Shape invariance in prepotential approach to exactly solvable models

Y uk a wa Institute Kyoto YITP-08-85 Shap e in v ariance in prep oten tial approac h to exactly solv able mo dels Cho on-Lin Ho ∗ Y ukawa Institute for The or e tic al Physics, Kyoto U n iversity, Kyoto 606-8502, Jap an and Dep artment of Physics, T amk ang University, T amsui 251, T aiwa n, R epublic of C hina † (Dated: Jan 9, 2009) Abstract In sup ersymmetric quan tum m ec hanics, exact- s olv abilit y of one-dimensional quan tum systems can b e classified only w ith an additional assum ption of int egrabilit y , the so-called shap e in v ariance condition. In th is pap er w e sh o w that in the prep oten tial appr oac h we prop osed previously , sh ap e in v ariance is automatical ly satisfied and n eeds not b e assumed. P ACS n umber s: 0 3.65.Ca , 03.65.Ge, 02.30.Ik Keywords: Prep otential, exact solv ability , shap e inv ariance, sup ersymmetry ∗ email: hcl@mail.tku.edu.tw T el: +886-2 -262 1 5656 F ax: + 886- 2-262 0-99 17 † Permanen t addres s 1 I. INTR O DUCTION It is generally kno wn that exactly solv able systems are v ery rare in an y branch of ph ysics. Th us an y new metho d to construct exactly solv able mo dels w ould b e of in terest to the comm unit y concerned. It is therefore v ery in teresting to realize that most exactly solv able one-dimensional quan tum systems can b e obtained in the framew ork of supersymmetric quan tum mec hanics (SUSYQM) [1, 2 ]. Ho we v er, in SUSYQM, exact-solv abilit y can b e classified only with an a dditional assumption of inte grabilit y , so called shap e in v ariance (SI) condition [3]. Hence in SUSYQM the SI condition m ust b e tak en as a sufficien t condition for in tegrability at the outset. What is more, the transformatio n of the original co or dinate, sa y x , to a new one z = z ( x ) needed in solving the SI condition is not naturally determined within the f ramew ork of SUSYQM in most cases, but ha ve to b e t ak en as giv en from the kno wn solutions of t he resp ectiv e mo dels. It w o uld b e more satisfactory if the exact-solv abilit y of a quantal system, including the required change of co ordinates, could b e determined with the simplest, and the most na tural requiremen ts. In [4, 5, 6] a unified a ppro ac h to b oth the exactly a nd quasi-exactly solv a ble systems is presen ted. This is a simple constructiv e approach , based o n the so-called prep oten tial [7, 8, 9, 10, 11, 12, 1 3 , 14, 15], whic h gives the p oten t ia l as w ell as the eigenfunctions and eigen v alues simu ltaneously . The nov el feature of the approac h is that b o th exact and quasi-exact solv abilities can b e solely classified by tw o integers, the degrees of tw o p oly- nomials whic h determine the change of v ariable and the zero-th order prep otential. Hence this a pproac h treats bo th quasi-exact and exact solv abilities o n the same fo oting, and it pro vides a simple w ay to determine the required change of co ordinates z ( x ). All the we ll- kno wn exactly solv able mo dels giv en in [1, 2], most quasi-exactly solv able mo dels discussed in [16, 17, 18, 19, 2 0], and some new quasi-exactly solv able ones (also for non-Hermitian Hamiltonians), can b e generated b y appropriately c ho osing the tw o p olynomials. Since all the w ell-kno wn o ne-dimensional