Higher-order SUSY, exactly solvable potentials, and exceptional orthogonal polynomials
aa r X i v : . [ m a t h - ph ] A ug HIGHER-ORDER SUSY, EXACTLY SOLVABLEPOTENTIALS, AND EXCEPTIONALORTHOGONAL POLYNOMIALS
C. QUESNE
Physique Nucl´eaire Th´eorique et Physique Math´ematique,Universit´e Libre de Bruxelles, Campus de la Plaine CP229,Boulevard du Triomphe, B-1050 Brussels, [email protected]
Abstract
Exactly solvable rationally-extended radial oscillator potentials, whosewavefunctions can be expressed in terms of Laguerre-type exceptional orthog-onal polynomials, are constructed in the framework of k th-order supersym-metric quantum mechanics, with special emphasis on k = 2. It is shown thatfor µ = 1, 2, and 3, there exist exactly µ distinct potentials of µ th type andassociated families of exceptional orthogonal polynomials, where µ denotesthe degree of the polynomial g µ arising in the denominator of the potentials. Running head: Higher-order SUSYKeywords: quantum mechanics; supersymmetry; orthogonal polynomialsPACS Nos.: 03.65.Fd, 03.65.Ge 1
Introduction
In recent years, one of the most interesting developments in quantum mechanicshas been the construction of new exactly solvable potentials connected with the ap-pearance of families of exceptional orthogonal polynomials (EOP) in mathematicalphysics. In contrast with families of classical orthogonal polynomials, which startwith a constant, the exceptional ones start with some polynomial of degree m ≥ X families,corresponding to m = 1, were proposed in the context of Sturm-Liouville theory[1, 2]. They were then applied to quantum mechanics and proved to be related tosome exactly solvable rational extensions of known potentials, with the additionalinteresting property of being translationally shape invariant [3, 4] (although at thattime the list of additive shape-invariant potentials was thought to be complete [5,6, 7]).Shortly thereafter, the first examples of X EOP and of related shape-invariantpotentials were proposed in the framework of conventional supersymmetric quantummechanics (SUSYQM), suggesting a growing complexity with m [8]. Two distinctfamilies of Laguerre- and Jacobi-type X m families were then constructed for arbi-trary large m [9, 10] and their properties were studied [11, 12]. Several approacheswere considered in connection with the Darboux-Crum transformation [13, 14, 15],the Darboux-B¨acklund one [16, 17] or the prepotential method [18, 19]. Possibleapplications to position-dependent mass systems [20], to PT -symmetric potentials[4, 21], to conditionally-exactly solvable ones [22, 23], and to the Dirac and Fokker-Planck equations [24] were studied.Very recently, the X m EOP were generalized to multi-indexed families by meansof multi-step Darboux algebraic transformations [25] or the Crum-Adler mechanism[26].The purpose of the present work is to show how the related exactly solvablepotentials arise in an (essentially equivalent) simple extension to higher order of the2USYQM approach used in [4, 8]. For simplicity’s sake, we shall restrict ourselveshere to rational extensions of the radial oscillator connected with Laguerre-typeEOP.
Going from first- to second-order SUSYQM may be achieved in two slightly differentways, corresponding to the parasupersymmetric (PSUSY) scheme [27] or to thesecond-derivative (SSUSY) setting [28, 29, 30, 31, 32].In the former approach, one starts with a pair of first-order SUSYQM partners(in units ~ = 2 m = 1) H (+) = A † A = − d dx + V (+) ( x ) − E, H ( − ) = AA † = − d dx + V ( − ) ( x ) − E,A † = − ddx + W ( x ) , A = ddx + W ( x ) , V ( ± ) ( x ) = W ( x ) ∓ W ′ ( x ) + E, (2.1)which intertwine with the first-order differential operators A and A † as AH (+) = H ( − ) A and A † H ( − ) = H (+) A † . Here W ( x ) is the superpotential, which can beexpressed as W ( x ) = − φ ′ ( x ) /φ ( x ) in terms of a (nodeless) seed solution φ ( x ) of theinitial Schr¨odinger equation (cid:18) − d dx + V (+) ( x ) (cid:19) φ ( x ) = Eφ ( x ) , (2.2) E is the factorization energy, assumed smaller than or equal to the ground-stateenergy E (+)0 of V (+) , and a prime denotes a derivative with respect to x . We shall onlyconsider here the case where E < E (+)0 , in which occurrence φ ( x ) is nonnormalizable,and we shall assume