Prepotential approach to solvable rational extensions of Harmonic Oscillator and Morse potentials

Prep oten tial approac h to solv able rational extensions of Harmonic Oscillator and Morse p oten tials C.-L. Ho Dep art ment of Physics, T amkang University, T amsui 251, T aiwan, R.O. C. (Dated: Oct 14, 2011) W e show ho w the recentl y discov ered solv able ra tional extensions of Harmonic Oscillato r and Morse potentials can be constructed in a direct and systematic wa y , without the need of supersym- metry , shap e inv ariance, Darb oux-Crum and D arboux-B¨ ac klund transformations. I. INTRO DUCTION It is fair to say that in the last three years some o f the most interesting developments in mathematica l physics hav e bee n the discov eries of new t yp es of ortho g onal p olynomials, called the exceptio nal or thogonal p olynomials, and the quantal systems related to them [1- 18]. Unlike the classica l orthogona l po ly nomials, these new po lynomials have the remark able prop erties that they still form c o mplete sets with respe ct to some p ositive-definite measure, although they start with deg ree ℓ ≥ 1 p olynomials instead of a constant. Two families of such p olynomials , namely , the Laguerr e- a nd Ja cobi-t yp e X 1 po lynomials, co rresp onding to ℓ = 1, were first pro posed by G´ omez-Ullate et al. in [1], within the Sturm-Lioville theory , as s olutions o f second-order eigenv a lue e quations with rational coefficients. The results in [1] w ere reformulated in the f ramework of quan tum mechanics in [2], and in sup ersymmetric quantum mechanics using sup erpo ten tia l in [3]. Thes e quan tal sys tems turn out to be rationally extended systems of the traditional ones whic h are related to the classica l orthogo nal polyno mials. The most gener al X ℓ exceptional p olynomials, v a lid fo r a ll in tegral ℓ = 1 , 2 , . . . , were discovered by O dak e and Sasaki [4] (the case of ℓ = 2 was also discuss e d in [3]). Later, in [5] equiv alent but muc h simpler lo oking for ms of the Laguerr e- and Jacobi-type X ℓ po lynomials were presented. Suc h forms facilitate an in-depth study of some impor tan t prop erties o f the X ℓ po lynomials. V er y recently , such systems hav e b een generaliz e d to multi-indexed ca s es [16, 1 7]. Other r ational extensions of solv able systems, which a re not rela ted to the exceptional p olynomials, are possible [19 – 23]. In these sys tems, the poly nomial part of the w ave functions start with degr ee zero. One of the simplest example of such systems was discus sed in [20], which w as later shown to b e a ce r tain sup ersymmetric partner o f the harmonic o scillator in [21]. Extending the sup erp otent ial scheme o f [21], Gandati and B´ erard were able to genera te an infinite set of solv a ble rational extension for transla tionally shape- in v ariant p otentials of the second ca tegory [22]. More recently , ratio nal extensions of the Morse and Kepler -Coulomb p otentials hav e also be e n obtained by means o f the Darb oux-B¨ ac k lund tr ansformation in [23]. F rom the viewp oin t of the genera lized Crum’s theorem, solv able ra tionally extended sys tems r elated to the ex- ceptional po lynomials