Kent on 3–7 July 2017. This open problem was also communicated in a newsletter of the SIAM Activity Group on Orthogonal Polynomials and Special Functions [27]. In that open problem, we introduced (µ,ν) two polynomial families of significance in quantum mechanics, Hn (z; α, θ ) and (µ,ν) Gn (z; σ ), and enquired about their properties (e.g. weight function, generating function, orthogonality, asymptotics, etc.). In [25], Van Assche, reported on some (µ,ν) of the properties of the first family, Hn (z; α, θ ), including its spectral nature and asymptotics. These results are stated in Section 3 below. On the other hand, (µ,ν) by the time OPSFA-14 concluded, the second family, Gn (z; σ ) has already been identified by Y. T. Li as a special case of the Wilson polynomials [28] and it was written down explicitly in Section 3 of [25]. Therefore, the polynomial (µ,ν) Gn (z; σ ) is not in the scope of this present study. In [29] an investigation of (µ,ν) the polynomial family Hn (z; α, θ ) by D. D. Tcheutia led to a clear conclusion about its novelty. (µ,ν) In addition to the polynomial Hn (z; α, θ ) presented in Section 3 below, we (µ,ν) also introduce in Section 4 the polynomial Qn (z; α, θ ) that we have encountered recently while solving a physically relevant Heun-type differential equation [30]. We tried our best to give as much information as possible about the properties of these polynomials using numerical means. For example, we could use any one of three numerical routines (analytic continuation, dispersion correction, and Stieltjes imaging methods) to obtain the continuous weight function, employing only the recursion relation coefficients [31, 32]. We could also use the J -matrix method to obtain the discrete spectrum [8, 9, 33, 34]. These numerical results are to be used only as a guide or suggestion for further and more rigorous mathematical analysis. Due to the relevance of these polynomials to the solution of some quantum mechanical problems with nonconventional but useful potentials, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g. in terms of hypergeometric functions). In the following section, we give an illustrative example of how these polynomials appear in the solution of the Schr¨odinger wave equation in quantum mechanics.
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Two of these polynomials are defined in Sections 3 and 4 whereas in Section 5 we introduce a modification to the three-term recursion relations of the hypergeometric polynomials in the Askey scheme [20]. 2. The Schr¨odinger wave equation of a physical system and its corresponding orthogonal polynomial In this section, we give an illustrative example showing how the three-term recursion relation of one of the new family of orthogonal polynomials appear in the solution of the one-dimensional Schr¨odinger wave equation of a quantum mechanical system with a nonconventional but significant interaction potential. The potential function of interest is V+ − V− sin(π x/L) + V1 sin(π x/L), (3) V (x) = V0 + cos2 (π x/L) where −L/2 ≤ x ≤ L/2. This potential represents an infinite potential well of width L with {V± , V0 , V1 } being a set of real parameters such that V+ +|V− | ≥ −2(π/4L)2 . If V1 6 = 0, then this potential does not have a known exact solution in the conventional formulations of quantum mechanics. The special case with V± = 0 is an infinite square well potential with sinusoidal rather than flat bottom. This has been studied using the TRA by our group in [14, 35]. On the other hand, if V1 = 0 then the potential becomes the Scarf I potential (an alternative form of the trigonometric P¨oschl–Teller potential) which is exactly solvable by the conventional methods [36, 37]. Due to total confinement inside this infinite potential well, the energy spectrum of the system is bounded from below but not from above and it consists of an infinite number of discrete eigenvalues. The proper set of basis functions to expand the physical wave function ψE (x) in a series as shown by Eq. (1), which is also compatible with the tridiagonal representation of the TRA, is
φn (x) = (1 − y)
µ+ 21 /2
(1 + y)
ν+ 21 /2
Pn(µ,ν) (y),
(4)
(µ,ν)
where y(x) = sin(πx/L), Pn (y) is the Jacobi polynomial and the real parameters µ and ν are greater than −1. In the atomic units h¯ = m = 1, the Schr¨odinger wave equation for a point particle of mass m and energy E under the influence of the potential (3) reads as follows 1 d2 + V (x) − E ψE (x) = 0. (5) − 2 dx 2 We rewrite this equation in terms of the independent variable y where the potential function (3) reads V+ −V− y V (x) = V0 + + V1 y. 1 − y2
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
397
Substituting the expansion (1) with the basis elements (4) in (5) and using the differential equation of the Jacobi polynomial, the resulting equation becomes X n
fn
1 ν 2 − 1/4 µ+ν+1 2 1 µ2 − 1/4 + − n+ 2 1−y 2 1+y 2 2E u+ − u− y φn = 0, + 2 − u0 + u1 y + λ 1 − y2
(6)
where λ = π/L and ui = 2Vi /λ2 . Choosing the basis parameters as µ2 = 41 +u+ −u− , ν 2 = 14 + u+ + u− and using the recursion relation of the Jacobi polynomials, we obtain the following three-term recursion relation for the expansion coefficients of the wave function " # µ+ν+1 2 ν 2 − µ2 E − V0 1 n+ + fn (E) = fn (E) V1 u1 2 (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) fn−1 (E) (2n + µ + ν)(2n + µ + ν + 1) 2(n + 1)(n + µ + ν + 1) + fn+1 (E). (2n + µ + ν + 1)(2n + µ + ν + 2) +
(7)
If we write fn (E) = f0 (E)Pn (ε) then the solution space of this recursion relation for Pn (ε) will be divided into two subspaces depending on the values of the physical parameters: 1. (E − V0 )2 ≤ V12 : Comparing the recursion relation to (8) in Section 3 below, (µ,ν) we conclude q that Pn (ε) = Hn (z; α, θ ) with α 2 = 0, cos(θ ) = (E − V0 )/V1 , and z−1 =
2 λ2
V12 − (E − E0 )2 .
2. (E − V0 )2 > V12 : We compare the recursion relation (7) to (8′ ) in Section 3 (µ,ν) 2 below and conclude that Pn (ε) q = Hn (−iz; α, iθ ) with α = 0, cosh(θ ) = (E − V0 )/|V1 |, and z−1 =
2 λ2
(E − V0 )2 − V12 .
Therefore, the physical property of this system suggests that the polynomial (µ,ν) should be discrete with Hn (−iz; α, iθ ) having countably infinite spectrum. Moreover, it is easy to see that in the limit V1 → 0, relation (7) gives directly 1 2 µ+ν+1 2 E = V0 + λ n + , 2 2 (µ,ν) Hn (z; α, θ)
which is the well-known energy spectrum of the Scarf I potential [36, 37]. The same TRA treatment of other quantum mechanical systems with nonconventional potential functions yields similar results [12–19].
