Redalyc.Exact Spectrum and Wave Functions of the Hyperbolic Scarf

Revista Mexicana de Física ISSN: 0035-001X [email protected] Sociedad Mexicana de Física A.C. México

Alvarez–Castillo, D.E.; Kirchbach, M. Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of finite Romanovski polynomials Revista Mexicana de Física, vol. 53, núm. 2, diciembre, 2007, pp. 143-154 Sociedad Mexicana de Física A.C. Distrito Federal, México

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Exact spectrum and wave functions of the hyperbolic Scarf potential in terms of ﬁnite Romanovski polynomials

D.E. Alvarez-Castillo and M. Kirchbach Instituto de F´ısica, Universidad Autonoma´ de San Luis Potos´ı, Av. Manuel Nava 6, San Luis Potos´ı, S.L.P. 78290, Mexico. Recibido el 1 de septiembre de 2006; aceptado el 24 de febrero de 2007

The Schrodinger¨ equation with the hyperbolic Scarf potential reported so far in the literature is somewhat artificially manipulated into the form of the Jacobi equation with an imaginary argument and parameters that are complex conjugate to each other. Instead we here solve the former equation anew and make the case that it reduces straightforward to a particular form of the generalized real hypergeometric equation whose solutions are referred to in the mathematics literature as the finite Romanovski polynomials, in reference to the observation that for any parameter set only a finite number of such polynomials appear to be orthogonal. This is a qualitatively new integral property that does not copy any of the features of the Jacobi polynomials. In this manner, the finite number of bound states within the hyperbolic Scarf potential is brought into correspondence with a finite system of a new class of orthogonal polynomials. This work adds a new example to the circle of the problems on the Schrodinger¨ equation. The techniques used by us extend the teachings on the Sturm-Liouville theory of ordinary differential equations beyond their standard presentation in the textbooks on mathematical methods in physics. Keywords: Schrodinger equation; Scarf potentials; Romanovski polynomials.

La solucion´ a la ecuacion´ de Schrodinger¨ con el potencial de Scarf hiperbolico´ reportada hasta ahora en la literatura f´ısica esta´ manipulada artificialmente para obtenerla en la forma de los polinomios de Jacobi con argumentos imaginarios y parametros´ que son complejos conju- gados entre ellos. En lugar de eso, nosotros resolvimos la nueva ecuacion´ obtenida y desarrollamos el caso en el que realmente se reduce a una forma particular de la ecuacion´ hipergeometrica´ generalizada real, cuyas soluciones se refieren en la literatura matematica´ como los polinomios finitos de Romanovski. La notacion´ de finito se refiere a que, para cualquier parametro´ fijo, solo un numero´ finito de dichos polinomios son ortogonales. Esta es una nueva propiedad cualitativa de la integral que no surge como copia de ninguna de las caracter´ısticas de los polinomios de Jacobi. De esta manera, el numero´ finito de estados en el potencial de Scarf hiperbolico´ es consistente en correspondencia a un sistema finito de polinomios ortogonales de una nueva clase.

Descriptores: Ecuacion´ de Schrodinger; potenciales de Scarf; polinomios de Romanovski.

PACS: 02.30.Gp; 03.65.Ge; 12.60.Jv

1. Introduction then employing the relevant group algebra in order to con- struct the solutions as the group representation spaces [8, 9]. The exactly solvable Schrodinger¨ equations occupy a pole Finally, there is also the most recent and powerful method position in quantum mechanics insofar as most of them re- of super-symmetric quantum mechanics (SUSYQM) which late directly to physical systems. Suffices to mention as considers the special class of Schrodinger¨ equations (in units prominent examples the quantum Kepler or Coulomb prob- of ~ = 1 = 2m) that allow for a factorization according lem and its importance in the description of the discrete spec- to [10-12], trum of the hydrogen atom [1], the harmonic-oscillator, the µ 2 ¶ Hulthen, and the Morse potentials with their relevance to vi- d (H(z) − en) ψn(z) = − 2 + v(z) − en ψn(z) = 0 , brational spectra [2, 3]. Another good example is given by dz the Poschl-Teller¨ potential [4] which appears as an effective + − H(z) = A (z)A (z) + e0 , mean field in many-body systems with δ-interactions [5]. In µ ¶ d terms of path integrals, the criteria for exact resolvability of A±(z) = ± + U(z) . (1) the Schrodinger¨ equation is the existence of exactly solvable dz corresponding path integrals [6]. H(z) There are various methods of finding the exact solutions Here, stands for the (one-dimensional) Hamiltonian, U(z) A±(z) of the Schrodinger¨ equation (SE) for the bound states, an is- is the so called super-potential, and are the lad- der operators connecting neighboring solutions. The super- sue on which we shall focus in the present work. The tra- potential allows us to recover the ground state wave function, ditional method, to be pursued by us here, consists in re- ψgst(z), as ducing SE by an appropriate change of the variables to that R − z U(y)dy very form of the generalized hypergeometric equation [7] ψgst(z) ∼ e . (2) whose solutions are polynomials, the majority of them being well known. The second method suggests to first un- The excited states are then built up on top of ψgst(z) through veiling the dynamical symmetry of the potential problem and the repeated action of the A+(z) operators. 144 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

FIGURE 2. The hyperbolic Scarf potential (Scarf II) for the set of FIGURE 1. The trigonometric Scarf potential (Scarf I) for the set parameters, a = 10, b = 5, and α = 1. The horizontal lines of parameters, a = 10, b = 5, and α = 1. The horizontal lines represent the energies, en, of the bound states. represent the discrete levels. Scarf I is transformed into the so-called hyperbolic Scarf po- (a,b) 1.1. The trigonometric Scarf potential tential (Scarf II), here denoted by vh (z) and displayed in Fig. 2, Super-symmetric quantum mechanics manages a family of (a,b) 2 2 2 2 exactly solvable potentials (see Refs. 11 to 13 for details), vh (z) = a + (b − a − aα)sech αz one of which is the so called trigonometric Scarf potential + b(2a + α)sechαz tanh αz . (7) (Scarf I) [14], here denoted by vt(z) and given by The latter potential has also been found independently within v(a,b)(z) = − a2 + (a2 + b2 − aα) sec2 αz t the framework of super-symmetric quantum mechanics while − b(2a + α) tan αz sec αz. (3) exploring the super-potential [11, 13, 15]:

