It is shown that for any exactly solvable potential considered here, belonging to the class of the first generation, there are always posibilities to construct the class of second generation potentials, which are either exactly solvable or quasi-exactly solvable. The alternatives exactly solvable or quasi-exactly solvable depend on the choice of the "base" related to the set of excited eigenstates corresponding to the first generation potentials.Pokazuje se da za svaki ovdje razmatrani egzaktno-rješivi potencijal, koji pripada razredu prve generacije, uvijek postoji mogućnost složiti razred druge generacije potencijala koji su ili egzaktno rješivi ili kvazi-egzaktno rješivi. Inačice egzaktno rješivi ili kvazi-egzaktno rješivi ovise o izboru “baze” koja se odnosi na skup viših svojstvenih stanja koji je vezan s prvom generacijom potencijala

Chuan, Cao Xuan

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ISSN1330–0016CODEN FIZBE7ON QUASI-EXACTLY SOLVABLE POTENTIALSCAO XUAN CHUAN01 Parvis du Breuil, 92160 Antony, FranceReceived 10 October 2001; Accepted 21 January 2002Online 25 May 2002It is shown that for any exactly solvable potential considered here, belonging tothe class of the first generation, there are always posibilities to construct the classof second generation potentials, which are either exactly solvable or quasi-exactlysolvable. The alternatives exactly solvable or quasi-exactly solvable depend on thechoice of the “base” related to the set of excited eigenstates corresponding to thefirst generation potentials.PACS number: 03.64.Ge UDC 530.145Keywords: Schrödinger equation, exactly solvable potentials, quasi-exactly solvable po-tentials, supersymmetry1. IntroductionExactly solvable potentials are referred to those for which the Schrödinger equa-tion can be solved analytically, generating the sets of eigenvalues, En, and eigen-functions, φn.For quasi-exactly solvable potentials, only part of these sets can be reachedanalytically, the remaining must be dealt by other means, mostly numerical.The interest in investigating the second-generation potentials is to wonder whatwould be the possible links between these two classes of potentials, or, in otherwords, would it be possible to construct the second generation potentials which areexactly or quasi-exactly solvable from the first ones?During the past decade, one may observe the first attempts in that direction,undertaken from different angles of attack, by taking advantage of the recent ad-vances in supersymmetry (SU(2)) [1–6].Assumig that the couple of partner potentials are exactly solvable and definedFIZIKA B 11 (2002) 1, 9–18 9cao xuan chuan: on quasi-exactly solvable potentialsbyV (1)∓ = u2 ∓ u′ ,where u(x) is the usual superpotential, the second generation potentials are definedbyV (2)∓ = v(2)2 ∓ v(2)′ , (1)where v(2) is the second generation superpotential,v(2) = u − φ′nφn, (2)while φn represents the nth excited eigenstate related to V (1)−.When n = 0 (the ground state), this definition does not bring anything new, butwhen n > 0, the second-generation potential v(2)n must contain singularities whichreflect the presence of the “nodes” of the eigenstates (theorem of Sturm-Liouville).As a consequence, the second-generation potentials V (2)∓ will also involve singu-larities. Thus, the next question which arises naturally is what can one expect fromthe construction in Eq. (1)?The present work will focus on two aspects which may serve for future reseachwork:(1) How and to which extent the second-generation potential V (2)∓ is solvable?