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3. A polynomial that generalizes the Jacobi polynomial (µ,ν)
The four-parameter polynomial, which we designate as Hn (z; α, θ ), satisfies the following three-term recursion relation: µ+ν+1 2 2 (µ,ν) − α z sin θ (cos θ)Hn (z; α, θ) = n+ 2 ν 2 − µ2 + Hn(µ,ν) (z; α, θ ) (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) + H (z; α, θ ) (2n + µ + ν)(2n + µ + ν + 1) n−1 2(n + 1)(n + µ + ν + 1) (µ,ν) H (z; α, θ ), (8) + (2n + µ + ν + 1)(2n + µ + ν + 2) n+1 where z is real, 0 < θ ≤ π and n = 1, 2, 3, . . .. It is a polynomial of degree n in z. Setting z ≡ 0 turns (8) into the recursion relation of the Jacobi polynomials (µ,ν) Pn (cos θ). Physical requirements dictate that µ and ν be greater than −1. The polynomial of the first kind satisfies this recursion relation together with (µ,ν) H0 (z; α, θ) = 1 and (µ,ν)
H1
(z; α, θ) =
µ−ν 1 + (µ + ν + 2) 2 2 1 × cos(θ ) − (µ + ν + 1)2 − α 2 z sin(θ ) , 4
(9)
(µ,ν)
which is obtained from (8) by setting n =0 and H−1 (z; α, θ ) ≡ 0. We divide 2 the recursion relation (8) by n + µ+ν+1 sin(θ ) and then define the orthogonal 2 √ (µ,ν) (µ,ν) version of this polynomial as H˜ n (z; α, θ ) = λn Hn (z; α, θ ), where (n + 1)(n + µ + ν + 1)(2n + µ + ν + 3) (2n + µ + ν + 3) − 4α 2 λn+1 , (10) = λn (n + µ + 1)(n + ν + 1)(2n + µ + ν + 1) (2n + µ + ν + 1) − 4α 2 (µ,ν) and λ0 = 1. Then H˜ n (z; α, θ) satisfies the following symmetric three-term recursion relation (µ,ν) (µ,ν) zH˜ n(µ,ν) (z; α, θ) = an H˜ n(µ,ν) (z; α, θ ) − bn−1 H˜ (z; α, θ ) − bn H˜ (z; α, θ ), (11) n−1
n+1
where
an = cos θ +
µ2 − ν 2 (2n + µ + ν)(2n + µ + ν + 2) (" # ) . µ+ν+1 2 2 n+ − α sin θ , 2
(12a)
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
bn2 =
26 (n + 1)(n + µ + 1)(n + ν + 1) (2n + µ + ν + 1)(2n + µ + ν + 3) (n + µ + ν + 1)(sin θ )−2 (2n + µ + ν + 2)−2 . × (2n + µ + ν + 1)2 − 4α 2 (2n + µ + ν + 3)2 − 4α 2
399
(12b)
The observation that the asymptotics (n → ∞) of these recursion coefficients go as O(1/n2 ) leads to the conclusion that the Jacobi matrix associated with (µ,ν) H˜ n (z; α, θ), which reads a0 b0 b0 a1 b1 b1 a2 b2 (13) J = b2 a3 b3 × × × × × × × ×
0
0
is a compact operator (in fact, a trace class operator) [25, 38]. This implies that the spectrum of J is a countable set {zk , k, ∈ N} with one accumulation point at P∞ zero (i.e. limk→∞ zk = 0). The trace class condition implies that |z | is finite k k=0 (µ,ν) ˜ [38]. Consequently, the orthogonality measure for Hn (z; α, θ ) is a discrete mea(µ,ν) sure supported on this countable set. The asymptotic limn→∞ zn H˜ n (1/z; α, θ ) is an entire function with zeros at the points 1/zk [25]. Recently, Tcheutia [29] managed to extend the Maple implementation retode of Koepf and Schmersau [23] to deal with classical orthogonal polynomials on quadratic and q(µ,ν) quadratic lattices and concluded that “the polynomial family Hn (z; α, θ ) is not related to a known classical orthogonal polynomial sequence” and that its “recurrence equation may lead to a new family of orthogonal polynomial sequences”. In [4, 12–18], we presented examples of physical systems associated with (µ,ν) the orthogonal polynomial Hn (z; α, θ ). We derived the corresponding three-term recursion relation and showed how the polynomial argument and parameters are related to the potential function parameters and the energy. Using this association, we demonstrate how to obtain the physical properties of the quantum mechanical systems (e.g. bound state energy spectrum, scattering phase shift, density of states, etc.) from those of the polynomial (such as the weight function, asymptotics, zeros, etc.). If the parameters of the problem which correspond to the recursion relation (8) violate the condition that |cos θ| ≤ 1 then we rewrite (8) with θ 7→ iθ and z 7→ iz giving
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A. D. ALHAIDARI
µ+ν+1 2 2 − α z sinh θ = n+ 2 ν 2 − µ2 + Hn(µ,ν) (−iz; α, iθ ) (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) Hn−1 (−iz; α, iθ ) + (2n + µ + ν)(2n + µ + ν + 1) 2(n + 1)(n + µ + ν + 1) (µ,ν) + H (−iz; α, iθ ). (2n + µ + ν + 1)(2n + µ + ν + 2) n+1
(cosh θ)Hn(µ,ν) (−iz; α, iθ)
(8′ )
4. A polynomial arising from the solution of a Heun-type ordinary differential equation Recently, while transforming the Schr¨odinger wave equation to a Heun-type equation, we encountered a set of new or generalized orthogonal polynomials [30]. (µ,ν) One of these polynomials has four parameters and we designate it as Qn (z; α, θ ). This polynomial satisfies the following three-term recursion relation −1 µ+ν+1 2 2 z sin θ (cos θ)Q(µ,ν) (z; α, θ) = n + − α n 2 ν 2 − µ2 + Q(µ,ν) (z; α, θ ) n (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) (z; α, θ ) Q + (2n + µ + ν)(2n + µ + ν + 1) n−1 2(n + 1)(n + µ + ν + 1) (µ,ν) + Qn+1 (z; α, θ ), (14) (2n + µ + ν + 1)(2n + µ + ν + 2)
where z is real, 0 < θ ≤ π and n = 1, 2, 3, . . .. Note the inverse power of the square bracket. This constitutes a major difference from the recurrence relation (8). (µ,ν) (µ,ν) Here as well, Q0 (z; α, θ) = 1 and Q1 (z; α, θ ) is obtained from (14) by setting (µ,ν) n = 0 and Q−1 (z; α, θ) ≡ 0. (µ,ν) Numerical experimentations suggest that the polynomial Qn (z; α, θ ) has a purely continuous spectrum over the entire real line. However, it has another version, which (µ,ν) we refer to as Vn (z; α, β), whose recursion relation is obtained from (14) by the replacement θ 7 → iθ and z 7 → iz giving −1 (µ,ν) µ+ν+1 2 2 2 2 1 + β Vn (z; α, β) = z 1 − β n+ −α 2 2 ν 2 − µ2 β + V (µ,ν) (z; α, β) (2n + µ + ν)(2n + µ + ν + 2) n
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
4(n + µ)(n + ν)β (µ,ν) V (z; α, β) (2n + µ + ν)(2n + µ + ν + 1) n−1 4(n + 1)(n + µ + ν + 1)β (µ,ν) + V (z; α, β), (2n + µ + ν + 1)(2n + µ + ν + 2) n+1
401
+
(15)
where z ≥ 0 and β = e−θ with θ > 0. Using only numerical means and without rigorous mathematical proof, we found that if α is pure imaginary, then the spectrum is purely continuous. However, if α is real then the spectrum is a mix of a continuous positive spectrum and a discrete negative spectrum of finite size N + 1, . This behaviour where N is the largeset integer less than or equal to |α| − µ+ν+1 2 2 mimics that of the continuous dual Hahn polynomial Sn (x ; a, b, c) and the Wilson polynomial Wn (x 2 ; a, b, c, d) for negative a. Here α 2 plays the role of −a. In this case, the polynomial satisfies a generalized orthogonality relation that consists of a continuous part and a discrete part similar to those of the Wilson polynomials and the continuous dual Hahn polynomials [5, 20]. This is also suggested by physical arguments concerning quantum systems with a mix of discrete bound states and continuous scattering states (see, for example, Section II of [7]). Consequently, the orthogonality will be of the following form Z ∞ ρ(z)Vn(µ,ν) (z; α, β)Vm(µ,ν) (z; α, β) dz 0