exactly solv able mo dels obtained in SUSYQM, b y taking SI condition as a sufficie n t condition, can also b e deriv ed without t he SI condition in the prep oten tial approac h, one wonde rs what role the SI condition pla ys in the latter approac h. In this pap er w e w ould lik e to show that the SI condition is only a neces sary con- dition in the prep oten tial approach to exactly solv able systems. Therefore, unlik e SUSUQM, shap e in v ariance needs not b e assumed in the prep o ten tial approac h.. 2 This pap er is org a nized as follo ws. In Sect. I I we give a brief r eview of the prep oten tial approac h to exactly solv able mo dels with b oth sinus o idal and non-sin usoidal co ordinates. The idea of SI as a sufficien t condition of in tegrability in SUSYQM is sk etch ed in Sect. I I I. Sect. IV and V then demonstrate that in the prepo ten tial approac h for mo dels with sin usoidal and non-sin usoidal co ordinates, SI is automatically satisfied and needs not b e impo sed. Sect. VI concludes the pap er. I I. PREPOTENTI AL APP R OA CH The main ideas of the prep otential approach can b e summarize d as follows (w e adopt t he unit system in whic h ~ and the mass m of the particle are suc h that ~ = 2 m = 1). Consider a w av e function φ N ( x ) ( N : non-negativ e in teger) whic h is defined as φ N ( x ) ≡ e − W 0 ( x ) p N ( z ) , (1) with p N ( z ) ≡    1 , N = 0; Q N k =1 ( z − z k ) , N > 0 . (2) Here z = z ( x ) is some real function of the basic v ariable x , W 0 ( x ) is a regular function of z ( x ), and z k ’s are the r o ots of p N ( z ). The v ariable x is defined on the full line, half- line, or finite in terv al, as dictated b y the c ho ice of z ( x ). The function p N ( z ) is a p olynomial in an ( N + 1)- dimensional Hilb ert space with the basis h 1 , z , z 2 , . . . , z N i . W 0 ( x ) defines the ground state wa v e f unction. The w av e function φ N can b e recast as φ N = exp ( − W N ( x, { z k } )) , (3) with W N giv en b y W N ( x, { z k } ) = W 0 ( x ) − N X k =1 ln | z ( x ) − z k | . (4) Op erating on φ N b y the op erator − d 2 /dx 2 results in a Sc hr¨ odinger equation H N φ N = 0, where H N = − d 2 dx 2 + V N , (5) V N ≡ W ′ 2 N − W ′′ N . (6) 3 Here prime represen ts differen tiatio n with respect to x . It is seen that the p oten tial V N is defined b y W N , and we shall call W N the N th o rder prepo ten tial. F rom Eq. (4), one finds that V N has the form V N = V 0 + ∆ V N : V 0 = W ′ 2 0 − W ′′ 0 , ∆ V N = − 2  W ′ 0 z ′ − z ′′ 2  N X k =1 1 z − z k + X k,l k 6 = l z ′ 2 ( z − z k )( z − z l ) . (7) Th us the form of V N , and consequen tly its solv ability , are determined b y the c hoice of W 0 ( x ) and z ′ 2 (or equiv a len tly by z ′′ = ( dz ′ 2 /dz ) / 2). Let W ′ 0 z ′ = P m ( z ) and z ′ 2 = Q n ( z ) b e p o lynomials of degree m and n in z , respectiv ely . In [4 ], it w as sho wn that if the degree of W ′ 0 z ′ is no higher than one ( m ≤ 1), and the degree of z ′ 2 no higher than t wo ( n ≤ 2), then in V N ( x ) t he par ameter N and the ro ots z k ’s, whic h satisfy the so-called Bethe ansatz equations (BAE) t o make the p ot en tial a na lytic, will only app ear in an additive constan t and not in an y term in volvin g p o we r s of z . Such system is then exactly solv able. If the degree of one of the t wo p olynomials