(2) In the context of SU(2), how the concept of shape invariance and doubledegeneracy, which is associated with the first generation, can be conserved orbroken with the second generation?Since it will be convenient to approach this problem within the frame of thetheory of mixing function, it is useful to devote the first part of this work to themethodological aspect with a brief reminder of the guidelines. The details can befound in earlier references [7–10].In the second part, these results will serve to analyse the situation by showingthat one must discern two categories of the “bases”, for which the second-generationpotentials may either be exactly or non-exactly solvable, with partial or total break-ing of the symmetry, and conservation or non-conservation of the shape invariance.In order to show how the theory can be implemented, the third part describesthe details of an example. The case of the harmonic oscillator is chosen because ofits potential role in various models of many fields of physics.Next, returning to more general considerations, in the last part, the proof ofa theorem will be given according to which, under certain conditions, it is alwayspossible to construct a new type of quasi-exactly solvable potentails, regardless ofthe choice of the “base”.10 FIZIKA B 11 (2002) 1, 9–18cao xuan chuan: on quasi-exactly solvable potentials2. TheoryLet the first-order differential matrix equationφ′ + Fφ = 0, with φ = (φ1, φ2)+ and F =(u d0 u), (3)u(x) and d(x) being arbitrary analytic functions. By construction,φ2 ' exp(∫−u dx).Let the mixing function beX(x), with φ1 = Xφ2 . (4)It has already been shown that if X is a solution of the second-order differentialequationX ′′ − 2uX ′ − EnXn = 0 , (5)with En constant, the Schrödinger equation corresponding to the first componentφ1φ′′1 − V (1)−φ1 = Enφ1 , with φ1 = Xn exp(−∫udx)(6)must be exactly solvable with eigenspectrum {En} and eigenfunctions {φ1,n}.Likewise, letφ̄′ + F̄ φ̄ = 0, with φ̄ = (φ̄1, φ2) and F̄ =( −u d̄0 u), (7)and introduce a second mixing function Y , with φ̄1 = Y φ2, then the Schrödingerequationφ̄′′1 − V (1)+ φ̄1 = Ēφ̄1 (8)is also exactly solvable if Y satisfies the second differential equationY ′′ − 2uY ′ − (2u′ + Ē)Y = 0 . (9)Noting that Eq. (9) is simply the result of differentiation of Eq. (5), one can inferY = X ′n and Ē = En . (10)In the context of SU(2), u(x) is playing the role of the superpotential of the firstgeneration. The second condition in Eq. (10) merely expresses the double degener-acy of V (1)− and V (1)+.FIZIKA B 11 (2002) 1, 9–18 11cao xuan chuan: on quasi-exactly solvable potentials3. The second generationLet |n〉 be any excited state of the first component V (1)−. The second generationcouple can be constructed according to Eqs. (1) and (2). The following results areobtained and already described in earlier references:(a) The first component of the second generation couple V (2)−, up to a constant,is similar to V (1)−, i.e.,V (2)− = V (1)− + const ,which means that regardless of the choice of the base, if V (1)− is exactlysolvable, so will be V (2)−, too.(b) The eigenspectrum of the second generation couple isE(2)m = E(1)m − E(1)n , (11)where E(1)m and E(1)n are eigenvalues corresponding to the first component.(c) The second component V (2)+ of this second generation couple is of the formV (2)+ = V (1)+ + 2X ′nXn(X ′nXn− 2u)− En . (12)Its eigenfunction can be written asφ̄(2)1,m = Ym exp(−∫udx), (13)whereYm = X ′m −X ′nXnXm . (14)The solution is not defined for m = n, and normalisation requirements willput further constraints on this solution.For instance, the presence of the singularity contained in the analytic expressingof Eq. (14) can be removed only if some kind of “factorisation” is possible in thesenseXm = XnFn(x) .Fn(x) is an unspecified function.When this is not the case, while the first component V (2)− is always exactlysolvable for the reasons invoked in (a), the second component V (2)+ is not becauserelation (14) becomes invalid. The singularities appearing in V (2)+ are generally12 FIZIKA B 11 (2002) 1, 9–18cao xuan chuan: on quasi-exactly solvable potentials“strong” in the sense of Eq. (12), which, on the other hand, require that at thelocations of these singularities (x = xi), φ̄(2)(xi) = 0.The whole situation radically changes furthermore when one introduces anotherparameter t and constructs the second-generation superpotential v(2)t asv(2)t = u − tX ′nXn.The couple becomesV (2)− = V (1)− + t(t − 1)(X ′nXn)2+ tEn ,V (2)+ = V (1)+ + t(t + 1)(X ′nXn)2− 4tuX′nXn− tEn .