It has been used in the construction of a periodic potential and U(z) = a tanh αz + bsechαz . (8) employed in one-dimensional crystal models in solid state physics. Upon the above substitutions, and taking α = 1 for simplic- The exact solution of the Schrodinger¨ equation with the ity, the energy changes to trigonometric Scarf potential (displayed in Fig. 1) is well ² = e − a2 = −(a − n)2 , n = 0, 1, 2, ... < a . (9) known [11, 13] and given in terms of the Jacobi polynomi- n n als, P β,α(x), as n It is important to notice that, while the trigonometric q 1 1 Scarf potential allows for an infinite number of bound states, γ δ γ− 2 ,δ− 2 ψn(x) = (1 − x) (1 + x) Pn (x), the number of discrete levels within the hyperbolic Scarf potential is finite, a difference that will be explained in Sec. 3 x = sin αz, below. Yet the most violent changes seem to be suffered by γ− 1 ,δ− 1 γ− 1 δ− 1 w 2 2 (x) = (1 − x) 2 (1 + x) 2 , the Jacobi weight function in Eq. (4) and are due to the open- ing of the finite interval [−1, +1] toward infinity 1 1 γ = (a − b), δ = (a + b) . (4) α α x= sin αz ∈ [−1, 1] −→ x= sinh αz ∈ [−∞, +∞] . (10)

γ− 1 ,δ− 1 Here, w 2 2 (x) stands for the weight function from 1 1 In this case, the wave functions become [11, 16, 17], γ− 2 ,δ− 2 which the Jacobi polynomials Pn (x) are obtained via a −1 ∗ 2 − 2 −b tan x η ,η the Rodrigues formula. ψn(−ix) = (1 + x ) e cnPn (−ix) , The corresponding energy spectrum is obtained as 1 η = ib − a − . (11) 2 2 2 ²n = en + a = (a + nα) . (5) Here, cn is some state dependent complex phase to be ﬁxed 1.2. The hyperbolic Scarf potential later on. The latter equation gives the impression that the exact solutions of the hyperbolic Scarf potential rely exclusively By means of the substitutions upon those peculiar Jacobi polynomials with imaginary arguments and complex indices. We here draw attention to the a −→ ia , α −→ −iα , b −→ b , (6) fact that this need not be so.

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. . . 145

1.3. The goal physics, Scarf II finds application in studies of the non- perturbative sector of gauge theories by means of toy mod- The goal of this work is to solve the Schrodinger¨ els such as the scalar field theory in (1+1) space-time di- equation with the hyperbolic Scarf potential mensions. Here, one encounters the so called “kink-like” anew and to make the case that it reduces in solutions which are nothing more than static solitons. The a straightforward manner to a particular form spatial derivative of the kink-like solution is viewed as the of the generalized real hypergeometric equation ground state wave function of an appropriately constructed whose solutions are given by a finite set of real Schrodinger¨ equation, which is then employed in the calcu- orthogonal polynomials. In this manner, the fi- lation of the quantum corrections to the first order. In Ref. 17 nite number of bound states within the hyper- it was shown that specifically Scarf II is amenable to a stable bolic Scarf potential is brought in correspon- renormalizable scalar field theory. dence with a finite system of orthogonal poly- The paper is organized as follows. In the next section nomials of a new class. we first highlight in brief the basics of the generalized hy- These polynomials were discovered in 1884 by the En- pergeometric equation and then relate it to the Schrodinger¨ glish mathematician Sir Edward John Routh [18] and re- equation with the hyperbolic Scarf potential. The solutions discovered 45 years later by the Russian mathematician are obtained in terms of finite Romanovski polynomials and Vsevolod Ivanovich Romanovski in 1929 [19] within the are presented in Sec. 3. Section 4 is devoted to the disguise of context of probability distributions. Though they have been the Romanovski polynomials as non–spherical angular func- studied on few occasions in the current mathematical litera- tions. The paper ends with a brief summary. ture where they are termed to as “finite Romanovski” [20-23] or “Romanovski-Pseudo-Jacobi” polynomials [24], they have been completely ignored by the textbooks on mathematical 2. Master formulas for the polynomial so- methods in physics and, surprisingly enough, by the standard lutions to the generalized hypergeometric mathematics textbooks as well [7,25-28]. The notion “finite” equation refers to the observation that, for any given set of parameters (i.e. in any potential), only a finite number of polynomials All classical orthogonal polynomials appear as solutions of appear orthogonal. the so called generalized hypergeometric equation (the pre- The Romanovski polynomials happen to be equal (up to sentation in this section closely follows Ref. 22): a phase factor) to Jacobi polynomials with imaginary argu- 00 0 ments and parameters that are complex conjugate to each σ(x)yn(x) + τ(x)yn(x) − λnyn(x) = 0, (12) other, much like the sinh z = i sin iz relationship. Although σ(x) = ax2 + bx + c, τ(x) = xd + e , one may (but does not have to) deduce the local characteristics of the latter, such as generating function and recurrence λn = n(n − 1)a + nd. (13) relations, from those of the former, the finite orthogonality theorem is qualitatively new. It does not copy any of the prop- There are various methods for finding the solution, here de- erties of the Jacobi polynomials, but requires an independent noted by Ã ¯ ! proof. ¯ Our work adds a new example to the circle of typical d e ¯ yn(x) ≡ Pn ¯x , (14) quantum mechanical problems [29]. The techniques used by a b c ¯ us here extend the study of the Sturm-Liouville theory of ordinary differential equations beyond that of the usual text- with the symbol books. Ã ¯ ! ¯ A final comment on the importance of the potential in d e ¯ P ¯x , Eq. (7). The hyperbolic Scarf potential finds various applica- n a b c ¯ tions in physics ranging from electrodynamics and solid state physics to particle theory. In solid state, physics Scarf II is in which the equation parameters have been made explicit, used in the construction of more realistic periodic potentials standing for a polynomial of degree n, λn being the eigen- in crystals [30] than those built from the trigonometric Scarf value parameter, and n = 0, 1, 2,.... In Ref. 22 a master potential. In electrodynamics, Scarf II appears in a class of formula for the (monic, P¯n) polynomial solutions has been problems with non-central potentials (see Sec. 4). In particle derived by Koepf and Masjed-Jamei; according to them, one finds