+
N X k=0
ω(k)Vn(µ,ν) (zk ; α, β) Vm(µ,ν) (zk ; α, β) = λn δn,m ,
(16)
where λn > 0, ρ(z) and ω(k) are the positive definite continuous and discrete weight functions, respectively. It remains an open problem to determine these weight functions analytically. The finite discrete spectrum {zk }N k=0 could be determined from (µ,ν) the condition that forces the asymptotics (n → ∞) of Vn (z; α, β) to vanish (see, for example, Section III and the Appendix in [1] where a detailed explanation is given and a procedure to obtain the spectrum from the asymptotics is carried out, and also see [2]). (µ,ν) Here we can also define the orthonormal version of the polynomial Qn (z; α, θ ) √ (µ,ν) (µ,ν) ˜ n (z; α, θ) = λn Qn (z; α, θ ) but now with as Q (n + 1)(n + µ + ν + 1)(2n + µ + ν + 3) (2n + µ + ν + 1) − 4α 2 λn+1 . (17) = λn (n + µ + 1)(n + ν + 1)(2n + µ + ν + 1) (2n + µ + ν + 3) − 4α 2
The recursion coefficients of the associated symmetric three-term recursion relation are # " µ2 − ν 2 µ+ν+1 2 1 cos θ + n+ − α2 , an = sin θ (2n + µ + ν)(2n + µ + ν + 2) 2 (18a)
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A. D. ALHAIDARI
(n + 1)(n + µ + 1)(n + ν + 1)(n + µ + ν + 1) (2 sin θ )2 (2n + µ + ν + 1) (2n + µ + ν + 1)2 − 4α 2 (2n + µ + ν + 3)2 − 4α 2 × . (18b) (2n + µ + ν + 2)2 (2n + µ + ν + 3) Note, however, that the associated Jacobi matrix (13) in this case is not compact since the asymptotics of these recursion coefficients go as an ∼ n2 and bn ∼ n2 . bn2 =
5. Deformation of the Askey scheme Our recent studies in [30] and [39] seem to suggest that the type of deformation in the recursion relation like that of the Jacobi polynomial in Eq. (8) is in fact common to a larger class of orthogonal polynomials: the Askey scheme of hypergeometric orthogonal polynomials [20]. This scheme consists of two chains of hypergeometric orthogonal polynomials. One of them is a continuous set with the Wilson polynomials at the top of the chain that contains the continuous dual Hahn, continuous Hahn, Meixner–Pollaczek, Jacobi, Laguerre, etc. polynomials. The other is a discrete set with the Racah polynomials at the top of the chain that includes, the dual Hahn, Hahn, Meixner, Krawtchouk, Charlier, etc. The polynomials in each chain are obtained from that at the top by certain limits and specializations of the hypergeometric functions (e.g. 4 F3 → 3 F2 → 2 F1 → 1 F1 ). We write the three-term recursion relation of the original polynomials in the scheme generically as follows γ
γ
γ
xPnγ (x) = anγ Pnγ (x) + bn−1 Pn−1 (x) + cnγ Pn+1 (x),
(19)
where γ stands for a finite set of parameters and x is in a continuous discrete γ γ and/or γ set (finite or countably infinite). The recursion coefficients an , bn , cn depend on n and γ but are independent of x. The monic version of the polynomial can be written as γ Fn+1 γ γ γ γ γ Qn (x) = Fn Pn (x), where and F0 = 1. γ = cn Fn γ
γ
Multiplying (19) by Fn , turns it into the following three-term recursion for Qn (x), γ
γ
γ γ
γ
γ
xQγn (x) = anγ Qγn (x) + Bn−1 Qn−1 (x) + Qn+1 (x),
(20)
where Bn = bn cn . For these polynomials to be orthogonal with respect to a positive γ definite weight function, Favard’s theorem required that Bn be positive definite, γ γ which in turn requires that bn cn > 0 in (19). This is true for all polynomials in the Askey scheme and for all polynomial families introduced in this study. As an γ example of (19) in the Askey scheme, we can take the Laguerre polynomial Ln (x) γ γ where x is continuous with x ≥ 0, γ > −1 and an = 2n + γ + 1, bn = −(n + γ + 1), γ cn = −(n + 1). Now, the deformation of the recursion relation (19) is introduced by modifying it such that it reads x P˜nγ (x; λ, α) = anγ + λ (n + σ )2 − α 2 P˜nγ (x; λ, α) γ
γ
γ
+ bn−1 P˜n−1 (x; λ, α) + cnγ P˜n+1 (x; λ, α),
(21)
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
403
where λ is the deformation parameter and σ is a function of the parameter set γ . As an example, the recursion relation (8) above is obtained by deforming that of (µ,ν) the Jacobi polynomial Pn (cos θ ) with x = cos θ, λ = z sin θ and 2σ = µ + ν + 1. Moreover, in [30] and [39], we also encountered modified versions of orthogonal polynomials in the Askey scheme while searching for series solutions of the following second-order linear differential equation x(1 − x)(r − x)
d 2 y(x) + dx 2
a b c dy(x) − − +d x 1−x r −x dx A B C + − − + xD − E y(x) = 0, x 1−x r −x
(22)
where {a, b, c, d, r, A, B, C, D, E} are real parameters with r 6= 0, 1. For d = 0, the equation has four regular singular points at x = {0, 1, r, ∞} and one of its solutions, which we referred to as a “generalized solution” [30], is written as a series of square integrable basis functions similar to (1) with the expansion coefficients being a modified version of the Wilson polynomial W˜ n (z2 ; κ, τ, η, ξ ) which satisfies the deformed recursion relation (21) with x = z2 ,
λ = −r,
2σ = κ + τ + η + ξ − 1,
(23) 1 α 2 = (κ + τ + η + ξ − 1)2 − (κ + η)(τ + ξ ), 4 and the polynomial parameters {κ, τ, η, ξ } are related to the differential equation parameters in a particular way. In [40] the authors refer to W˜ n (z2 ; κ, τ, η, ξ ) as the “Racah-Heun polynomial” but none of its analytic properties was given. On the other hand, for d 6 = 0, Eq. (22) has four singular points with three regular at x = {0, 1, r} and one irregular at infinity. In [39], we obtained a series solution of this differential equation where the expansion coefficients are a modified version of the continuous Hahn polynomial p˜ n (z; κ, τ, η, ξ ) which satisfies the deformed recursion relation (21) with x = κ + iz, 2σ = κ + τ + η + ξ − 1,
λ = ±d −1 , 1 α 2 = (a + b + c − 1)2 − D. 4
(24)
The polynomial parameters {κ, τ, η, ξ } are related to the differential equation parameters {a, b, c, d, r, A, B, C, D, E} via one of two alternative ways depending on the ± sign of λ. It is worth noting that deformations (perturbations) of the three-term recursion relation of a given known polynomial sequence is not new. In an interesting work, P. Maroni looked at the result of a deformation affecting a finite γ γ γ N set of the recursion coefficients, an , bn , cn n=0 in which the type of perturbation γ N γ γ N was additive for an n=0 and multiplicative for bn , cn n=0 (see Section 4.3 of [41]).