exceeds the corresp onding upp er limit, the resulted system is quasi-exactly solv able. The t ransformed co ordinates z ( x ) suc h t hat the degree of z ′ 2 is no higher than tw o are called sinus oidal co ordinates. There are six t yp es of one-dimensional exactly solv able mo dels whic h are based on suc h co ordinates, namely , the shifted-oscillator, three-dimensional oscillator, Morse, Scarf type I and I I, and generalized P¨ osc hl-T eller mo dels as list in [1]. In [6], the prep otential approac h to exactly solv a ble systems w as extended to systems based on non-sin usoidal transformed v ariable z ( x ) whic h is a solution of z ′ = λ − z 2 . With this, the remaining f our types of exactly solv able systems listed in [1], namely , the Coulom b, Ec k art, and Rosen-Morse t yp e I and I I mo dels, are also co vere d b y the prep otential approac h. A. Sin usoidal co ordinates F or exactly solv able mo dels with sin usoidal co ordinates we tak e m = 1 and n = 2, i.e., P 1 ( z ) = az + b , and Q 2 ( z ) = α z 2 + β z + γ , where a, b, α, β and γ are real constan t s. With these c hoices w e obtain [4] V N = W ′ 0 2 − W ′′ 0 + αN 2 − 2 aN − 2 N X k =1 1 z − z k (  a − α 2  z k + b − β 4 − X l 6 = k Q 2 ( z k ) z k − z l ) . (8) 4 Demanding the residues at z k ’s v anish g iv es t he set of Bethe ansatz equ ations  a − α 2  z k + b − β 4 − X l 6 = k Q 2 ( z k ) z k − z l = 0 , k = 1 , 2 , . . . , N . (9) With this set o f ro ots z k , the last term in Eq. (8) v anishes, and w e obtain a p otential V N ( x ) = V 0 ( x ) − E N without simple p oles. Here V 0 ( x ) = W ′ 2 0 − W ′′ 0 do es not in volv e N and z k ’s, and can b e tak en as the exactly solv able p ot ential of the system with eigen-energies E N = 2 aN − αN 2 . In fact, V 0 ( x ) is exactly the sup ersymmetric form presen ted in [1] for the shifted-oscillator, three-dimensional oscillator, Morse, Scarf t yp e I and I I, and generalized P¨ osc hl-T eller mo dels (for easy comparison, w e note that α and a here equal ± α 2 and α A in [1]). B. Non-sin usoidal co ordinates As mentioned b efore, the Coulomb, Ec k art, and Ro sen-Morse type I and I I mo dels inv olv e a c hange of co ordinat es of the form z ′ = λ − z 2 whic h is non- sinus oidal. But with a sligh t extension of the metho ds in [4], all these four mo dels can be t r eated in a unified w a y in the prep oten t ia l approach [6]. The extension is simply to allow the co efficien ts in W 0 b e dep enden t on N . It turns out that W ′ 0 tak es the fo rm W ′ 0 ( N ) = − ( A + N α ) z + B A + N α , (10) where A and B are real parameters. Then the p otential V N b ecomes V N ( x ) = V ( x ) − E N , where V ( x ) = A ( A − 1) z 2 ( x ) − 2 B z ( x ) , (11) and E N = − B 2 ( A + N ) 2 − λ  A (2 N + 1) + N 2  . (12) No w V ( x ) is indep enden t of N , and can b e ta ken to b e the p oten tia l of a n exactly solv able system, with eigenv alues E N ( N = 0 , 1 , 2 , . . . ). The corresp onding w av e functions φ N are giv en b y (1): φ N ∼ e ( A + N ) R x dxz ( x ) − B A + N x p N ( x ) , N = 0 , 1 , . . . (13) 5 The BAE satisfied by the ro ots z k ’s are X l 6 = k z 2 k − λ z k − z l − ( A + N − 1) z k + B A + N = 0 , k = 1 , 2 , . . . , N . (14) Finally ,w e men tio n here that V ( x ) in (11) can be o btained, up to an additiv e constant, from W 0 ( N ) with an