Ã ¯ ! ¯ Xn µ ¶ d e ¯ n (n) P¯ ¯x = G (a, b, c, d, e)xk , n a b c ¯ k k k=0

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 146 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

Ã ³ ´ ¯ ! µ ¶n ¯ √ 2a 2√ae−bd d 2 b2 − 4ac (n) (k − n), 2 + 1 − 2a − n ¯ G = √ 2F1 2a b −4ac ¯ √ . (15) k b + b2 − 4ac d ¯b + b2 − 4ac 2 − a − 2n

Though the formal proof of this relation is a bit lengthy, The Pearson equation is solved by its verification with symbolic mathematical software such as Ã ¯ ! ¯ Maple is straightforward. Notice that the G(n) are not nor- d e ¯ k w(x) ≡ W ¯x malized. On the other side, Eq. (12) can be treated alterna- a b c ¯ tively as described in the textbook by Nikiforov and Uvarov µZ ¶ (d − 2a)x + (e − b) in Ref. 7, where it is cast into a self-adjoint form and its = exp dx , (17) weight function, w(x), satisfies the so called Pearson differ- ax2 + bx + c ential equation, and shows how one can calculate any weight function associated with any parameter set of interest (we again used a ∂ (σ(x)w(x)) = τ(x)w(x) . (16) symbol for the weight function that makes explicit the pa- dx rameters of the equation). The corresponding polynomials are now classified according to the weight function, and are built up from the Rodrigues representation as

Ã ¯ ! Ã ¯ ! ¯ ¯ d e ¯ d e ¯ 1 P ¯x = Πk=n(d + (n + k − 2)a)P¯ ¯x = Ã ¯ ! n a b c k=1 n a b c ¯ ¯ ¯ d e ¯ W ¯x a b c ¯ Ã Ã ¯ !! ¯ dn d e ¯ × (ax2 + bx + c)nW ¯x . dxn a b c ¯ (18)

The master formulas in the respective Eqs. (15) and (18), allow for the construction of all the polynomial solutions to the cases of the Jacobi polynomials. The Bessel polynomials are generalized hypergeometric equation. One identifies as spe- not orthogonal in the conventional sense, i.e. within a real cial cases the canonical parameterizations known as interval, and will be left out of consideration. Some of the properties of the fourth polynomials have • the Jacobi polynomials with a = −1, b = 0, c = 1, been studied in the specialized mathematics literature such d = −γ − δ − 2, and e = −γ + δ, as Refs. 20, 21, and 23. Their weight function is calculated from Eq. (17) as • the Laguerre polynomials with a = 0, b = 1, c = 0, d = −1, and e = α + 1, −1 w(p,q)(x) = (x2 + 1)−peq tan x . (19) • the Hermite polynomials with a = b = 0, c = 1, d = −2, and e = 0, This weight function was first reported by Routh [18], then and independently by Romanovski [19]. The polynomials as- • the Romanovski polynomials with a = 1, b = 0, c = 1, sociated with Eq. (19) are named after Romanovski, and will (p,q) d = 2(1 − p), and e = q with p > 0, be denoted by Rm (x). They have non-trivial orthogonality properties over the infinite interval [−∞, +∞]. Indeed, • the Bessel polynomials with a = 1, b = 0, c = 0, as long as the weight function decreases as x−2p, integrals of d = α + 2, and e = β. the type

All other parameterizations can be reduced to one of the Z+∞ above five by an appropriate shift of the variables. The first (p,q) (p,q) (p,q) w (x)Rm (x)Rm0 (x)dx (20) three polynomials are the only ones that are traditionally −∞ presented in the standard textbooks on mathematical methods in physics such as [25-28], while the fourth and fifth seem to will be convergent only if have escaped due attention. Notice that the Legendre, Gegen- bauer, and Chebychev polynomials appear as particular m + m0 < 2p − 1 , (21)

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. . . 147 meaning that only a finite number of Romanovski polyno- point canonical transformation [32, 33], taken by us as mials are orthogonal. This is the reason for the term “fi- z = f(x) = sinh−1 x, is obtained as: nite”Romanovski polynomials (details are given in Ref. 31). The differential equation satisfied by the Romanovski poly- d2g (x) dg (x) (1 + x2) n + x n nomials reads as dx2 dx d2R(p,q)(x) dR(p,q)(x) µ ¶ (1 + x2) n + (2(−p + 1)x + q) n −b2+a(a+1) b(2a+1) 2 + − x+² g (x)=0, (23) d x dx 1+x2 1+x2 n n (p,q) − (n(n − 1) + 2n(1 − p))Rn (x) = 0 . (22) −1 2 In the next section we shall show that the Schrodinger¨ equa- with gn(x) = ψn(sinh x), and ²n = en − a . Equa- tion with the hyperbolic Scarf potential reduces precisely to tion (19) suggests the following substitution in Eq. (23): that very Eq. (22). 2 β − α tan−1 x (β,α) g (x) = (1 + x ) 2 e 2 D (x) , 2.1. The real polynomial equation associated with the n n hyperbolic Scarf potential x = sinh z , −∞ < x < +∞. (24) The Schrodinger¨ equation for the potential of interest when rewritten in a new variable, x, introduced via an appropriate In effect, Eq. (23) reduces to the following equation for (β,α) Dn (x),

d2D(β,α)(x) dD(β,α)(x) (1 + x2) n + ((2β + 1)x − α) n dx2 dx Ã 2 ! (a + a2 + β − β2 − b2 + α ) + x(−b − 2ab + α − αβ) + β2 + ² + 4 2 D(β,α)(x) = 0 . (25) n 1 + x2 n