are obtainable from the corr esponding o rdinary sy stems, which are related to the classica l orthogo nal poly nomials, by deleting the low est energy levels including the gro und states [17]. The rationa l extensions of the harmonic and isoto nic oscillators considered in [20 – 22 ] ca n be obtained in the same w ay , with t he exception that the g round states were not deleted (se e App endix A o f [24]). So far most of the metho ds e mployed to gener ate solv able r ational extensions of ordinary systems with o r without the exceptional p olynomials ha ve in voked in one wa y or another the ideas of shape inv ar iance and/or the related Darb oux-Crum transformation (sup ersymmetry). This r equires an exactly solv able ordinar y system, s uch as the harmonic os cillator, to b e known in the first place, a nd the sup erp oten tial asso ciated with suc h system is modified for the extensio n. In [18] w e ha ve prop osed a simple constructiv e procedur e to gener a te the exceptional ortho gonal p olynomials without the need of shap e inv ar iance and Darbo ux-Crum transformation. Th us in o ur w o rk an exactly solv able ordina ry system and its as socia ted supe rpotential need not b e a ssumed a priori as in the other w or ks. The superp otential, as well as the po ten tia l, the e ig enfunctions and eig en v alues of the new system are all derived from first principle in our metho d. T o disting uish the different ro les the sup erp otent ial play in our approach and in those employing Darb oux- Crum transformation or sup ersymmetry , we prefer to ca ll the sup erp o ten tial “ prepotential”, and o ur pro cedure the “prep otent ial approach”. It is the purp ose of this paper to demonstrate that the solv able rational extensions of the harmonic oscillator given in [20 – 22] a nd the Morse p oten tial in [23] can also b e genera ted very simply in t he pr epotential approach, without the need of sup e rsymmetry , shape inv ariance , Darboux- Crum and Darboux- B¨ acklund transfor mations. These t wo systems are in the same class as the ra tionally extended Jacobi system discussed in Sect. 4.4 of [1 8]. 2 II. PREPOTENTIAL APPRO ACH The main ideas of the prep o ten tial approach are s ummarized here. W e r efer the reader to [18] for the details of the pro cedure. W e a dopt the unit sy stem in which ~ and the mass m of the pa r ticle are such that ~ = 2 m = 1. Consider a wa ve function φ ( x ) which is written in terms of a function W ( x ) as φ ( x ) ≡ exp( W ( x )). The function W ( x ) is assumed to hav e the for m W ( x, η ) = W 0 ( x ) − ln ξ ( η ) + ln p ( η ) . (1) Here η ( x ) is a function of x which we sha ll choose to b e one of the sinusoidal co ordinates, i.e., co ordinates such that ˙ η ( x ) 2 , where the dot denotes deriv a tiv e with resp ect to x , is at most qua dratic in η . The functions W 0 ( x ) , ξ ( η ) and p ( η ) are functions to be deter mined later. W e shall assume ξ ( η ) to be a p olynomial in η . The wa ve function is φ ( x ) = e W 0 ( x ) ξ ( η ) p ( η ) . (2) Op erating on φ ( x ) b y the op erator − d 2 /dx 2 results in a Schr¨ odinger equation H φ = 0, where H = − d 2 /dx 2 + ¯ V , ¯ V ≡ ˙ W 2 + ¨ W . F or simplicity