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6. Conclusion Due to the relevance of these new (or modified) polynomials, along with their discrete versions, to the solution of various wave equations (with nonconventional but useful potentials) in relativistic and nonrelativistic physics, we encourage experts in the field of orthogonal polynomials to study them, derive their analytic properties and write them in closed form. The sought-after properties of these polynomials include the weight function, generating function, asymptotics, orthogonality, Rodrigues-type formula, forward/backward shift operator relations, zeros, etc. Acknowledgements I am very grateful to Howard Cohl for his extensive and thorough review of the work that resulted in a substantial improvement of its content and presentation. REFERENCES [1] A. D. Alhaidari and M. E. H. Ismail: Quantum mechanics without potential function, J. Math. Phys. 56 (2015), 072107. [2] A. D. Alhaidari and T. J. Taiwo: Wilson–Racah Quantum System, J. Math. Phys. 58 (2017), 022101. [3] A. D. Alhaidari, H. Bahlouli and M. E. H. Ismail: The Dirac–Coulomb Problem: a mathematical revisit, J. Phys. A 45 (2012), 365204. [4] A. D. Alhaidari: Solution of the nonrelativistic wave equation using the tridiagonal representation approach, J. Math. Phys. 58 (2017), 072104. [5] T. S. Chihara: An Introduction to Orthogonal Polynomials, Mathematics and its Applications, Vol. 13 (Gordon and Breach Science Publishers, New York-London-Paris, 1978). [6] M. E. H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge 2009. [7] A. D. Alhaidari and H. Bahlouli: Tridiagonal representation approach in Quantum Mechanics, Phys. Scripta 94 (2019), 125206. [8] E. J. Heller and H. A. Yamani: New L2 approach to quantum scattering: Theory, Phys. Rev. A 9 (1974), 1201. [9] H. A. Yamani and L. Fishmann: J-Matrix method: extensions to arbitrary angular momentum and to Coulomb scattering, J. Math. Phys. 16 (1975), 410. [10] H. A. Yamani and W. P. Reinhardt: L2 discretizations of the continuum: radial kinetic energy and Coulomb Hamiltonian, Phys. Rev. A 11 (1975), 1144. [11] P. C. Ojha: SO(2, 1) Lie algebra, the Jacobi matrix and the scattering states of the Morse oscillator, J. Phys. A 21 (1988), 875. [12] A. D. Alhaidari: An extended class of L2-series solutions of the wave equation, Ann. Phys. 317 (2005), 152. [13] A. D. Alhaidari: Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment, Ann. Phys. 323 (2008), 1709. [14] A. D. Alhaidari and H. Bahlouli: Extending the class of solvable potentials: I. The infinite potential well with a sinusoidal bottom, J. Math. Phys. 49 (2008), 082102. [15] A. D. Alhaidari: Extending the class of solvable potentials: II. Screened Coulomb potential with a barrier, Phys. Scripta 81 (2010), 025013. [16] H. Bahlouli and A. D. Alhaidari: Extending the class of solvable potentials: III. The hyperbolic single wave, Phys. Scripta 81 (2010), 025008. [17] A. D. Alhaidari: Four-parameter 1/r2 singular short-range potential with a rich bound states and resonance spectrum, Theor. Math. Phys. 195 (2018), 861.
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[18] A. D. Alhaidari and T. J. Taiwo: Four-parameter potential box with inverse square singular boundaries, Eur. Phys. J. Plus 133 (2018), 115. [19] I. A. Assi, A. D. Alhaidari and H. Bahlouli: Solution of spin and pseudo-spin symmetric Dirac equation in 1+1 space-time using tridiagonal representation approach, Commun. Theor. Phys. 69 (2018), 241. [20] R. Koekoek, P. A. Lesky and R. F. Swarttouw: Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, Heidelberg 2010. [21] F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders (Editors): NIST Digital Library of Mathematical Functions, Release 1.0.22 of 2019-03-15 (http://dlmf.nist.gov). [22] CAOP Computer Algebra & Orthogonal Polynomials, http://www.caop.org/ (maintained by Wolfram Koepf, University of Kassel) [23] W. Koepf and D. Schmersau: Recurrence equations and their classical orthogonal polynomial solutions, Appl. Math. Comput. 128 (2002), 303. [24] Private communications with Yohji van Weert (November 2017 and January 2018). [25] W. Van Assche: Solution of an open problem about two families of orthogonal polynomials, SIGMA 15 (2019), 005. [26] A. D. Alhaidari: Open problem in orthogonal polynomials, arXiv:1709.06081v1. [27] A. D. Alhaidari: Orthogonal polynomials associated with nonconventional potentials, The Electronic News Net of the SIAM Activity Group on Orthogonal Polynomials and Special Functions 24, Number 5 (2017), pp. 7–10, communicated by W. van Assche. [28] Y. T. Li: Private communication with the author. [29] D. D. Tcheutia: Recurrence equations and their classical orthogonal polynomial solutions on a quadratic or q-quadratic lattice, J. Differ. Equ. Appl. 25 (2019), 969. [30] A. D. Alhaidari: Series solutions of Heun-type equation in terms of orthogonal polynomials, J. Math. Phys. 59 (2018), 113507. [31] H. A. Yamani, M. S. Abdelmonem and A. D. Al-Haidari: Extracting density information from finite Hamiltonian matrices, Phys. Rev. A 62 (2000), 052103. [32] A. D. Alhaidari: Density of states engineering: normalized energy density of states band structure using the tridiagonal representation approach, Can. J. Phys. 96 (2018), 275. [33] M. E. H. Ismail and E. Koelink: The J-matrix method, Adv. Appl. Math. 46 (2011), 379. [34] M. E. H. Ismail and E. Koelink: Spectral properties of operators using tridiagonalization, Anal. Appl. 10 (2012), 327. [35] S. A. Al-Buradah, H. Bahlouli and A. D. Alhaidari: Energy spectrum of a generalized Scarf potential using the Asymptotic Iteration Method and the Tridiagonal Representation Approach, J. Math. Phys. 58 (2017), 083501. [36] R. De, R. Dutt, U. Sukhatme: Mapping of shape invariant potentials under point canonical transformations, J. Phys. A 25 (1992), L843. [37] F. Cooper, A. Khare and U. Sukhatme: Supersymmetry in Quantum Mechanics, World Scientific, Singapore 2004. [38] W. Van Assche: Compact Jacobi matrices: from Stieltjes to Krein and M (a; b), Annales de la Facult´e des sciences de Toulouse: Math´ematiques, S´erie 6, S5 (1996), p. 195–215. [39] A. D. Alhaidari: Series solution of a ten-parameter second order differential equation with three regular and one irregular singularities, arXiv:1811.11266 [math-ph], to be published in Theor. Math. Phys. 201 (2019). [40] F. A. Gr¨unbaum, L. Vinet and A. Zhedanov: Tridiagonalization and the Heun equation, J. Math. Phys. 58 (2017), 031703. [41] P. Maroni: Une th´eorie alg´ebrique des polynˆomes orthogonaux. Application aux polynˆomes orthogonaux semi-classiques, in “Orthogonal polynomials and their applications” edited by C. Brezinski, L. Gori and A. Ronveaux (Erice, 1990), 95-130, IMACS, Ann. Comput. Appl. Math. 9, Baltzer, Basel, 1991.
REPORTS ON MATHEMATICAL PHYSICS
No. 3
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS A BDULAZIZ D. A LHAIDARI Saudi Center for Theoretical Physics, P.O. Box 32741, Jeddah 21438, Saudi Arabia (e-mail: [email protected]) (Received May 7, 2019 — Revised June 13, 2019) Using an algebraic method for solving the wave equation in quantum mechanics, we encountered new families of orthogonal polynomials on the real line. The properties of the physical system (e.g. energy spectrum, phase shift, density of states, etc.) are obtained from the properties of these polynomials. One of these new families is composed of four-parameter polynomials describing a discrete spectrum of the corresponding quantum mechanical system. Another that appeared while solving a Heun-type equation has a mix of continuous and discrete spectra. Based on these results, we introduce a modification of the hypergeometric polynomials in the Askey scheme. Up to now, all of these polynomials are defined only by their threeterm recursion relations and initial values. However, their other properties like weight functions, generating functions, orthogonality, Rodrigues-type formula, etc., are yet to be derived analytically. Obtaining these properties is an open problem in orthogonal polynomials. Keywords: tridiagonal representation, orthogonal polynomials, potential functions, asymptotics, recursion relation, spectrum formula.