(15)Note that when t = 1, the relation (12) is recovered. The problem becomes evenmore complicated because neither potential is exactly solvable, except perhaps forsome very special choices of the base.Thus, before engaging oneself in this direction, it will be useful to return to thesimplest case when t = 1, in order to see how the theory can be implimented inpractice.4. ExampleConsider the case of the harmonic oscillator potential for which the superpo-tential of the first generation can be defined asu(x) =12x .Obviously, the solutions of Eq. (5) can be represented by the usual Hermite poly-nomials Xn = Hn(z). Since the study of the first component V (2)− does not bringanything new, the second component, V (2)+, is more interesting. Its analytic ex-pression for the first values of n of the bases aren = 1 V (2)+1 =14x2 +2x2− 12n = 2 V (2)+2 =14x2 +4x2x2 − 1(2x2 − 1 − 1)− 32n = 3 V (2)+3 =14x2 + 6x2 − 1x2 − 3(3x2x2 − 1x2 − 3 − 1)− 52. . .(16)Below are some interesting remarks which may be useful in practice. Some of themhave already been mentioned in Ref. [6].FIZIKA B 11 (2002) 1, 9–18 13cao xuan chuan: on quasi-exactly solvable potentials(a) V (2)+n (−x) = V (2)+n (x) (even parity).(b) At infinity, they all tend to the classical harmonic-oscillator potential.(c) If one chooses the base n = 1, then it can be verified that V (2)−1 and V(1)−1differ only by a constant.(d) The couple V (2)−1 and V(2)+1 is shape-invariant when t /= 1, while it is notwhen t = 1.The discussion will now be focused on the special case t = 1, with n = 1 (i.e.,Xn = Hn). The eigenfunctions of V(2)−1 can be written asφm = NmXm exp(−∫u dx),where Nm is the normalization constant, with the eigenspectrum E(2)m = E(1)m −E(1)nin a general case. Depending on the choice of the base |n〉, one thus may havenegative eigenvalues. For instance, if n = 1, one has only one such state. Theexistence of such states was pointed out earlier [1], and it is beleived to have animpact on the construction of certain field theories.For V (2)−1 , the running index m in Eq. (16) may be any integer (m = 0, 1, 2, . . .),but for V (2)+1 , only the solutions with m odd (m = 1, 3, 5, . . .) can be accepted,because the validity of Eq. (14) can be assured only in this case.Therefore, the pairings of the eigenstates of V (2)−1 and V(2)+1 are reduced to onehalf.This constitutes an example of partial pairing of the states, for which the usualdefinition of the Witten index, specifying a clear-cut distinction between breakingand non-breaking of symmetry, would become inadequate. On the other hand, forthis case, the principle of “nodal structure” [12,13] will be more appropriate.Keeping the same base (n = 1), but with t /=1, the principle of shape invariancementioned above in (d) can be explicitly written asV(2)+1 (t − 1, x) = V (2)−1 (t, x) + t − 1 . (17)Both V (2)−1 and V(2)+1 are exactly solvable with the solutions expressed in terms ofthe generalized Laguerre polynomials Lnm, providing the parameter t be an integer[14,15].When both t and n are different from 1, the presence of singularities of type“strong” mentioned above, will split the domain of the definition of them intoseparated and unrelated regions, implying, therefore, different Hilbert spaces. Onecan, therefore, expect total breaking of symmetry.However, the situation is not so dim since some information can still be extractedwith the proof of the following theorem concerning the quasi-exact solvability ofthe first component V (2)− under certain conditions, while the second componentmust be approached in another way.14 FIZIKA B 11 (2002) 1, 9–18cao xuan chuan: on quasi-exactly solvable potentials5. Quasi-exact solvabilityTheorem: If equation X ′′ − 2uX ′ − EnXn = 0 (Eq. (5)) is exactly solvablewith the set of solutions {Xn}, then, regardless of the choice of the base, the sameequation in which u is substituted by v(2)(t, x), t being a parameter, will have atleast two exact solutions.Proof: The equation can be written in the formX ′′ − 2(u − t X′nXn)X ′ − Ē X = 0 , (18)Ē being a constant.(a) Obviously, the first solution is X = const which leads to Ē = 0.