of pre sen tation, we shall often leave out the independent v aria ble of a function if no co nfusio n arises. Since W ( x ) de ter mines the potential ¯ V , it is ther efore c a lled th e prep otential. T o mak e ¯ V exactly solv able, we demand tha t: (1) W 0 is a r egular function o f x , (2) the function ξ ( η ) has no zero s in the the ordinary (or physical) domain o f η ( x ), and (3 ) the function p ( η ) do es not appe a r in ¯ V . F or ξ ( η ) = 1, the prep oten tial approa ch can gener ate exactly and qua s i-exactly so lv a ble systems asso ciated with the classical or thogonal p olynomials [25]. The presence o f ξ in the denominator s of φ ( x ) and V ( x ) thus gives a rationa l extension, or defo rmation, of the traditional system. W e therefore ca ll ξ ( η ) the deforming function. If the factor exp( W 0 ( x )) /ξ ( η ) in Eq. (2) is normaliza ble, then p ( η ) = constant (in this case we shall take p ( η ) = 1 for s implicit y ) is a dmiss ible. This gives the gr ound state φ 0 ( x ) = e W 0 ( x ) ξ ( η ) . (3) How ever, if exp( W 0 ( x )) /ξ ( η ) is non-normalizable, then φ 0 ( x ) canno t be the gro und state. In this ca s e, the ground state, like all the excited sta tes, m ust in volv e non-trivia l p ( η ) 6 = 1. Typically it is in such situation that the exceptional orthogo nal p olynomials arise [1 8]. The ca ses consider ed in [23], which we shall rederived by means o f the prep oten tial approach in this pap er, are s uc h tha t exp( W 0 ( x )) /ξ ( η ) is no rmalizable and thus φ 0 ( x ) is the gro und state. F ollowing the pro cedure in [18], we assume ξ ( η ) to sa tisfy the equation c 2 ( η ) ξ ′′ + c 1 ( η ) ξ ′ + e E ( η ) ξ = 0 . (4) Here the prime denotes deriv ative with r espective to η . W e cho ose c 2 ( η ) = ± ˙ η 2 , and c 1 is determined by c 1 ( η ) = ±  1 2 d dη  ˙ η 2  − 2 Q ( η )  , (5) where Q ( η ) ≡ ˙ W 0 ˙ η . The function e E ( η ) w as taken to b e a real constant in [18], but here we allow the p ossibilit y that e E ( η ) may be a function of η . Nonetheless, it turns o ut that all formulae presented in [18] rema in int act. By matching Eq. (4) with the (confluent) h yp ergeometric equation, one determines ˜ E , Q ( η ) a nd ξ ( η ). Integrating Q ( x ) = ˙ W 0 ˙ η then gives the prep oten tial W 0 ( x ): W 0 ( x ) = Z x dx Q ( η ( x )) ˙ η ( x ) = Z η ( x ) dη Q ( η ) ˙ η 2 ( η ) ; (6) The function p ( η ) is then given b y a linear co m bination o f ξ and ξ ′ : p ( η ) = ξ ′ ( η ) F ( η ) + ξ ( η ) G ( η ) . (7) 3 Here the tw o functions F ( η ) a nd G ( η ) a re determined by F ( η ) = c 2 ( η ) V ( η ) , (8) G ( η ) = ( c 1 − c ′ 2 ) V − c 2 V ′ , (9) with the function V ( η ) satisfying c 2 V ′′ + (2 c ′ 2 − c 1 ) V ′ + h c ′′ 2 − c ′ 1 + e E ± E i V = 0 . (10) By matching E q. (1 0) with the (confluen t) hypergeo metric equatio n, one determines V , and th us F ( η ) , G ( η ) , p ( η ) and E . Once a ll the r elev ant f unctions and par ameters are determined, we w ould hav e co nstructed an exactly solv able quantal system H φ = E φ defined by H = − d 2 /dx 2 + V ( x ), with the wa ve function (2) and the po ten tia l V ( x ) ≡ ˙ W 2 0 + ¨ W 0 + ξ ′ ξ  2 ˙ η 2  ξ ′ ξ  −  2 ˙ W 0 ˙ η + ¨ η  ± c 