1.
Introduction The wave function in quantum mechanics could be viewed as an element of an infinite-dimensional vector space with local unit vectors. Therefore, in one of the algebraic formulations of quantum mechanics, the wave function at an energy E, ψE (x), is written as a convergent sum over a complete set of square integrable basis functions in configuration space with coordinate x, X ψE (x) = fn (E)φn (x), (1) n
where {φn (x)} are the basis elements (local unit vectors) and {fn (E)} are the expansion coefficients (projections of the wave function along the unit vectors) as a function of energy. All physical information about the system, both structural and dynamical, are contained in these expansion coefficients. In our earlier publications [1–4], we developed the so-called “Tridiagonal Representation Approach (TRA)” — a new algebraic method for solving the wave equation (e.g. the Schr¨odinger equation or the Dirac equation). In the TRA, the basis elements are chosen such that the matrix representation of the wave operator is tridiagonal. Consequently, the resulting matrix wave equation reduces to a three-term recursion relation for {fn (E)}. If [393]
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we write fn (E) = f0 (E)Pn (ε), where the argument ε depends on the energy and physical parameters, then P0 (ε) = 1 and Favard’s theorem [5, 6] can be used to show that {Pn (ε)} is a complete set of orthogonal polynomials satisfying the said recursion relation. We have shown that the corresponding positive definite weight function is [f0 (E)]2 (see page 4 of [1] and page 3 of [7]). This was derived using the completeness R of the basis set {φn (x)} and normalizability of the wavefunction ψE (x) giving fn (E)fm (E) dε ∼ δn,m . Substituting fn (E) = f0 (E)Pn (ε), turns this integral into [1–7] Z (2) [f0 (E)]2 Pn (ε)Pm (ε) dε ∼ δn,m .
These polynomials are associated with the continuum scattering states of the system when E is a continuous set. On the other hand, the discrete bound states are associated with the discrete version of these polynomials. We reproduced all such polynomials that correspond to well-known physical systems and found new ones as well. For example, we reproduced earlier results by Reinhardt, Heller, Yamani and Fishmann [8–10] that the scattering states of the Coulomb problem are associated with the Meixner–Pollaczek polynomials whereas the bound states are associated with one of its discrete versions; the Meixner polynomials. Moreover, we also reproduced earlier findings by Ojha [11] that the scattering states of the Morse oscillator are associated with the continuous dual Hahn polynomials, whereas the finite number of bound states is associated with their discrete version, the dual Hahn polynomials. Additionally, the continuum scattering states of the hyperbolic P¨oschl–Teller potential correspond to the Wilson polynomials, whereas the finite number of bound states is associated with the Racah polynomials, which are the discrete version of the Wilson polynomials. On the other hand, while using the TRA [12–19], we have encountered a new family of exactly solvable problems that are associated with new orthogonal polynomials that seem to have been over-looked in the traditional mathematics and physics literature. These polynomials are defined, up to now, by their three-term recursion relations and initial value P0 (ε) = 1. However, their other important properties are yet to be derived analytically. These properties include the weight functions, generating functions, asymptotics, orthogonality, Rodrigues-type formulae, etc. The purpose of this outline is to shed light on the most important properties of these remarkable families of orthogonal polynomials, bring awareness, and raise enough interest in the community of orthogonal polynomials specialists in order to study them. In physics, any special functions or orthogonal polynomials that enter in the description of a physical system are considered to be of prime significance. The orthogonal polynomials presented here are associated with nonconventional but interesting and highly significant potential functions in quantum mechanics. All attempts to compare this family of polynomials with known ones failed to produce a match. For example, we tried using the table of recurrence formulae in Chihara’s book [5] and the properties of the hypergeometric orthogonal polynomials in the
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book of Koekoek, Lesky and Swarttouw [20]. We also studied the chapter on Orthogonal Polynomials in the Digital Library of Mathematical Functions [21] and compared with the information available in CAOP (Computer Algebra & Orthogonal Polynomials) [22]. Members of our group tried using computer algebra systems (such as rec2ortho of Koornwinder and Swarttouw or retode of Koepf and Schmersau [23]) to identify the polynomials from their recurrence relations. Additionally, we had various private communications with experts in the field including M. E. H. Ismail, W. Van Assche, and Y. T. Li. Some colleagues and students of Van Assche studied several of these polynomials and partial results were communicated [24] and reported in [25]. Furthermore, in the latter reference [25], Van Assche treated two polynomial families submitted by us as an open problem [26] to the 14th International Symposium on Orthogonal Polynomials, Special Functions and Applications (OPSFA14) held at the University of Kent on 3–7 July 2017. This open problem was also communicated in a newsletter of the SIAM Activity Group on Orthogonal Polynomials and Special Functions [27]. In that open problem, we introduced (µ,ν) two polynomial families of significance in quantum mechanics, Hn (z; α, θ ) and (µ,ν) Gn (z; σ ), and enquired about their properties (e.g. weight function, generating function, orthogonality, asymptotics, etc.). In [25], Van Assche, reported on some (µ,ν) of the properties of the first family, Hn (z; α, θ ), including its spectral nature and asymptotics. These results are stated in Section 3 below. On the other hand, (µ,ν) by the time OPSFA-14 concluded, the second family, Gn (z; σ ) has already been identified by Y. T. Li as a special case of the Wilson polynomials [28] and it was written down explicitly in Section 3 of [25]. Therefore, the polynomial (µ,ν) Gn (z; σ ) is not in the scope of this present study. In [29] an investigation of (µ,ν) the polynomial family Hn (z; α, θ ) by D. D. Tcheutia led to a clear conclusion about its novelty. (µ,ν) In addition to the polynomial Hn (z; α, θ ) presented in Section 3 below, we (µ,ν) also introduce in Section 4 the polynomial Qn (z; α, θ ) that we have encountered recently while solving a physically relevant Heun-type differential equation [30]. We tried our best to give as much information as possible about the properties of these polynomials using numerical means. For example, we could use any one of three numerical routines (analytic continuation, dispersion correction, and Stieltjes imaging methods) to obtain the continuous weight function, employing only the recursion relation coefficients [31, 32]. We could also use the J -matrix method to obtain the discrete spectrum [8, 9, 33, 34]. These numerical results are to be used only as a guide or suggestion for further and more rigorous mathematical analysis. Due to the relevance of these polynomials to the solution of some quantum mechanical problems with nonconventional but useful potentials, we call upon experts in the field of orthogonal polynomials to study them, derive their properties and write them in closed form (e.g. in terms of hypergeometric functions). In the following section, we give an illustrative example of how these polynomials appear in the solution of the Schr¨odinger wave equation in quantum mechanics.