If the potential equation (25) is to coincide with the Ro- manovski equation (22), then Finally, the third restriction leads to a condition that fixes the • first the coefficient in front of the 1/(x2 + 1) term Scarf II energy spectrum as in (25) must vanish, ² = −(a − n)2 . (30) • the coefficients in front of the first derivatives must be n equal, i.e. 2(−p + 1) + q = (2β + 1)x − α, In this way, the polynomials that enter the solution of the • the eigenvalue constants should also be equal, i.e. Schrodinger¨ equation will be 2 ²n + β = −n((n − 1) + 2(1 − p)).

(β,α) 1 The ﬁrst condition restricts the parameters of the Dn (x) (β=−a,α=2b) (p=a+ 2 ,q=−2b) D (x) ≡ Rn (x). (31) polynomials to n α2 a + a2 − b2 + + β − β2 = 0 , (26) They are obtained by means of the Rodrigues formula from (a+ 1 ,−2b) 4 the weight function w 2 (x) as α −b − 2ab + − αβ = 0 . (27) 2 1 m (a+ 2 ,−2b) 1 d Rn (x) = Solving Eqs. (26) and (27) for α and β results in (a+ 1 ,−2b) m w 2 (x) dx

β = −a , α = 2b . (28) 2 m (a+ 1 ,−2b) × (1 + x ) w 2 (x), The second condition relates the parameters α and β to p and (a+ 1 ,−2b) 2 −a− 1 −2b tan−1 x q, and amounts to w 2 (x) = (1 + x ) 2 e . (32) 1 β = −a = −p + , −α = q = −2b. (29) 2 As a result, the wave function of the nth level, ψn, takes the form q 1 a+ 1 ,−2b −1 def 1 2 −(a+ ) −2b tan−1 x ( 2 ) ψn(z = sinh x): = gn(x) = q (1 + x ) 2 e Rn (x), d sinh−1 x dx 1 d sinh−1 x = √ dx . (33) 1 + x2

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 148 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

The orthogonality integral of the Schrodinger¨ wave functions gives rise to the following orthogonality integral of the Ro- manovski polynomials:

Z+∞ Z+∞ 1 −1 (a+ 1 ,−2b) (a+ 1 ,−2b) 2 −(a+ 2 ) −2b tan x 2 2 ψn(z)ψn0 (z)dz = (1 + x ) e Rn (x)Rn0 (x)dx , (34) −∞ −∞

which coincides in form with the integral in Eq. (20) and is convergent for n < a. The fact that only a finite number of the Romanovski polynomials by a phase factor found as in in Romanovski polynomials are orthogonal is reflected by the Ref. 35: Ã ¯ ! finite number of bound states within the potential of interest, ¯ a number that depends on the potential parameters, in accord n 2(1 − p) iq ¯ i Pn ¯ix with Eq. (21). −1 0 1 ¯ Ã ¯ ! As to the complete Scarf II spectrum, it was constructed ¯ in Ref. 9 using the dynamical symmetry approach [8]. There, 2(1 − p) q ¯ = Pn ¯x . (38) the Scarf II potential was found to possess SU(1, 1) as a sym- 1 0 1 ¯ metry group. The bound states have been assigned to the discrete unitary irreducible representations of SU(1, 1). The Because of this relationship, the Romanovski polyno- scattering and resonant states (which are beyond the scope of mials have been termed to as “Romanovski-Pseudo- the present study) have been related to the continuous unitary Jacobi” by Lesky [24]. The relationship in Eq. (38) (p,q) and non-unitary representations of SU(1, 1), respectively. tells that the Rn (x) properties translate into those of −p−i q ,−p+i q A comment is in order on the relation between the Ro- P 2 2 (ix) and visa versa, and that it is a matter of manovski polynomials and the Jacobi polynomials of imag- convenience to prefer one polynomial over the other. When inary arguments and parameters that are complex conjugate it comes to recurrence relations, generating functions, etc., it to each other. Recall the real Jacobi equation, is perhaps more convenient to favor the Jacobi polynomials, although the generating function of the Romanovski polyno- d2P γ,δ(x) dP γ,δ(x) mials is equally well calculated directly from the correspond- (1 − x2) n + (γ − δ − (γ + δ + 2)x) n dx2 dx ing Taylor series expansion [31]. However, concerning the orthogonality integrals, the advantage is clearly on the side − n(n + γ + δ + 1)P γ,δ(x) = 0 . (35) n of the real Romanovski polynomials. This is so because the As mentioned above, the real Jacobi polynomials are orthog- complex Jacobi polynomials are known for their highly non- onal within the [−1, 1] interval with respect to the weight- trivial orthogonality properties, which depend on the inter- function in Eq. (4). Transforming to a complex argument, play between integration contour and parameter values [36]. x → ix, and parameters, γ = δ∗ = c + id, Eq. (35) trans- For this reason, in random matrix theory [37], the problem forms into on the gap probabilities in the spectrum of the circular Jacobi ensemble is treated in terms of the Cauchy random ensemble, d2P c+id,c−id(ix) a venue that heads one again to the Romanovski polynomials (1 + x2) n dx2 (notice that for p = 1, q = 0 the weight function in Eq. (19) dP c+id,c−id(ix) reduces to the Cauchy distribution). + (−2d + 2(c + 1)x) n In summary, for all the reasons given above, the Ro- dx manovski polynomials qualify as the most adequate real de- c+id,c−id + n(n + 2c + 1)Pn (ix) = 0 . (36) grees of freedom in the mathematics of the hyperbolic Scarf potential. Correspondingly, the weight function turns out to be