1  ± ˜ E . (11) Lastly , w e note tha t the functions p E (here we add a subscript to distinguish p co r resp o nding to a pa rticular eigenv a lue E ) are or thogonal , i.e., Z dη p E ( η ) p E ′ ( η ) W 2 ( x ( η )) ˙ η ∝ δ E , E ′ (12) in the η -spa ce with the weight function W ( x ) ≡ exp Z x dx ˙ W 0 − ˙ ξ ξ !! = e W 0 ( x ) ξ ℓ ( η ( x )) . (13) II I. HA RMONIC OSCILLA TOR Let us cho ose η ( x ) = x ∈ ( −∞ , ∞ ). Then ˙ η 2 = 1. F or c 2 and c 1 , we take the uppe r signs in c 1 and c 2 (it turns out that the lower signs give the same mo de l) . Thus c 2 ( η ) = 1 and c 1 = − 2 Q ( η ). Eq. (4 ) b ecomes ξ ′′ − 2 Q ( η ) ξ ′ + ˜ E ξ = 0 . (14) Comparing E q. (14) with the Hermite equa tion H ′′ ℓ ( η ) − 2 η H ′ ℓ ( η ) + 2 ℓH ℓ ( η ) = 0 , ℓ = 0 , 1 , 2 , . . . , (15) where H ℓ ( η ) is the Hermite p olynomial, we would hav e ξ ( η ) ≡ ξ ℓ ( η ; α ) = H ℓ ( η ) , ˜ E = 2 ℓ, Q ( η ) = η . (16) But this c hoice is not viable, as ξ ℓ ( η ; α ) = H ℓ ( η ) has zeros in the ordinary do main ( −∞ , ∞ ), which we want to av oid. A s imple wa y to solve this is to make the zeros of Hermite p olynomials lie on the imaginary axis. This is achieved if we set η → iη in E q. (15), giving H ′′ ℓ ( iη ) + 2 η H ′ ℓ ( iη ) − 2 ℓ H ℓ ( iη ) = 0 , ℓ = 0 , 1 , 2 , . . . . (17) Matching Eq. (1 4) with (17) gives ξ ( η ) ≡ ξ ℓ ( η ; α ) = H ℓ ( iη ) , ˜ E = − 2 ℓ, Q ( η ) = − η . (18) One notes that the H ermite po lynomials ar e o dd fun ctions in η for odd ℓ . So in t his case ξ ℓ has a zer o at η = 0. F urther study o f this case reveals that the wa ve functions are not no rmalizable. So here we sha ll only consider the case with even ℓ = 2 m ( m = 1 , 2 , . . . ) . By Eq. (6 ), the form of Q ( η ) leads to W 0 ( x ) = − x 2 2 . (19) 4 W e shall ignor e the co ns tan t o f integration as it can b e absorb ed in to the normalization consta n t. As noted in Sect. II, in this ca s e p ( η ) = 1 is admissible, as exp( W 0 ( x )) /ξ ( η ) is nor malizable. So the energ y and eig enfunction of the gro und state of this sys tem are E 0 = 0 and φ 0 ( x ) = exp( W 0 ( x )) /ξ ( η ). Below we deter mine the energies and eigenfunctions of the excited states. With the solutio ns in Eq. (1 8), Eq . (10) b ecomes V ′′ − 2 η V ′ + ( E − 2 ℓ − 2 ) V = 0 . (20) Comparing E qs. (20) and (15) (with ℓ r eplaced b y n = 0 , 1 , 2 , . . . ), we get V ( η ) = H n ( η ) , E ≡ E ℓ,n = 2( n + ℓ + 1 ) , ℓ = 2 m. (21) F rom Eqs. (8) and (9), one even tually o bta ins p ( η ) ≡ p ℓ,n ( η ) = ξ ′ F + ξ G = H n ( η ) ξ ′ ℓ ( η ) + [2 η H n ( η ) − H ′ n ( η )] ξ ℓ ( η ) = H n ( η ) H ′ ℓ ( iη ) + H n +1 ( η ) H ℓ ( iη ) . (22) Use has b een made o f the identit y H ′ n = 2 η H n − H n +1 in obtaining the last line in E q. (22). W e note that p ℓ,n ( η ) is a p olynomial of degree ℓ + n + 1 . By E q. (12), one finds that p ℓ,n ( η ; α )’s are orthogo na l in the sense Z ∞ −∞ dη e − η 2 ξ 2 ℓ p ℓ,n ( η ; α ) p ℓ,k ( η ; α ) ∝ δ nk . (23) The exactly solv able po ten tial is g iv en by Eq . (11) with W 0 ( x ) and ξ ℓ ( η ; α ) given by E q s. (19) and (18), resp ectively . Explicitly , the po ten tial is V ( x ) = x 2 − 1 + 2 ξ ′ ℓ ξ ℓ  ξ ′ ℓ ξ ℓ  + 2 η  + 2 ℓ. (24) The complete eig enfunctions