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Two of these polynomials are defined in Sections 3 and 4 whereas in Section 5 we introduce a modification to the three-term recursion relations of the hypergeometric polynomials in the Askey scheme [20]. 2. The Schr¨odinger wave equation of a physical system and its corresponding orthogonal polynomial In this section, we give an illustrative example showing how the three-term recursion relation of one of the new family of orthogonal polynomials appear in the solution of the one-dimensional Schr¨odinger wave equation of a quantum mechanical system with a nonconventional but significant interaction potential. The potential function of interest is V+ − V− sin(π x/L) + V1 sin(π x/L), (3) V (x) = V0 + cos2 (π x/L) where −L/2 ≤ x ≤ L/2. This potential represents an infinite potential well of width L with {V± , V0 , V1 } being a set of real parameters such that V+ +|V− | ≥ −2(π/4L)2 . If V1 6 = 0, then this potential does not have a known exact solution in the conventional formulations of quantum mechanics. The special case with V± = 0 is an infinite square well potential with sinusoidal rather than flat bottom. This has been studied using the TRA by our group in [14, 35]. On the other hand, if V1 = 0 then the potential becomes the Scarf I potential (an alternative form of the trigonometric P¨oschl–Teller potential) which is exactly solvable by the conventional methods [36, 37]. Due to total confinement inside this infinite potential well, the energy spectrum of the system is bounded from below but not from above and it consists of an infinite number of discrete eigenvalues. The proper set of basis functions to expand the physical wave function ψE (x) in a series as shown by Eq. (1), which is also compatible with the tridiagonal representation of the TRA, is
φn (x) = (1 − y)
µ+ 21 /2
(1 + y)
ν+ 21 /2
Pn(µ,ν) (y),
(4)
(µ,ν)
where y(x) = sin(πx/L), Pn (y) is the Jacobi polynomial and the real parameters µ and ν are greater than −1. In the atomic units h¯ = m = 1, the Schr¨odinger wave equation for a point particle of mass m and energy E under the influence of the potential (3) reads as follows 1 d2 + V (x) − E ψE (x) = 0. (5) − 2 dx 2 We rewrite this equation in terms of the independent variable y where the potential function (3) reads V+ −V− y V (x) = V0 + + V1 y. 1 − y2
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Substituting the expansion (1) with the basis elements (4) in (5) and using the differential equation of the Jacobi polynomial, the resulting equation becomes X n
fn
1 ν 2 − 1/4 µ+ν+1 2 1 µ2 − 1/4 + − n+ 2 1−y 2 1+y 2 2E u+ − u− y φn = 0, + 2 − u0 + u1 y + λ 1 − y2
(6)
where λ = π/L and ui = 2Vi /λ2 . Choosing the basis parameters as µ2 = 41 +u+ −u− , ν 2 = 14 + u+ + u− and using the recursion relation of the Jacobi polynomials, we obtain the following three-term recursion relation for the expansion coefficients of the wave function " # µ+ν+1 2 ν 2 − µ2 E − V0 1 n+ + fn (E) = fn (E) V1 u1 2 (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) fn−1 (E) (2n + µ + ν)(2n + µ + ν + 1) 2(n + 1)(n + µ + ν + 1) + fn+1 (E). (2n + µ + ν + 1)(2n + µ + ν + 2) +
(7)
If we write fn (E) = f0 (E)Pn (ε) then the solution space of this recursion relation for Pn (ε) will be divided into two subspaces depending on the values of the physical parameters: 1. (E − V0 )2 ≤ V12 : Comparing the recursion relation to (8) in Section 3 below, (µ,ν) we conclude q that Pn (ε) = Hn (z; α, θ ) with α 2 = 0, cos(θ ) = (E − V0 )/V1 , and z−1 =
2 λ2
V12 − (E − E0 )2 .
2. (E − V0 )2 > V12 : We compare the recursion relation (7) to (8′ ) in Section 3 (µ,ν) 2 below and conclude that Pn (ε) q = Hn (−iz; α, iθ ) with α = 0, cosh(θ ) = (E − V0 )/|V1 |, and z−1 =
2 λ2
(E − V0 )2 − V12 .
Therefore, the physical property of this system suggests that the polynomial (µ,ν) should be discrete with Hn (−iz; α, iθ ) having countably infinite spectrum. Moreover, it is easy to see that in the limit V1 → 0, relation (7) gives directly 1 2 µ+ν+1 2 E = V0 + λ n + , 2 2 (µ,ν) Hn (z; α, θ)
which is the well-known energy spectrum of the Scarf I potential [36, 37]. The same TRA treatment of other quantum mechanical systems with nonconventional potential functions yields similar results [12–19].
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3. A polynomial that generalizes the Jacobi polynomial (µ,ν)
The four-parameter polynomial, which we designate as Hn (z; α, θ ), satisfies the following three-term recursion relation: µ+ν+1 2 2 (µ,ν) − α z sin θ (cos θ)Hn (z; α, θ) = n+ 2 ν 2 − µ2 + Hn(µ,ν) (z; α, θ ) (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) + H (z; α, θ ) (2n + µ + ν)(2n + µ + ν + 1) n−1 2(n + 1)(n + µ + ν + 1) (µ,ν) H (z; α, θ ), (8) + (2n + µ + ν + 1)(2n + µ + ν + 2) n+1 where z is real, 0 < θ ≤ π and n = 1, 2, 3, . . .. It is a polynomial of degree n in z. Setting z ≡ 0 turns (8) into the recursion relation of the Jacobi polynomials (µ,ν) Pn (cos θ). Physical requirements dictate that µ and ν be greater than −1. The polynomial of the first kind satisfies this recursion relation together with (µ,ν) H0 (z; α, θ) = 1 and (µ,ν)
H1
(z; α, θ) =
µ−ν 1 + (µ + ν + 2) 2 2 1 × cos(θ ) − (µ + ν + 1)2 − α 2 z sin(θ ) , 4
(9)
(µ,ν)
which is obtained from (8) by setting n =0 and H−1 (z; α, θ ) ≡ 0. We divide 2 the recursion relation (8) by n + µ+ν+1 sin(θ ) and then define the orthogonal 2 √ (µ,ν) (µ,ν) version of this polynomial as H˜ n (z; α, θ ) = λn Hn (z; α, θ ), where (n + 1)(n + µ + ν + 1)(2n + µ + ν + 3) (2n + µ + ν + 3) − 4α 2 λn+1 , (10) = λn (n + µ + 1)(n + ν + 1)(2n + µ + ν + 1) (2n + µ + ν + 1) − 4α 2 (µ,ν) and λ0 = 1. Then H˜ n (z; α, θ) satisfies the following symmetric three-term recursion relation (µ,ν) (µ,ν) zH˜ n(µ,ν) (z; α, θ) = an H˜ n(µ,ν) (z; α, θ ) − bn−1 H˜ (z; α, θ ) − bn H˜ (z; α, θ ), (11) n−1
n+1
where
an = cos θ +
µ2 − ν 2 (2n + µ + ν)(2n + µ + ν + 2) (" # ) . µ+ν+1 2 2 n+ − α sin θ , 2
(12a)
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
bn2 =
26 (n + 1)(n + µ + 1)(n + ν + 1) (2n + µ + ν + 1)(2n + µ + ν + 3) (n + µ + ν + 1)(sin θ )−2 (2n + µ + ν + 2)−2 . × (2n + µ + ν + 1)2 − 4α 2 (2n + µ + ν + 3)2 − 4α 2
399
(12b)
The observation that the asymptotics (n → ∞) of these recursion coefficients go as O(1/n2 ) leads to the conclusion that the Jacobi matrix associated with (µ,ν) H˜ n (z; α, θ), which reads a0 b0 b0 a1 b1 b1 a2 b2 (13) J = b2 a3 b3 × × × × × × × ×
0
0