−1 wc+id,c−id(ix) = (1 + x2)ce−2d tan x , (37) 3. The polynomial construction

1 (a+ 2 ,−2b) and it coincides with the weight function of the Romanovski The construction of the Rn (x) polynomials is now polynomials in Eq. (19) upon the identiﬁcations c = −p, and straightforward and based upon the Rodrigues representation c+id,c−id q = −2d. This means that Pn (ix) will differ from in Eq. (18), where we plug in the weight function from

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. . . 149

Eq. (19). In carrying out the differentiations, we ﬁnd the lowest four (unnormalized) polynomials to be

1 (a+ 2 ,−2b) R0 (x) = 1, (39)

1 (a+ 2 ,−2b) R1 (x) = −2b + (1 − 2a)x, (40)

1 (a+ 2 ,−2b) 2 2 2 R2 (x) = 3 − 2a + 4b − 8b(1 − a)x + (6 − 10a + 4a )x , (41)

1 (a+ 2 ,−2b) 3 2 2 R3 (x) = −266 + 12ab − 8b + [−3(−15+16a−4a )+12(3−2a)b ]x + (−72b + 84ab − 24a2b)x2 + 2(−2 + a)(−15 + 16a − 4a2)x3. (42)

As illustration, in Fig. 3 we show the Scarf II wave functions of the first and third levels. The finite orthogonality of the Romanovski polynomials becomes especially transparent in the interesting limiting case of the sech2z potential (it appears in the non-relativistic reduction of the sine-Gordon equation), where one easily finds that the 1 (a+ 2 ,0) normalization constants Nn are given by the following expressions: µ ¶ √ 1 2 2 (a+ 2 ,0) (2a − 1) πΓ(a − 1) N1 = 1 , a > 1 , 2Γ(a + 2 ) µ ¶ √ 1 2 (a+ 2 ,0) 2 π(a − 1)Γ(a − 2) 2 N2 = 1 (3 − 2a) , a > 2 , Γ(a − 2 ) µ ¶ √ 1 2 (a+ 2 ,0) 3 π(a − 2)Γ(a − 3) 2 2 N3 = 1 (4a − 16a + 15) , a > 3 etc. (43) Γ(a − 2 )

Software like Maple or Mathematica are quite useful for the 4. Romanovski polynomials and non- graphical study of these functions. The latter expressions spherical angular functions in electrody- show that, for positive integer values of the a parameter, namics with non-central potentials a = n, only the ﬁrst (n − 1) Romanovski polynomials are orthogonal (the convergence of the integrals requires n < a), In recent years, there have been several studies of the bound as it should be in accordance with Eqs. (21), (9). The gen- states of an electron within a compound Coulomb- and non- eral expressions for the normalization constants of any Ro- central potential (see Refs. 38 and 39, and references therein). manovski polynomial are deﬁned by integrals of the type Let us assume the following potential: Z+∞ V2(θ) 2 −(p−n) q tan−1 x V (r, θ) = V (r) + ,V (θ) = −c cot θ , (44) (1 + x ) e dx C r2 2 −∞ where VC(r) denotes the Coulomb potential and θ is the polar and are analytic for (p − n) integer or semi-integer. angle (see Fig. 4). The corresponding Schrodinger¨ equation reads

· · ¸ ¸ 1 ∂ ∂ 1 ∂ ∂ 1 ∂2 − r2 + sin θ + + V (r, θ) Ψ(r, θ, ϕ) = EΨ(r, θ, ϕ) , (45) r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂φ2

and is solved as usual by separating the variables: and Ψ(r, θ, φ) = R(r)Θ(θ)Φ(ϕ) . (46) d2Θ(θ) dΘ(θ) + cot(θ) The radial and angular differential equations for R(r) and dθ2 dθ Θ(θ) are then found to be · ¸ m2 + l(l + 1) − V2(θ) − Θ(θ) = 0 , (48) d2R(r) 2 dR(r) sin2 θ + dr2 r dr · ¸ with l(l + 1) being the separation constant. From now on we l(l + 1) + (V (r) + E) − ) R(r) = 0, (47) shall focus our attention on Eq. (48). It is obvious that for C r2 V2(θ) = 0, and upon changing variables from θ to cos θ, it

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 150 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

new variable, z, introduced via θ → f(z). This transforna- tion leads to the new equation: · · ¸ d2 f 00(z) d + − + f 0(z) cot f(z) + [−V (f(z) dz2 f 0(z) dz 2 ¸ ¸ m2 + l(l + 1) − f 0 2(z) ψ(z) = 0, (50) sin2 f(z)

with f 0(z)≡df(z)/dz, and ψ(z) defined as ψ(z)=Θ(f(z)). Next, one requires the coefficient in front of the first derivative to vanish, which transforms Eq. (50) into a 1d Schrodinger¨ equation. This restricts f(z) to satisfying the differential equation:

f 00(z) = f 0(z) cot f(z) , (51) f 0(z)

which is solved by f(z) = 2 tan−1 ez. With this relation and after some algebraic manipulations, one ﬁnds that 1 sin θ = , cos θ = − tanh z , (52) cosh z and consequently,

f 0(z) = sin f(z) = sech z. (53) FIGURA 3. Wave functions for the ﬁrst and third levels within the hyperbolic Scarf potencial. Equation (52) implies sinh z = − cot θ, or, equivalently, θ = cot−1(− sinh z). Upon substituting Eq. (53) into Eqs. (44), and (50), one arrives at · d2ψ(z) 1 + l(l + 1) dz2 cosh2 z ¸ 1 + c tanh z − m2 ψ(z) = 0 , cosh z ¡ ¢ ψ(z):def= Θ θ = cot−1(− sinh z) ¡ ¢ Θ(θ):def= ψ z = sinh−1(− cot θ) . (54)