and energies are φ 0 ( x ; α ) ∝ e − x 2 2 ξ ℓ , E 0 = 0 , (25) φ ℓ,n ( x ; α ) ∝ e − x 2 2 ξ ℓ p ℓ,n ( η ( x ); α ) , E ℓ,n = 2( n + ℓ + 1 ) , n = 0 , 1 , 2 . . . . (26) Using the identit y for ℓ = 2 m [26, 27], i.e., H 2 m ( iη ) = ( − 1) m 2 2 m m ! L ( − 1 2 ) m ( − η 2 ) , (27) where L ( α ) ℓ ( η ) is the Laguer re polyno mial, and the ident ity d dη L ( α ) ℓ ( η ) = − L ( α +1) ℓ − 1 ( η ) , (28) we can reduce E q. (26) to φ 2 m,n ∼ e − x 2 2 L ( − 1 2 ) m ( − η 2 )  1 2 L ( − 1 2 ) m ( − η 2 ) H n +1 ( η ) + η L ( 1 2 ) m − 1 ( − η 2 ) H n ( η )  . (29) This result is identical with that given in [23]. 5 IV. MORSE POTENTIAL Now w e co nsider ra tional extension o f the Morse po ten tia l. It turns out that in this cas e ˜ E cannot b e a constant. Let us choo s e η ( x ) = e − x ∈ (0 , ∞ ), with ˙ η 2 = η 2 . F or definiteness w e sha ll take the upp er signs for c 2 and c 1 , as the low er signs lead to the same results. So we hav e c 2 ( η ) = η 2 and c 1 = ( η − 2 Q ( η )). Equation determining ξ is η 2 ξ ′′ ( η ) + ( η − 2 Q ( η )) ξ ′ ( η ) + ˜ E ( η ) ξ ( η ) = 0 . (30) In order to link Eq. (30) with the La guerre equation η L ′′ ( α ) ℓ + ( α + 1 − η ) L ′ ( α ) ℓ + ℓL ( α ) ℓ = 0 , ℓ = 0 , 1 , 2 , . . . , (31) we rewrite E q. (30) as η ξ ′′ ( η ) +  1 − 2 Q ( η ) η  ξ ′ ( η ) + ˜ E ( η ) η ξ ( η ) = 0 . (32) Directly matchin g this equation with Eq. (31) w ill leads to unnormaliz able w av e functions. So instead w e set η → − η in Eq. (32). This gives η ξ ′′ ( − η ) +  1 + 2 Q ( − η ) η  ξ ′ ( − η ) + ˜ E ( − η ) η ξ ( − η ) = 0 . (33) Comparing this eq ua tion with Eq. (31) leads to ξ ( − η ) ≡ ξ ℓ ( − η ; α ) = L ( α ) ℓ ( η ) , ˜ E ( − η ) = ℓη , Q ( − η ) = α 2 η − 1 2 η , (34) or equiv alently , ξ ℓ ( η ; α ) = L ( α ) ℓ ( − η ) , ˜ E ( η ) = − ℓη , Q ( η ) = − α 2 η − 1 2 η , (35) The form o f Q ( η ) leads to W 0 ( x ) = − α 2 ln η − η 2 . (36) According to the Kienast-Lawton-Hahn’s Theorem [26, 27], the deforming function ξ ℓ ( η ) will hav e no po sitiv e z e ros in (0 , ∞ ) if: (i) − 2 k − 1 < α < − 2 k with − ℓ < α < − 1, or (ii) ℓ is e ven with α < − ℓ . Again, in this case, p ( η ) = 1 is admissible. Thus the energy a nd eigenfunction of the ground state o f this sy stem are E 0 = 0 a nd φ 0 ( x ) = exp( W 0 ( x )) /ξ ( η ). W e now determine the energies a nd eigenfunctions of the excited sta tes. With the solutio ns in Eq. (3 5), Eq . (10) b ecomes V ′′ + ( − α + 3 − η ) V ′ +  E − α + 1 η − ( ℓ + 2)  V = 0 . (37) In order that E b e dep enden t on n , we try V = η γ U ( η ) wher e γ is a r eal pa rameter a nd U ( η ) a function of η . F rom Eq. (1 0) we get η U ′′ + (2 γ − α + 3 − η ) U ′ +  E − α + 1 + γ ( γ − α + 2) η − ( γ + ℓ + 2)  U = 0 . (38) Matching this equation with Eq . (31), we hav e ( n = 0 , 1 , 2 , . . . ) γ = − ( n + ℓ + 2) , E ℓ,n = α − 1 − γ ( γ − α + 2 ) = α − 1 − ( n + ℓ + 2 )( n + ℓ + α ) , (39) β = 2 γ − α + 2 = − α − 2( n + ℓ + 1) , U n ( η ) = L β n ( η ) , β > − 1 . 