(b) The second solution has the form X = X1−2tn . By substitution into Eq. (18)and after simplification, one obtains(1 − 2t)X1−2tn[En − Ē1 − 2t]= 0 ,which implies that Ē = (1 − 2t)En.Note that this derivation is independent of the choice of |n〉.Alternatively, one may also use the method described earlier [8] by letting X =F (x)/Xsn, F (x) being an unspecified function and s a parameter. Then, Eq. (5)can be cast into the formX ′′ − 2uX ′X= A(x) + B(x) , (19)whereA(x) =1Xsn[F ′′ − 2uF ′] − sFXs+1n[X ′′n − 2uX ′n],andB(x) = 2(t − s)F′FX ′nXn− s(s + 1 − 2t)(X ′nXn)2.Requiring B(x) = 0, one obtains a relationship between s and t, F (x) = Xqn, whereq is arbitrary. Therefore, the simplest case will be q = 1, or F (x) = Xn. The twoparameters, s and t, must satisfy the constraints(s + 1 − 2t) = 2(t − s) , (20)FIZIKA B 11 (2002) 1, 9–18 15cao xuan chuan: on quasi-exactly solvable potentialswhich can be solved with the two following real solutionss+ = 2t, and s− = 1 .In this case, the eigenvalues areE+ = (1 − 2t)En and E− = 0 ,withA+ = 2t ,corresponding to the two solutions mentioned above.6. ApplicationIn the context of the present discussion, the above theorem can be used byreturning to the system (1) in which the quantity u(x) is replaced by v(2)n (x).With the above results, one can see that the Schrödinger equation is solvable ifthe function X satisfies Eq. (19)φ′′1,± − V (2)− φ1,± = Eφ1,± .The two eigenfunctions can be written asφ1,+ = X1−tn exp(∫−u dx)and φ1,− = Xtn exp(∫−u dx).Normalisation requires that 0 < t < 1.Extension. The condition of exact solvability of Eq. (5) is sufficient but notnecessary, which means that the function X must not be limited to the set of usualorthogonal functions of mathematical physics. The details of this aspect will bepresented later in another report.7. ConclusionSeveral facets of the use of the method of mixing functions are discussed. Par-ticularly, it is quite adapted for analysing the problem of singular potentials ofthe second generation, which are constructed from the first ones that are assumedsupersymmetric and exactly solvable.For the special choice of the base, both components of the second generation,V (2)− and V (2)+, are exactly solvable, but with the loss of shape and a partialpairing of the states.16 FIZIKA B 11 (2002) 1, 9–18cao xuan chuan: on quasi-exactly solvable potentialsFor that example, but with the parameter t /= 1, the concept of the shape in-variance remains valid, and for the same choice, both components are also exactlysolvable under specific conditions of this parameter.But for other choices of the base, both components are not solvable exactly witha complete breaking of the symmetry.However, under certain different specific constraints on this parameter, the firstcomponent V (2)− can be made quasi-exactly solvable, with an exact “doublet”,regardless of the base.References[1] A. Jewicki and P. P. Rodrigues, Phys. Lett. B 146 (1984) 55.[2] J. Casahoran, Phys. Lett. A 136 (1991) 425.[3] J. Casahoran and S. Nam, Int. J. Mod. Phys. A 6 (1991) 2719.[4] P. K. Panigrahi and U. Sukhatme, Phys. Lett. A 178 (1993) 251.[5] F. Cooper, A. Khare and U. Sukhatme, Phys. Reports, 251(1995) 267.[6] M. Robnik, J. Phys. A: Math. Gen. 30 (1997) 1287.[7] X. C. Cao, Compt. Rendu Acad. Sci. (Paris) 311 (1994) 2625.[8] X. C. Cao, Int. J. Theor. Phys. 36 (1997) 61445.[9] X. C. Cao, Int. J. Theor. Phys. 38 (1999) 9445.[10] X. C. Cao, Fizika B (Zagreb) 9 (2000) 1.[11] W. Frank, D. Land and R. Spector, Rev. Mod. Phys. 43 (1971) 36.[12] E. Witten, Nucl. Phys. B 188 (1981) 513.[13] E. Gozzi, Phys. Rev. D 33 (1986) 3665.[14] L. Lanthouwers, J. Math. Phys. 16 (1975) 7.[15] R. S. Kaushal, Pramana J. Phys. 42 (1994) 4.FIZIKA B 11 (2002) 1, 9–18 17cao xuan chuan: on quasi-exactly solvable potentialsKVAZI-EGZAKTNO RJEŠIVI POTENCIJALIPokazuje se da za svaki ovdje razmatrani egzaktno-rješivi potencijal, koji pripadarazredu prve generacije, uvijek postoji mogućnost složiti razred druge generacijepotencijala koji su ili egzaktno rješivi ili kvazi-egzaktno rješivi. Inačice egzaktnorješivi ili kvazi-egzaktno rješivi ovise o izboru “baze” koja se odnosi na skup vǐsihsvojstvenih stanja koji je vezan s prvom generacijom potencijala.18 FIZIKA B 11 (2002) 1, 9–18

Kvazi-egzaktno rješivi potencijal

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