is a compact operator (in fact, a trace class operator) [25, 38]. This implies that the spectrum of J is a countable set {zk , k, ∈ N} with one accumulation point at P∞ zero (i.e. limk→∞ zk = 0). The trace class condition implies that |z | is finite k k=0 (µ,ν) ˜ [38]. Consequently, the orthogonality measure for Hn (z; α, θ ) is a discrete mea(µ,ν) sure supported on this countable set. The asymptotic limn→∞ zn H˜ n (1/z; α, θ ) is an entire function with zeros at the points 1/zk [25]. Recently, Tcheutia [29] managed to extend the Maple implementation retode of Koepf and Schmersau [23] to deal with classical orthogonal polynomials on quadratic and q(µ,ν) quadratic lattices and concluded that “the polynomial family Hn (z; α, θ ) is not related to a known classical orthogonal polynomial sequence” and that its “recurrence equation may lead to a new family of orthogonal polynomial sequences”. In [4, 12–18], we presented examples of physical systems associated with (µ,ν) the orthogonal polynomial Hn (z; α, θ ). We derived the corresponding three-term recursion relation and showed how the polynomial argument and parameters are related to the potential function parameters and the energy. Using this association, we demonstrate how to obtain the physical properties of the quantum mechanical systems (e.g. bound state energy spectrum, scattering phase shift, density of states, etc.) from those of the polynomial (such as the weight function, asymptotics, zeros, etc.). If the parameters of the problem which correspond to the recursion relation (8) violate the condition that |cos θ| ≤ 1 then we rewrite (8) with θ 7→ iθ and z 7→ iz giving
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µ+ν+1 2 2 − α z sinh θ = n+ 2 ν 2 − µ2 + Hn(µ,ν) (−iz; α, iθ ) (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) Hn−1 (−iz; α, iθ ) + (2n + µ + ν)(2n + µ + ν + 1) 2(n + 1)(n + µ + ν + 1) (µ,ν) + H (−iz; α, iθ ). (2n + µ + ν + 1)(2n + µ + ν + 2) n+1
(cosh θ)Hn(µ,ν) (−iz; α, iθ)
(8′ )
4. A polynomial arising from the solution of a Heun-type ordinary differential equation Recently, while transforming the Schr¨odinger wave equation to a Heun-type equation, we encountered a set of new or generalized orthogonal polynomials [30]. (µ,ν) One of these polynomials has four parameters and we designate it as Qn (z; α, θ ). This polynomial satisfies the following three-term recursion relation −1 µ+ν+1 2 2 z sin θ (cos θ)Q(µ,ν) (z; α, θ) = n + − α n 2 ν 2 − µ2 + Q(µ,ν) (z; α, θ ) n (2n + µ + ν)(2n + µ + ν + 2) 2(n + µ)(n + ν) (µ,ν) (z; α, θ ) Q + (2n + µ + ν)(2n + µ + ν + 1) n−1 2(n + 1)(n + µ + ν + 1) (µ,ν) + Qn+1 (z; α, θ ), (14) (2n + µ + ν + 1)(2n + µ + ν + 2)
where z is real, 0 < θ ≤ π and n = 1, 2, 3, . . .. Note the inverse power of the square bracket. This constitutes a major difference from the recurrence relation (8). (µ,ν) (µ,ν) Here as well, Q0 (z; α, θ) = 1 and Q1 (z; α, θ ) is obtained from (14) by setting (µ,ν) n = 0 and Q−1 (z; α, θ) ≡ 0. (µ,ν) Numerical experimentations suggest that the polynomial Qn (z; α, θ ) has a purely continuous spectrum over the entire real line. However, it has another version, which (µ,ν) we refer to as Vn (z; α, β), whose recursion relation is obtained from (14) by the replacement θ 7 → iθ and z 7 → iz giving −1 (µ,ν) µ+ν+1 2 2 2 2 1 + β Vn (z; α, β) = z 1 − β n+ −α 2 2 ν 2 − µ2 β + V (µ,ν) (z; α, β) (2n + µ + ν)(2n + µ + ν + 2) n
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
4(n + µ)(n + ν)β (µ,ν) V (z; α, β) (2n + µ + ν)(2n + µ + ν + 1) n−1 4(n + 1)(n + µ + ν + 1)β (µ,ν) + V (z; α, β), (2n + µ + ν + 1)(2n + µ + ν + 2) n+1
401
+
(15)
where z ≥ 0 and β = e−θ with θ > 0. Using only numerical means and without rigorous mathematical proof, we found that if α is pure imaginary, then the spectrum is purely continuous. However, if α is real then the spectrum is a mix of a continuous positive spectrum and a discrete negative spectrum of finite size N + 1, . This behaviour where N is the largeset integer less than or equal to |α| − µ+ν+1 2 2 mimics that of the continuous dual Hahn polynomial Sn (x ; a, b, c) and the Wilson polynomial Wn (x 2 ; a, b, c, d) for negative a. Here α 2 plays the role of −a. In this case, the polynomial satisfies a generalized orthogonality relation that consists of a continuous part and a discrete part similar to those of the Wilson polynomials and the continuous dual Hahn polynomials [5, 20]. This is also suggested by physical arguments concerning quantum systems with a mix of discrete bound states and continuous scattering states (see, for example, Section II of [7]). Consequently, the orthogonality will be of the following form Z ∞ ρ(z)Vn(µ,ν) (z; α, β)Vm(µ,ν) (z; α, β) dz 0
+
N X k=0
ω(k)Vn(µ,ν) (zk ; α, β) Vm(µ,ν) (zk ; α, β) = λn δn,m ,
(16)
where λn > 0, ρ(z) and ω(k) are the positive definite continuous and discrete weight functions, respectively. It remains an open problem to determine these weight functions analytically. The finite discrete spectrum {zk }N k=0 could be determined from (µ,ν) the condition that forces the asymptotics (n → ∞) of Vn (z; α, β) to vanish (see, for example, Section III and the Appendix in [1] where a detailed explanation is given and a procedure to obtain the spectrum from the asymptotics is carried out, and also see [2]). (µ,ν) Here we can also define the orthonormal version of the polynomial Qn (z; α, θ ) √ (µ,ν) (µ,ν) ˜ n (z; α, θ) = λn Qn (z; α, θ ) but now with as Q (n + 1)(n + µ + ν + 1)(2n + µ + ν + 3) (2n + µ + ν + 1) − 4α 2 λn+1 . (17) = λn (n + µ + 1)(n + ν + 1)(2n + µ + ν + 1) (2n + µ + ν + 3) − 4α 2
The recursion coefficients of the associated symmetric three-term recursion relation are # " µ2 − ν 2 µ+ν+1 2 1 cos θ + n+ − α2 , an = sin θ (2n + µ + ν)(2n + µ + ν + 2) 2 (18a)
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(n + 1)(n + µ + 1)(n + ν + 1)(n + µ + ν + 1) (2 sin θ )2 (2n + µ + ν + 1) (2n + µ + ν + 1)2 − 4α 2 (2n + µ + ν + 3)2 − 4α 2 × . (18b) (2n + µ + ν + 2)2 (2n + µ + ν + 3) Note, however, that the associated Jacobi matrix (13) in this case is not compact since the asymptotics of these recursion coefficients go as an ∼ n2 and bn ∼ n2 . bn2 =
5. Deformation of the Askey scheme Our recent studies in [30] and [39] seem to suggest that the type of deformation in the recursion relation like that of the Jacobi polynomial in Eq. (8) is in fact common to a larger class of orthogonal polynomials: the Askey scheme of hypergeometric orthogonal polynomials [20]. This scheme consists of two chains of hypergeometric orthogonal polynomials. One of them is a continuous set with the Wilson polynomials at the top of the chain that contains the continuous dual Hahn, continuous Hahn, Meixner–Pollaczek, Jacobi, Laguerre, etc. polynomials. The other is a discrete set with the Racah polynomials at the top of the chain that includes, the dual Hahn, Hahn, Meixner, Krawtchouk, Charlier, etc. The polynomials in each chain are obtained from that at the top by certain limits and specializations of the hypergeometric functions (e.g. 4 F3 → 3 F2 → 2 F1 → 1 F1 ). We write the three-term recursion relation of the original polynomials in the scheme generically as follows γ
γ
γ
xPnγ (x) = anγ Pnγ (x) + bn−1 Pn−1 (x) + cnγ Pn+1 (x),
(19)