y v alue of N . Particularly , the form adopted in sup ersymmetric quan tum mec hanics (e.g., in [1]) is obtained from the zero-th order prepo ten tial W 0 ( N = 0) with N = 0 [6]. I I I. SHAPE I NV ARIANCE IN SUPERSYMMETRIC QUANTUM MECHANICS F rom the discussions in the last section, w e see that in the prep otential a ppro ac h, exactly solv able mo dels are determined b y the zero-th order prep oten tial W 0 ( x ) in the sin usoidal cases, or W 0 ≡ W 0 ( N = 0) with N = 0 in the four non- sin usoidal cases. The p oten tial V 0 is completely determined by W 0 : V 0 = W ′ 0 2 − W ′′ 0 , and consequen tly , the Ha milto nian H 0 = − d 2 /dx 2 + V 0 is factorizable as H 0 = A + A with the first-order op erators A ≡ d dx + W ′ 0 , A + ≡ − d dx + W ′ 0 . (15) This fact is indeed the base of SUSYQM. In SUSYQM [1, 2] one considers the relation b et wee n the sp ectrum of H 0 and that of its so-called sup er-pa rtner Hamiltonian H 1 con- structed according to H 1 ≡ AA + = − d 2 /dx 2 + V 1 , where V 1 ≡ W ′ 2 0 + W ′′ 0 . In f o rming V 1 , it is equiv alent to using a prep oten tial − W 0 . The ground state of H 1 is therefore exp( W 0 ), and it follow s that the gro und states of H 0 and H 1 cannot b e b oth normalizable. Let us supp ose the ground state of H 0 , i.e. exp( − W 0 ), is norma lizable, and denote the normalized eigenfunctions of the Hamilto nians H 0 , 1 b y ψ (0 , 1) n with eigen v alues E (0 , 1) n , resp ectiv ely . Here the subscript n = 0 , 1 , 2 , . . . denotes the num b er of no des of the w av e function. It is easily pro v ed that V 0 and V 1 ha ve the same energy sp ectrum except for the ground state of V 0 with E (0) 0 = 0, whic h has no corresp onding lev el for V 1 [1, 2]. More explicitly , w e hav e the following sup ersymmetric relatio ns: E (1) n = E (0) n +1 , ψ (1) n =  E (0) n +1  − 1 / 2 Aψ (0) n +1 , Aψ (0) 0 = 0 , (16) ψ (0) n +1 =  E (1) n  − 1 / 2 A + ψ (1) n . 6 Hence A annihilates ψ (0) 0 , and conv erts an eigenfunction of an excited state of H 0 in to an eigenfunction o f H 1 with t he same energy , but with one less num b er of no des, while A + do es the reve rse. Consequen tly , if t he sp ectrum of one system is exactly kno wn, so is the sp ectrum of the ot her. This is, ho w eve r, all that sup ersymme try sa ys ab out t he tw o partner p otentials. If an y one of t he sp ectra is unkno wn, then sup ersymmetry is useless in solving them. It is therefore gratifying that most of the w ell-known one- dimensional exactly solv able mo dels pro cess a prop erty called shap e in v ariance. With hindsigh t, one can then imp o se shap e in v ariance as an additional requiremen t along with sup ersymmetry to classify exactly solv able systems ha ving suc h prop erty . This has b een done and most exactly solv able systems a r e then unified within the framew ork of SUSYQM [1, 2]. Shap e in v ariance means that the t wo sup er-partner p oten tials V 0 and V 1 are related by the relation V 1 ( x ; λ 0 ) = V 0 ( x ; λ 1 ) + R ( λ 0 ) , (17) where λ 0 is a set of parameters of the original V 0 , λ 1 = f ( λ 0 ) is a function of λ 0 , and R ( λ 0 ) is a constan t whic h dep ends only λ 0 . This implies W ′ 2 0 ( x, λ 0 ) + W ′′ 0 ( x, λ 0 ) = W ′ 2 0 ( x, λ 1 ) − W ′′ 0 ( x, λ 1 ) + R ( λ 0 ) . (18) Eq. (17) implies that V 1 has the same shap e as that