Taking into consideration Eqs. (7),(44), and (52), one real- izes that the latter equation is precisely the one-dimensional Schrodinger¨ equation with the hyperbolic Scarf potential where

FIGURE 4. The non-central potential V (r, θ), displayed here in its 2 2 2 p l(l + 1) = −(b − a(a + 1)) , m = −²n = (a − n) , intersection with the x = 0 plane, i.e. for r = y2 + z2, and θ = tan−1 y/z. The polar angle part of its exact solutions is ex- m > 0 , c = −b(2a + 1). (55) pressed in terms of the Romanovski polynomials. Taking into account Eq. (9) above, the first Eqs. (55) is satis- becomes the associated Legendre equation. Correspondingly, fied by Θ(θ) approaches the associated Legendre functions, a = b = l(l + 1) . (56) V2(θ)→0 m Θ(θ) −→ Pl (cos θ) , (49) an observation that will become important below. As long as the integer number m satisfies In order to solve Eq. (48) we follow the prescription given in Ref. 38 and begin by substituting the polar angle by the m = a − n = l(l + 1) − n > 0,

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. . . 151 then 1 ≤ m ≤ l(l + 1). In this way, the solution for Θ becomes ¡ −1 ¢ Θ(θ) = ψn=l(l+1)−m z = sinh (− cot θ)

2 − l(l+1) −l(l+1) tan−1(− cot θ) = (1 + cot θ) 2 e

1 (l(l+1)+ 2 ,−2l(l+1)) × Rl(l+1)−m (− cot θ) . (57) Therefore, the complete angular part of the solution is given by ¡ ¢ −1 i²nϕ Θ(θ)Φ(ϕ)=ψn=l(l+1)−m z= sinh (−cot θ) e . (58) m From now on, we shall introduce Zl (θ, ϕ), a new notation, according to ¡ ¢ m −1 i²nϕ Zl (θ, ϕ)=ψn=l(l+1)−m z= sinh (− cot θ) e , (59) m and refer to Zl (θ, ϕ) as non-spherical angular functions. In m Fig. 5, we display two of the lowest |Zl (θ, ϕ)| functions for illustrative purposes. A more extended sampler can be m found in Ref. 42. A comment is in order on |Zl (θ, ϕ)|. In that regard, it is important to become aware of the fact al- ready mentioned above that the Scarf II potential possesses su(1, 1) as a potential algebra, a result reported by Refs. 9 and 34 among others. There, it was pointed out that the respective Hamiltonian, H, equals H = −C − 1/4, with C being the su(1, 1) Casimir operator, whose eigenvalues in our convention are j(j − 1) with j > 0 versus j(j + 1) and j < 0 in the convention of [9, 34]. As a consequence, the bound state solutions to Scarf II are assigned to infinite discrete, + (m0) irreducible unitary representations, {Dj (θ, ϕ)}, of the SU(1, 1) group. The SU(1, 1) labels m0, and j are mapped onto ours via 1 1 m0 = a + = l(l + 1) + , 2 2 j = m0 − n , m0 = j, j + 1, j + 2, .... (60) FIGURE 5. Graphical presentation of the non-spherical- angular functions |Z1(θ, ϕ)| (left), and |Z1(θ, ϕ)| (right) to the 0 1 2 meaning that both j and m are half-integers. The represen- V2(r, θ) potential in Eq. (44). They portray in polar coordi- 0 0 tations are infinite because, for a fixed j value, m is bounded nate space in turn the components |D+ (m =5/2)(θ, ϕ)|, and 0 j=3/2 m = j + m0=13/2 from below to min , but it is not bound from above. |D ( )(θ, ϕ)| of the respective infinite discrete unitary In terms of the SU(1, 1) labels the energy rewrites as j=3/2 2 SU(1, 1) representation. ²n = −(j − 1/2) . The condition a > n translates now as j > 1/2. As a result, Θ(θ) becomes ¡ −1 ¢ Θ(θ) = ψn=m0−j sinh (− cot θ) 4.1. Romanovski polynomials and associated Legendre q functions. = (1 + cot2 θ)−m0+1/2e−2b tan−1(− cot θ)

(m0,−2b) It is quite instructive to consider the case of a vanishing × Rm0−j (− cot θ) V2(θ), i.e. c = 0, and compare Eq. (54) to Eq. (7) for b = 0. + (m0=l(l+1)+1/2) −im0ϕ In this case = Dj=m+1/2 (θ, ϕ)e . (61) Here we kept the parameter b general because its value 2 2 does not affect the SU(1, 1) symmetry. Within this con- l = a , m = (l − n) , (62) ¡ −1 ¢ text, |ψm0−j sinh (− cot θ | can be viewed as the abso- lute value of a component of an irreducible SU(1, 1) rep- which allows one to relate n to l and m as m = l − n. As resentation [40,41] in terms of the Romanovski polynomials. long as the two equations are equivalent, their solutions dif- m The |Zl (θ, ϕ)| functions are then images in polar coordinate fer at most by a constant factor. This allows us to establish a + (m0=l(l+1)+1/2) space of the |Dj=m+1/2 | components. relationship between the associated Legendre functions and

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 152 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