where γ stands for a finite set of parameters and x is in a continuous discrete γ γ and/or γ set (finite or countably infinite). The recursion coefficients an , bn , cn depend on n and γ but are independent of x. The monic version of the polynomial can be written as γ Fn+1 γ γ γ γ γ Qn (x) = Fn Pn (x), where and F0 = 1. γ = cn Fn γ
γ
Multiplying (19) by Fn , turns it into the following three-term recursion for Qn (x), γ
γ
γ γ
γ
γ
xQγn (x) = anγ Qγn (x) + Bn−1 Qn−1 (x) + Qn+1 (x),
(20)
where Bn = bn cn . For these polynomials to be orthogonal with respect to a positive γ definite weight function, Favard’s theorem required that Bn be positive definite, γ γ which in turn requires that bn cn > 0 in (19). This is true for all polynomials in the Askey scheme and for all polynomial families introduced in this study. As an γ example of (19) in the Askey scheme, we can take the Laguerre polynomial Ln (x) γ γ where x is continuous with x ≥ 0, γ > −1 and an = 2n + γ + 1, bn = −(n + γ + 1), γ cn = −(n + 1). Now, the deformation of the recursion relation (19) is introduced by modifying it such that it reads x P˜nγ (x; λ, α) = anγ + λ (n + σ )2 − α 2 P˜nγ (x; λ, α) γ
γ
γ
+ bn−1 P˜n−1 (x; λ, α) + cnγ P˜n+1 (x; λ, α),
(21)
OPEN PROBLEM IN ORTHOGONAL POLYNOMIALS
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where λ is the deformation parameter and σ is a function of the parameter set γ . As an example, the recursion relation (8) above is obtained by deforming that of (µ,ν) the Jacobi polynomial Pn (cos θ ) with x = cos θ, λ = z sin θ and 2σ = µ + ν + 1. Moreover, in [30] and [39], we also encountered modified versions of orthogonal polynomials in the Askey scheme while searching for series solutions of the following second-order linear differential equation x(1 − x)(r − x)
d 2 y(x) + dx 2
a b c dy(x) − − +d x 1−x r −x dx A B C + − − + xD − E y(x) = 0, x 1−x r −x
(22)
where {a, b, c, d, r, A, B, C, D, E} are real parameters with r 6= 0, 1. For d = 0, the equation has four regular singular points at x = {0, 1, r, ∞} and one of its solutions, which we referred to as a “generalized solution” [30], is written as a series of square integrable basis functions similar to (1) with the expansion coefficients being a modified version of the Wilson polynomial W˜ n (z2 ; κ, τ, η, ξ ) which satisfies the deformed recursion relation (21) with x = z2 ,
λ = −r,
2σ = κ + τ + η + ξ − 1,
(23) 1 α 2 = (κ + τ + η + ξ − 1)2 − (κ + η)(τ + ξ ), 4 and the polynomial parameters {κ, τ, η, ξ } are related to the differential equation parameters in a particular way. In [40] the authors refer to W˜ n (z2 ; κ, τ, η, ξ ) as the “Racah-Heun polynomial” but none of its analytic properties was given. On the other hand, for d 6 = 0, Eq. (22) has four singular points with three regular at x = {0, 1, r} and one irregular at infinity. In [39], we obtained a series solution of this differential equation where the expansion coefficients are a modified version of the continuous Hahn polynomial p˜ n (z; κ, τ, η, ξ ) which satisfies the deformed recursion relation (21) with x = κ + iz, 2σ = κ + τ + η + ξ − 1,
λ = ±d −1 , 1 α 2 = (a + b + c − 1)2 − D. 4
(24)
The polynomial parameters {κ, τ, η, ξ } are related to the differential equation parameters {a, b, c, d, r, A, B, C, D, E} via one of two alternative ways depending on the ± sign of λ. It is worth noting that deformations (perturbations) of the three-term recursion relation of a given known polynomial sequence is not new. In an interesting work, P. Maroni looked at the result of a deformation affecting a finite γ γ γ N set of the recursion coefficients, an , bn , cn n=0 in which the type of perturbation γ N γ γ N was additive for an n=0 and multiplicative for bn , cn n=0 (see Section 4.3 of [41]).
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6. Conclusion Due to the relevance of these new (or modified) polynomials, along with their discrete versions, to the solution of various wave equations (with nonconventional but useful potentials) in relativistic and nonrelativistic physics, we encourage experts in the field of orthogonal polynomials to study them, derive their analytic properties and write them in closed form. The sought-after properties of these polynomials include the weight function, generating function, asymptotics, orthogonality, Rodrigues-type formula, forward/backward shift operator relations, zeros, etc. Acknowledgements I am very grateful to Howard Cohl for his extensive and thorough review of the work that resulted in a substantial improvement of its content and presentation. REFERENCES [1] A. D. Alhaidari and M. E. H. Ismail: Quantum mechanics without potential function, J. Math. Phys. 56 (2015), 072107. [2] A. D. Alhaidari and T. J. Taiwo: Wilson–Racah Quantum System, J. Math. Phys. 58 (2017), 022101. [3] A. D. Alhaidari, H. Bahlouli and M. E. H. Ismail: The Dirac–Coulomb Problem: a mathematical revisit, J. Phys. A 45 (2012), 365204. [4] A. D. Alhaidari: Solution of the nonrelativistic wave equation using the tridiagonal representation approach, J. Math. Phys. 58 (2017), 072104. [5] T. S. Chihara: An Introduction to Orthogonal Polynomials, Mathematics and its Applications, Vol. 13 (Gordon and Breach Science Publishers, New York-London-Paris, 1978). [6] M. E. H. Ismail: Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge 2009. [7] A. D. Alhaidari and H. Bahlouli: Tridiagonal representation approach in Quantum Mechanics, Phys. Scripta 94 (2019), 125206. [8] E. J. Heller and H. A. Yamani: New L2 approach to quantum scattering: Theory, Phys. Rev. A 9 (1974), 1201. [9] H. A. Yamani and L. Fishmann: J-Matrix method: extensions to arbitrary angular momentum and to Coulomb scattering, J. Math. Phys. 16 (1975), 410. [10] H. A. Yamani and W. P. Reinhardt: L2 discretizations of the continuum: radial kinetic energy and Coulomb Hamiltonian, Phys. Rev. A 11 (1975), 1144. [11] P. C. Ojha: SO(2, 1) Lie algebra, the Jacobi matrix and the scattering states of the Morse oscillator, J. Phys. A 21 (1988), 875. [12] A. D. Alhaidari: An extended class of L2-series solutions of the wave equation, Ann. Phys. 317 (2005), 152. [13] A. D. Alhaidari: Analytic solution of the wave equation for an electron in the field of a molecule with an electric dipole moment, Ann. Phys. 323 (2008), 1709. [14] A. D. Alhaidari and H. Bahlouli: Extending the class of solvable potentials: I. The infinite potential well with a sinusoidal bottom, J. Math. Phys. 49 (2008), 082102. [15] A. D. Alhaidari: Extending the class of solvable potentials: II. Screened Coulomb potential with a barrier, Phys. Scripta 81 (2010), 025013. [16] H. Bahlouli and A. D. Alhaidari: Extending the class of solvable potentials: III. The hyperbolic single wave, Phys. Scripta 81 (2010), 025008. [17] A. D. Alhaidari: Four-parameter 1/r2 singular short-range potential with a rich bound states and resonance spectrum, Theor. Math. Phys. 195 (2018), 861.
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