of V 0 , but is defined b y pa rameters λ 1 instead of λ 0 . F ro m (18) one deduces that the gro und state w a ve function of V 1 is ψ (1) 0 ∼ exp( − W 0 ( x, λ 1 ) with energy R 0 ( λ 0 ). Then fr o m (16) w e know the energy o f the first excited stat e of V 0 to b e R ( λ 0 ), and the wa v e f unction ψ (0) 1 ∼ A + ψ (1) 0 . By rep eated use of the shap e in v ariance condition, one can construct the partner V 2 of V 1 , V 3 of V 2 , etc. The ground state w av e function of V n ( n = 0 , 1 , . . . ) is ψ ( n ) 0 ∼ exp( − W 0 ( x, λ n ), where λ n = f n ( λ 0 ), with energy P n − 1 k =0 R ( λ k ). Then ag ain fro m (16) w e kno w that the w a ve f unction of the n th state of H 0 is ψ (0) n ∼ ( A + ) n ψ ( n ) 0 , with energy E (0) n = n − 1 X k =0 R ( λ k ) , n = 0 , 1 , . . . (19) So with shap e inv aria nce one obtains the complete spectrum of H 0 . It is now obv ious that SI is a sufficien t condition o f in tegrability in SUSYQM. T o classify shap e-in v ariant exactly solv able mo dels in SUSYQM, o ne m ust solv e the SI condition (18) 7 to get all the functional forms of W 0 ( x ), λ 1 = f ( λ 0 ), and R ( λ 0 ). This general problem is v ery difficult and, to the best of our kno wledge, is still unsolv ed. F urther constrain t s on the p ossible class o f shap e in v ariant p otentials are required. P articularly , in order to obtain the we ll-kno wn exactly solv able mo dels one m ust assume that (again with hindsight) the parameters o f the t w o partner p otentials are related by simply a translational shift, i.e. λ 1 = f ( λ 0 ) = λ 0 + m differ fro m λ 0 only by a set of constants m . Ev en with this simplification, the required c hange of co ordinates z = z ( x ) needed in solving the SI condition cannot b e determined naturally in t he approac h of SUSYQM, but has to b e tak en as given from the kno wn solutions of the resp ectiv e mo dels. On the ot her ha nd, in the prepo ten tial a pproac h SI needs not b e imp osed, and W 0 and z ( x ) a re determined by simply pic king tw o p olynomials with the appropriate degrees. In this sense it app ear s to us that the prep o t en tial appro ac h is conceptually m uc h simpler. Nev ertheless, putting the differences of the t wo approa c hes aside, one could not help but w onder what role SI plays in the prep oten tial a ppro ac h. Belo w w e would lik e to demonstrate that for the exactly solv able models obtained in the prepotential approac h, SI is automat- ically satisfied. W e shall discuss the cases with sinus oidal and non-sinus oidal co ordinat es separately . IV. SHAPE INV ARIANCE IN PREPOTENTIAL APPR O ACH: SINUSOID AL COORDINA TES Our strategy is to sho w that, with z ( x ) and W 0 ( x ) giv en in Sect. I I(A) and (B) that pro duce the ten well-kno wn exactly solv able mo dels, the SI condition (18) is alw ays satisfied, i.e. one can alwa ys find the set of new parameters λ 1 in terms of the old ones λ 0 . In the pro cess, w e demonstrate that the change in the parameters of the shap e-inv ariant p ot entials are translational. In this section, we first consider the cases inv olving sinus oidal co or dinates. F or exactly solv able systems, w e mu st take W ′ 0 z ′ = P 1 ( z ). Lab elling the corresp onding pa r ameters of 8 the t wo shap e-in v ariant p oten tia ls b y k = 0 , 1, we hav e z ′ 2 = Q 2 ( z ) = αz 2 + β z + γ ; (20) P ( k ) 1 ( z ) = a k z + b k , , k = 0 , 1 (21) W ′ 0 ( λ k ) = P ( k ) 1 ( z ) p Q 2 ( z ) , λ k = ( a k , b k ) . (22) Note that z ( x ) is the same for the shap e-in v ariant p otentials. Then the SI condition ( 1 8) leads to  P (0)2 1 − P (1)2 1  + Q 2 d dz  P (0) 1 + P (1) 1  − 1 2 dQ 2 dz  P (0) 1 + P (1) 1  = R ( λ 0 ) Q 2 . (23) Equating the co efficien ts of the p ow ers of z , one arriv es at the follow ing equations relating the parameters a 2 0 − a 2 1 = Rα , 2 ( a 0 b 0 − a 1 b 1 ) + β 2 ( a 0 + a 1 ) − α ( b 0 + b 1 ) = R β , (24) b 2 0 − b 2 1 + γ ( a 0 + a 1 ) − β 2 ( b 0 + b 1 ) = R γ . F or simplicit y w e write R fo r R ( λ 0 ). W e men tion here that the signs of a and b are fixed b y the normalization o f the w av e functions. This means they are the same for the tw o shap e-in v ariant par tner p otentials. W e w ould lik e to solve (24) for λ 1 = ( a 1 , b 1 ) and R in terms of λ 0 = ( a 0 , b 0 ). T o facilitate solution, w e find it con v enien t to first determine all inequiv alen t t yp es of sin usoidal co ordinates. A. Inequiv alen t sin usoidal co ordinates Dep ending on the presence of the parameters α , β and γ , there are three inequiv alen t cases of sinusoidal co ordinates: (i) z ′ 2 = γ 6 = 0, (ii) z ′ 2 = β z + γ ( β 6 = 0), and (iii) z ′ 2 = αz 2 + β z + γ ( α 6 = 0). By an appropriate shifting and/or scaling, these cases can b e recast in to three canonical forms. The form giv en for case (i) is already the canonical form of this case. W e shall take γ > 0 as γ ≤ 0 leads to phy sically unin teresting c hange of v ariable. This case giv es rise to the shifted oscillator. 9 By shifting z to ˆ z ≡ z + γ /β in case ( ii) , w e get the canonical form ˆ z ′ 2 = β ˆ z . F or phy sical systems w e require β > 0. This case corresponds to the t hree-dimensional oscillator. Case (iii) can b e recast a s ˜ z ′ 2 = α ˜ z 2 + ˜ γ , where ˜ z ≡ z + β / 2 α and ˜ γ ≡ ∆ / 4 α with the discriminan t ∆ ≡ 4 αγ − β 2 . F or the case ∆ = 0 (the exp onential case) and α > 0, the system th us generated is related to the Morse p otential. F or ∆ 6 = 0, w e ha ve t w o situations. If α > 0 (the h yp erb olic case), the canonical fo rm is ˆ z ′ 2 = α ( ˆ z 2 ± 1), where ˆ z ≡ p 4 α 2 / | ∆ | ˜ z , and the plus (minus ) sign corresp onds to ∆ > 0 (∆ < 0). The plus sign give s rise to the Scarf I I mo del, while the minus sign correspo nds to the generalized P¨ osc hl-T eller mo del. F or α < 0 (the trigonometric case), the canonical form is ˆ z ′ 2 = | α | ( ± 1 − ˆ z 2 ), where again ˆ z ≡ p 4 α 2 / | ∆ | ˜ z , and the plus (minus) sign corresp onding t o ∆ < 0 (∆ > 0). With the plus sign w e get the Scarf I mo del, while the minus sign do es not lead to a n y viable system a s the transformation is imaginary . F rom the ab ov e discussions, we see that w e need only to discuss the three inequiv alen t canonical cases, namely , (i) z ′ 2 = γ 6 = 0, (ii) z ′ 2 = β z ( β > 0), and (iii) z ′ 2 = α ( z 2 + δ ) ( δ = 0 , ± 1 for α > 0 , and δ = − 1 if α < 0). B. Case (i): z ′ 2 = γ > 0 F or this case, it is easy to c hec k that a 0 ( a 1 ) m ust not v anish, or it will lead to v anishing p oten tial. F urthermore, w e must ha v e a 0 > 0 and a 1 > 0 in order that the w a v e f unctions b e normalizable. The SI conditions ( 2 4) b ecome ( a 0 + a 1 )( a 0 − a 1 ) = 0 , (25) a 0 b 0 − a 1 b 1 = 0 , (26) b 2 0 − b 2 1 + γ ( a 0 + a 1 ) = R γ . (27) Equations (2 5) and (26) require a 1 = a 0 , b 1 = b 0 , or a 1 = − a 0 , b 1 = − b 0 . In the latter solution the signs of a 1 and b 1 are differen t fro m those of a 0 and b 0 , and hence the w av e functions of one of the tw o systems cannot be nor malizable if those of the other system can. In fact, for this case we hav e R = 0 from (27). This means t he ground states of the tw o systems ha ve the same energy . But the flip of b o t h signs of a and b of W 0 means that the ground states of the t w o systems hav e the forms exp( − W o ) and exp(+ W 0 ). They cannot b e b oth normalizable. This is exactly the result in SUSYQM. 10 So we are left with the choice a 1 = a 0 , b 1 = b 0 . F rom (27) w e ha v e R = 2 a 0 . Th us R is a constant, and fr o m (19) it implies oscillator-lik e sp ectrum, i.e. E n = na 0 . This giv es the shifted oscillator. The ab o ve discussion sho ws that in t his case SI is a necessary condition. The parameters of t he tw o partner systems a r e related by ( a 1 , b 1 ) = ( a 0 , b 0 ), and the shift parameter is R = 2 a 0 . C. Case (ii): z ′ 2 = β z ( β > 0 ) Normalizabilit y of wa v e functions in this case require that a > 0 and b < 0. Now the SI conditions (24) ar e ( a 0 + a 1 )( a 0 − a 1 ) = 0 , (28) 2 ( a 0 b 0 − a 1 b 1 ) + β 2 ( a 0 + a 1 ) = R β , (29) ( b 0 + b 1 )  b 0 − b 1 − β 2  = 0 . (30) P ossible solutions of these equations a re a 0 ± a 1 = 0, b 0 + b 1 = 0 or b 0 − b 1 − β / 2 = 0. T o k eep the signs of a and b unc hanged, w e can only take ( a 1 , b 1 ) = ( a 0 , b 0 − β / 2) as the viable solution. Then from (30) w e get R = 2 a 0 , whic h again give s an oscillator- like sp ectrum. This is just the case of the three-dimensional oscillator. D. Case (iii): z ′ 2 = α ( z 2 + δ ) Next w e consider the case with z ′ 2 = α ( z 2 + δ ) ( δ = 0 , ± 1 for α > 0, and δ = − 1 if α < 0). As men tioned b efore, this case co v ers t he Morse, generalized P¨ osc hl-T eller, and the Scarf I and I I pot en tials. The SI conditions (24) are a 2 0 − a 2 1 = Rα , (31) 2 ( a 0 b 0 − a 1 b 1 ) − α ( b 0 + b 1 ) = 0 , (32) b 2 0 − b 2 1 + αδ ( a 0 + a 1 ) = R αδ. (33) T o solve a 1 , b 1 and R in terms of a 0 and b 0 , w e eliminate Rα in (33) using (31) t o g et ( b 0 + b 1 ) ( b 0 − b 1 ) + δ ( a 0 + a 1 ) ( a 1 − a 0 + α ) = 0 . (34) 11 F rom (34) w e can hav e four p ossible sets of solutions: a 0 + a 1 = 0 , b 0 + b 1 = 0; (35) a 0 + a 1 = 0 , b 0 − b 1 = 0; (36) a 0 − a 1 = α, b 0 + b 1 = 0; (37) a 0 − a 1 = α, b 0 − b 1 = 0 . (38) The first three sets o f solutions inv olv e c hange o f signs of a and/or b , and so a re not viable as discussed b efo r e. Th us fo r this case w e m ust take ( a 1 , b 1 ) = ( a 0 − α, b 0 ) whic h also satisfies (32). Eq. (31) then g iv es R ( λ 0 ) = a 2 0 − a 2 1 α = 2 a 0 − α. (39) F rom (19) the energies ar e E n = a 2 0 − a 2 n α = a 2 0 − ( a 0 − nα ) 2 α , n = 0 , 1 , . . . (4 0) This is exactly the results in SUSYQM [1 ]. T o conclude this section, w e hav e sho wn that SI is automatically satisfied in the prepo- ten tial approac h for the sinus o idal cases. V. SHAPE INV ARIANCE IN PREPOTENTIAL APP R OA CH: NON- SINUSOIDAL COORDINA TES In t his case, W ′ 0 = − Az + B / A and z ′ = α ( λ − z 2 ). Here λ 