TABLE I. Characteristics of the orthogonal polynomial solutions to the generalized hypergeometric equation. Notion Symbol w(x) Interval Number of orth. polynomials ν,µ ν µ Jacobi Pn (x) (1 − x) (1 + x) [−1, 1] infinite 2 Hermite H(x) e−x [−∞, ∞] infinite 2 Laguerre Lα,β (x) xβ e−αx [0, ∞] infinite (p,q) 2 −p q tan−1 x Romanovski Rn (x) (1 + x ) e [−∞, ∞] finite the Scarf II wave functions. Considering Eqs. (24), and (33) which is automatically fulfilled for any m > 0. The presence together with Eqs. (52), one finds cot θ = − sinh z which of the additional factor of (1+x2)−1 guarantees convergence produces the following intriguing relationship between the also for m = 0. Equation (66) reveals that, for parameters associated Legendre functions and the Romanovski polyno- attached to the degree of the polynomial, an infinite num- mials: ber of Romanovski polynomials can appear orthogonal, al-

l (l+ 1 ,0) though not precisely with respect to the weight function that m 2 − 2 2 Pl (cos θ) ∼ (1 + cot θ) Rl−m (− cot θ) , defines their Rodrigues representation. The study presented l − m = n = 0, 1, 2, ...l. (63) here is similar to Ref. 43. There, the exact solutions of the Schrodinger¨ equation with the trigonometric Rosen-Morse Substituting the latter expression into the orthogonality inte- potential have been expressed in terms of Romanovski poly- gral between the associated Legendre functions, nomials (not recognized as such at that time), and also with parameters that depended on the degree of the polynomial. Zπ m m 0 Also in this case, the n-dependence of the parameters, and P (cos θ)P 0 (cos θ)d cos θ = 0 , l 6= l , (64) l l the corresponding varying weight function make it possible 0 to satisfy Eq. (21) for an infinitely many polynomials. results in the following integral: Zπ l+l0 1 0 1 (l+ 2 ,0) (l + 2 ,0) 2 − 2 5. Summary (1 + cot θ) Rl−m (− cot θ)Rl0−m 0 In this work we presented the classification of the orthog- ×(− cot θ)d cos θ = 0 , l 6= l0. (65) onal polynomial solutions to the generalized hypergeometric equation in the schemes of Koepf–Masjed-Jamei [22], on Rewriting in conventional notations, the latter expression be- the one hand, and Nikiforov-Uvarov [7], on the other. We comes found among them the real polynomials that define the solu- Z+∞q tions of the bound states within the hyperbolic Scarf poten- 1 l+ 1 ,0 (l+ ,0) ( 2 ) 2 tial. These so called Romanovski polynomials have the re- w (x)Rn=l−m(x) markable property that, for any given set of parameters, only −∞ q a finite number of them are orthogonal. In such a manner, the 0 0 1 (l + 1 ,0) dx (l + 2 ,0) 2 0 finite number of bound states within Scarf II were mapped × w (x)R 0 0 (x) = 0 , l 6= l , n =l −m 1 + x2 onto a finite set of orthogonal polynomials of a new type. x = sinh z , l − n = l0 − n0 = m ≥ 0 . (66) We showed that the Romanovski polynomials also define the angular part of the wave function of the non-central po- This integral describes orthogonality between an infinite set tential considered in Sec. 4. Yet, in this case, the polynomial of Romanovski polynomials with different polynomial pa- parameters turned out to be dependent on the polynomial de- rameters (they would define wave functions of states bound gree. We identified these non-spherical angular solutions to in different potentials). This new orthogonality relationship the non-central potential under investigation as images in po- does not contradict the finite orthogonality in Eq. (21), which lar coordinate space of components of infinite discrete uni- is valid for states belonging to same potential (equal poly- tary SU(1, 1) representations. In the limit of the vanishing nomial parameters). Rather, for different potentials, Eq. (21) non-central piece of the potential, we established a non-linear can be fulfilled for an infinite number of states. To see this, let relationship between the Romanovski polynomials and the us consider for simplicity n = n0 = l − m, i.e., l = l0. Given associated Legendre functions. On the basis of the orthogo- p = l + 1/2, the condition in Eq. (21) defines normalizability nality integral for the latter, we derived a new such integral and takes the form for the former. µ ¶ 1 The presentation contains all the details which to our un- 2(l − m) < 2 l + − 1 = 2l , (67) 2 derstanding are essential for reproducing our results. By this

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 EXACT SPECTRUM AND WAVE FUNCTIONS OF THE HYPERBOLIC SCARF POTENTIAL IN TERMS. . . 153 means, we worked out two problems which could be used in Acknowledgments the class on quantum mechanics and on mathematical meth- We are indebted to Dr. Jose-Luis Lopez´ Bonilla for bring- ods in physics as well and which allow us to practice perform- ing the important Refs. 20 and 39 to our attention. We fur- ing with symbolic software. The appeal of the two examples thermore wish to thank Drs. Hans-Jurgen¨ Weber and Alvaro is that they simultaneously relate to relevant peer research. Perez´ Raposo for many insightful discussions on the orthogo- The hyperbolic Scarf potential and its exact solutions are nality issue. We beneﬁted from the lectures on supersymmet- interesting mathematical entities on their own, with several ric quantum mechanics at the 35th Latin American School applications in physics, ranging from super-symmetric quan- (ELAF), “Super-symmetries in Physics and Applications”, tum mechanics over soliton physics to ﬁeld theory. We expect held in Mexico in the Summer of 2004. future research to reveal additional, interesting properties and Work supported by Consejo Nacional de Ciencia y problems related to the hyperbolic Scarf potential and its ex- Technolog´ıa (CONACyT) Mexico under grant number C01- act real polynomial solutions. 39280.