6 Putting all these r esults into F ( η ) and G ( η ) gives p ( η ) ≡ p ℓ,n ( η ; α ) = η − n − ℓ − 1 P ℓ,n ( η ; α ) P ℓ,n ( η ; α ) ≡ η L ( β ) n ξ ′ ℓ −  ℓL ( β ) n + ( n + 1 ) L ( β ) n +1  ξ ℓ . (4 0 ) P ℓ,n ( η ; α ) is a p olynomial o f degree ℓ + n + 1. It is also e asy to check that p ℓ,n ( η ; α )’s are or thogonal with r espect to the w eight function e − η η − α ξ 2 ℓ . (41) The exactly solv able p otential is given by V ( x ) = 1 4 e − 2 x + 1 2 ( α − 4 ℓ − 1 ) e − x + α 2 4 + 2 ξ ′ ℓ ξ ℓ  e − 2 x  ξ ′ ℓ ξ ℓ + 1  + αe − x  . (42) The complete eig enfunctions ar e φ 0 ( x, α ) ∝ e − η 2 η − α 2 ξ ℓ , (43) φ ℓ,n ( x ; α ) ∝ e − η 2 η − α 2 ξ ℓ p ℓ,n ( η ; α ) , (44) where p ℓ,n ( η ; α )’s ar e given in (40). The corr esponding eig en-energies a re E 0 = 0 and E ℓ,n in (39 ), re spectively . F o r the w av e functions to be re g ular at x = 0, one m ust ha ve n < − α/ 2 − ℓ − 1. This mea ns the system admits o nly finite nu mber of bo und states. With the help of the identities (28) and L ( α ) ℓ ( η ) − L ( α − 1) ℓ ( η ) = L ( α ) ℓ − 1 ( η ) , (45) η L ( α +1) ℓ − 1 ( η ) − αL ( α ) ℓ − 1 ( η ) = − ℓL ( α − 1) ℓ ( η ) , (46) one ca n recast P ℓ,n ( η ; α ) in (40) in to P ℓ,n ( η ; α ) = − h ( α + ℓ ) L ( α ) ℓ − 1 ( − η ) L ( β ) n ( η ) + ( n + 1) L ( α ) ℓ ( − η ) L ( β ) n +1 ( η ) i . (47) This expression is exa c tly the same a s that given in [23] in the case ℓ = 2 m , with the identification α = − 2 ( a + ℓ + 1) , n → k, (48) β = − α − 2 ( n + ℓ + 1 ) = 2( a − k ) . (49) V. SUMMAR Y W e hav e s ho wn how the recently discov er ed solv a ble rational ex tens ions of Har monic Oscillator and Mors e po - ten tials can be constructed in a direct and systematic wa y , without the need of sup ersymmetry , shap e inv a r iance, Darb oux-Crum and Darb oux-B¨ acklund transformations . In our approach, the prep o ten tial, the deforming function, the potential, the e ig enfunctions and eigenv alues are all derived within the s a me framework. With the results given here and in [18], r ational extensions of all well-known one- dimensional solv able quant al systems based on sinusoidal co ordinates have b een ge ner ated b y the prep oten tial approach. One would like to apply the s ame approa c h to find rationa l ex tensions of the o ther solv a ble mo dels based on non- sin usoidal co ordinates, following the w ork of the third pap er in [25]. Unfor tunately , such wa y of rational extensions only lead to qua si-exactly solv able systems, b ecause the energy quan tum n umber n will a ppear in the x -dependent terms in V ( x ). A non-trivia l generaliza tion of the present approa c h may be in or der, which w e hop e to rep ort in the near future. Ackno wl edgmen ts This work is supp orted in part b y the National Science Council (NSC) of the Republic o f China under Grant NSC NSC-99-21 12-M-032 - 002-MY3. 7 [1] D. G´ omez-Ullate, N . K amran and R . Milson, J. Math. Anal. Ap pl. 359 , 352 (2009); D. G´ omez-Ullate, N. Kamran and R. Milson, J. Ap p ro x . Theory 162 , 987 (2010). [2] C. Quesne, J. Phys. A41 , 3920 01 (2008) ; B. Bagchi, C. Quesne and R. Royc h oudh ury , Pramana J. Phys. 73 , 337 (2009). [3] C. Quesne, SIGMA 5 , 084 (2009). [4] S. Odake and R. Sasaki, Phys. Lett. B679 , 414 (2009); S. Odake and R. Sasaki, Phys. Lett. B 684 , 173 (2009); S. Odake and R. Sasaki, J. Math. Phys. 51 , 053513 (2010). [5] C-L. 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