0 = ( A, B ). As in the la st section, w e sho w t ha t one can alw ay s find a set o f new parameter λ 1 = ( A ′ , B ′ ) in terms of λ 0 that solv es the SI condition (18). In fact, from (18) o ne finds A ( A + α ) = A ′ ( A ′ − α ) , (41) B = B ′ , (42) B 2 A 2 − αλA = B ′ 2 A ′ 2 + α λA ′ + R . (43) Solutions of (41) are A ′ = − A and A ′ = A + α . The first solution has the sign o f A c hanged, and will lead to non-normalized w a ve functions. Hence the viable solution is 12 λ 1 = ( A ′ , B ′ ) = ( A + α, B ). Once again, the ch ange in the para meters A and B of t he shap e-in v ariant p oten tials are translational. Finally , fro m (43) w e find R ( λ 0 ) = B 2  1 A 2 − 1 ( A + α ) 2  − αλ (2 A + α ) . (44) This agrees with the results in SUSYQM [1]. Th us we hav e sho wn that in the prep oten tial a pproac h for mo dels based on non-sin usoidal co ordinates, SI is a lso a necessary consequence of the fo rms of W 0 and z ′ . VI. SUMMAR Y A unified approa c h to b oth the exactly and quasi-exactly solv able systems has b een pro- p osed previously based on the so-called prep oten tia l in [4, 5, 6]. In t his appro ac h solv abilit y of a quan tal system can b e solely classifie d b y t wo inte gers, the degrees of t w o p olyno- mials whic h determine the c hange of v ariable and the zero-th o rder prep oten tial. All the w ell-kno wn exactly solv able models obtained in SUSYQM can b e easily constructed by ap- propriately c ho osing the tw o p olynomials. But all these exactly solv able mo dels are o btained in SUSYQM only b y taking the SI condition as a sufficien t condition. The requiremen t to get exactly solv able mo dels in the prep oten tia l approach app ears to b e mu c h simpler, and definitely without the need of SI condition. In this pap er w e ha ve show n that the SI condition is in fact only a necessary condition in the prep oten tial approac h to exactly solv able systems, and hence needs not b e a ssumed. In the pro cess, we ha ve demonstrated that the change in the parameters of the w ell-kno wn shap e-in v ariant p oten tials ar e indeed tra nslational, a result whic h w as also assumed in SUSYQM. Ac kno w ledgmen ts This w ork is supported in part b y the National Science Council (NSC) of the Republic of China under Grant Nos. NSC 96-21 12-M-032 -007-MY3 and NSC 95-2 9 11-M-032 -001-MY2. P art of the w ork was done during m y visit to the Y uk a w a Institute for Theoretical Ph ysics (YITP) at the Ky o to Univ ersity supported under NSC Gran t No. 97-2918-I- 032-00 2 . I w ould like to thank R. Sasaki and the staff and mem b ers o f YITP for their hospitalit y . I 13 am also grateful to Y. Hosotani and Y. Matsuo for useful discussion and hospitalit y . [1] F. Co op er, A. Kh are and U. Sukh atme, Ph ys. Rep. 251 , 267 (1995). [2] G. Junker, Sup ersymm etric Metho ds in Quantum an d Statistical Physic s , (Spr inger-V erlag, Berlin, 1996). [3] L. Genden s h tein, J E TP Lett. 38 , 356 (1983 ). [4] C.-L. Ho, An n . Phys. 323 , 2241 (2008 ). [5] C.-L. Ho, Pr ep oten tial approac h to exact and qu asi-exact solv abilit ies of Hermitian and non- Hermitian Hamilt onians (T a lk presen ted at “Conference in Honor of CN Y ang’s 85t h Birth- da y”, 31 Oct - 3 No v, 2007, Singap ore). arXiv:080 1.094 4 [hep-th]. 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