1. G.F. Torres del Castillo and J.L. Calvario Acocal,´ Rev. Mex. Fis. 19. V.I. Romanovski, Compt. Rend. Ac. Sci. Paris 188 (1929) 1023. 44 (1998) 344. 20. A. Zarzo Altarejos, Differential Equations of the Hypergeomet- 2. V.C.Aguilera-Navarro, E. Ley-Koo, and S. Mateos-Cortes,´ Rev. ric Type, (in Spanish), Ph.D. thesis, Faculty of Science, Univer- Mex. Fis. 44 (1998) 413. sity of Granada (1995). 3. F. Iachello and R.D. Levine, Algebraic Theory of Molecules 21. M. Majed-Jamei, Integral Transforms and Special Functions 13 (Oxford Univ. Press, New York, 1992). (2002) 169. 4. G. Poschl¨ and E. Teller, Z. Phys. 83 (1933) 143. 22. W. Koepf and M. Masjed-Jamei, Integral Transforms and Spe- 5. F. Calogero and A. Degasperis, Phys. Rev. A 11 (1975) 265; B. cial Functions 17 (2006) 559. Yoon and J.W. Negele, Phys. Rev. A 16 (1977) 145. 23. Yu. A. Neretin, Beta-Integrals and Finite Orthogonal Systems 6. Ch. Grosche, J. Phys. A: Math. Gen. 38 (2005) 2947. of Wilson Polynomials, ArXiv: math.CA/0206199. 7. A.F. Nikiforov and V.B. Uvarov, Special Functions of Mathe- 24. W. Lesky and Z. Angew, Math. Mech. 76 (1996) 181. matical Physics (Birkhauser¨ Verlag, Basel, 1988). 25. P. Dennery, Andre´ Krzywicki, Mathematics for Physicists 8. Y. Alhassid, F. Gursey,¨ and F. Yachello, Phys. Rev. Lett. 50 (Dover, New York, 1996); G.B. Arfken and H.J. Weber, Math- (1983) 873. ematical Methods for Physicists, 6th ed. (Elsevier-Academic Press, Amsterdam, 2005). 9. G. Levai, F. Cannata, and A. Ventura, J. Phys. A:Math. Gen. 34 (2001) 839. 26. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables 10. P.M. Morse and H. Feshbach, Methods of Theoretical Physics, (Dover, 2nd edition, New York, 1972). Part I (McGraw-Hill Book Company, Inc., New York, 1953). 27. G. Szego,¨ Orthogonal Polynomials (American Math. Soc., Vol. 11. F. Cooper, A. Khare, and U.P. Sukhatme, Supersymmetry in XXIII, Prov., RI, 1939). Quantum Mechanics (World Scientiﬁc, Singapore, 2001); U.J. Dabrowska, A. Khare, and U.P. Sukhatme, J. Phys. A: Math. 28. M.E.H. Ismail, Classical and Quantum Orthogonal Polynomi- Gen. A 21 (1988) L195. als in One Variable (Cambridge Univ. Press, 2005). 12. C.V. Sukumar, J. Phys. A: Math. Gen. 18 (1998) 2917; C.V. 29. S. Flugge,¨ Practical Quantum Mechanics (Springer, New York, Sukumar, AIP proceedings 744, eds. R. Bijker et al. Supersym- 1971) Vol. 1 . metries in Physics and Applications, (New York, 2005) p. 167. 30. H. Li and D. Kusnezov, Phys. Rev. Lett. 83 (1999) 1283. 13. G. Levai,´ J. Phys. A: Math. Gen. 22 (1989) 689. 31. A. Raposo, H.-J. Weber, D.E. Alvarez-Castillo, and M. Kirch- 14. F. Scarf, Phys. Rev. 112 (1958) 1137. bach, C. Eur. Phys. J. 5 (2007) 253. 15. B. Bagchi and R. Roychoudhury, J. Phys. A: Math. Gen. 33 32. F. Cooper, J.N. Ginocchio, and A. Wipf, J. Phys. A: Math. Gen. (2000) L1. 22 (1989) 370. 16. N. Cotfas, Contribution to 6th Int. Workshop “Lie Theory 33. R. De, R. Dutt, U. Sukhatme, J. Phys. A:Math. Gen. 25 (1992) and Its Applications in Physics”, Varna, Aug. 15-21 (2005); L843. N. Cotfas, Shape Invariant Hypergeometric Type Operators 34. C.V. Sukumar, J. Phys. A:Math.Gen. 19 (1986) 229. with Application to Quantum Mechanics, math-ph/0603032; 35. G. Levai,´ F. Cannata, and A. Ventura, Phys. Lett. A 300 (2002) http://fpcm5.ﬁzica.unibuc.ro/˜ ncotfas 271; G. Levai´ and M. Znojil, J. Phys. A:Math.Gen. 35 (2002) 17. G. Flores-Hidalgo and N.F. Svaiter, Phys. Rev. D 66 (2002) 8793. 025031. 36. A.B.J. Kuijlaars, A. Martinez-Finkelshtein, and R. Orive, 18. E.J. Routh, Proc. London Math. Soc. 16 (1884) 245. Transactions on Numerical Analysis 19 (2003) 1.

Rev. Mex. F´ıs. E 53 (2) (2007) 143–154 154 D.E. ALVAREZ-CASTILLO AND M. KIRCHBACH

37. N.S. Witte and P.J. Forrester, Nonlinearity 13 (2000) 40. B.G. Wybourne, Classical Grops for Physicists (Wiley, New 1965; P.J. Forrester, Random Matrices in Log Gases, Yor, 1974). http://www.ms.unimelb.edu.au/˜ mathpjf/mathpjf.html (book in 41. Y.S. Kim and M. Noz, Theory and Applications of the Poincare´ preparation). Group (D. Reidel, Dordrecht, 1986).

38. Ranabir Dutt, Asim Gangopadhyaya and Uday P. Sukhatme, 42. D.E. Alvarez-Castillo, Exactly Solvable Potentials and Ro- Am. J. Phys. 65 (1997) 400. manovski Polynomials in Quantum Mechanics, M.S. thesis, In- stitute of Physics, Autonomous University of San Luis Potosi, 39. B. Gon¨ ul¨ and M. Kocak, Mod. Phys. Lett. A 20 (2005) 355; M. Mexico, March, 2007. Kocak, I. Zorbo, and B. Gon¨ ul,¨ Mod. Phys. Lett. A 17 (2002) 43. C.B. Compean and M. Kirchbach, J. Math. Phys. A:Math.Gen. 2127. 39 (2006) 547.

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