that the same holds true for φ − ( x ). Then H (+) and H ( − ) turnout to be isospectral [5].Next, we consider a second pair of first-order SUSYQM partners ˜ H (+) and ˜ H ( − ) with the same characteristics and distinguish all related quantities from those per-taining to the first pair by tildes. If we choose ˜ V (+) ( x ) such that ˜ V (+) ( x ) = V ( − ) ( x ),then both pairs of SUSYQM partners ( H (+) , H ( − ) ) and ( ˜ H (+) , ˜ H ( − ) ) can be glued3ogether, so that we get a second-order PSUSY system or, equivalently, a reducibleSSUSY one.The latter is described in terms of two Hamiltonians h (1) = − d dx + V (1) ( x ) , h (2) = − d dx + V (2) ( x ) , (2.3)which intertwine with some second-order differential operators A † = d dx − p ( x ) ddx + q ( x ) , A = d dx + 2 p ( x ) ddx + 2 p ′ ( x ) + q ( x ) , (2.4)as A h (1) = h (2) A and A † h (2) = h (1) A † , so that the functions p ( x ), q ( x ) and thepotentials V (1 , ( x ) are constrained by the relations q ( x ) = − p ′ + p − p ′′ p + (cid:18) p ′ p (cid:19) − c p ,V (1 , ( x ) = ∓ p ′ + p + p ′′ p − (cid:18) p ′ p (cid:19) + c p , (2.5)where c is some integration constant.The relation between both approaches follows from the equations h (1) = H (+) + c , h (2) = ˜ H ( − ) − c , A † = A † ˜ A † , and A = ˜ AA . It turns out that h (1) and h (2) are bothpartners of some intermediate Hamiltonian h = H ( − ) + c = ˜ H (+) − c , that theconstant c is related to the two factorization energies through c = E − ˜ E , andthat the function p ( x ) can be expressed in terms of the two superpotentials as p ( x ) = ( W + ˜ W ).Instead of φ and ˜ φ , we may start from two seed solutions φ and φ of theinitial Schr¨odinger equation (2.2) with respective energies E and E (less than E (+)0 ) and such that φ − and φ − are nonnormalizable. Then, on choosing φ = φ and ˜ φ = Aφ = W ( φ , φ ) /φ , so that E = E and ˜ E = E , we get p ( x ) = − W ′ ( φ , φ )2 W ( φ , φ ) = − ( E − E ) φ φ W ( φ , φ ) , (2.6)where W ( φ , φ ) denotes the Wronskian of φ ( x ) and φ ( x ). Since, from (2.5), theSSUSY partner potential can be written as V (2) ( x ) = V (1) ( x ) + 4 p ′ ( x ), it is clearthat it can be completely determined from the knowledge of this Wronskian.4 Radial Harmonic Oscillator in Second-orderSUSYQM
Let us consider a radial oscillator potential V l ( x ) = 14 ω x + l ( l + 1) x , (3.1)where ω and l denote the oscillator frequency and the angular momentum quantumnumber, respectively, and the range of x is the half-line 0 < x < ∞ . As wellknown, the corresponding Schr¨odinger equation has an infinite number of bound-state wavefunctions, which, up to some normalization factor, can be written as ψ ( l ) ν ∝ x l +1 e − ωx L ( l + ) ν ( ωx ) ∝ η l ( z ) L ( α ) ν ( z ) , ν = 0 , , , . . . , (3.2)with z = ωx , α = l + , η l ( z ) = z (2 α +1) e − z , (3.3)and L ( α ) ν ( z ) some Laguerre polynomial. The associated bound-state energies aregiven by E ( l ) ν = ω (2 ν + l + ) = ω (2 ν + α + 1) . (3.4)Motivated by the experience gained in the first-order SUSYQM approach (seeEqs. (2.12), (2.16), and (2.18) of Ref. 7), as well as by subsequent developments[9, 10, 13], we may consider two different types of seed solutions φ ( x ) with propertiesas required in Sec. 2, namely φ I lm ( x ) = χ I l ( z ) L ( α ) m ( − z ) ∝ x l +1 e ωx L ( l + ) m ( − ωx ) , (3.5) φ II lm ( x ) = χ II l ( z ) L ( − α ) m ( z ) ∝ x − l e − ωx L ( − l − ) m ( ωx ) , (3.6)with χ I l ( z ) = z (2 α +1) e z , χ II l ( z ) = z − (2 α − e − z , (3.7)and corresponding energies E I lm = − ω ( α + 2 m + 1) , E II lm = − ω ( α − m − , (3.8)5espectively. Such seed solutions are related to the two families L1 and L2 ofLaguerre-type X m EOP. Note that for type II, α must be greater than m .Our purpose is to construct some rationally-extended radial oscillator potentials V l, ext ( x ) with a given l by using two seed solutions φ and φ , as explained in Sec. 2.Taking into account that the order of φ and φ is irrelevant, there are three types ofpossibilities for the pair ( φ , φ ) and it turns out that in each case, we have to startfrom a potential V (+) ( x ) = V l ′ ( x ) with some different l ′ . The corresponding Wron-skian W ( φ ( x ) , φ ( x )) can be written in terms of some µ th-degree polynomial in z , g µ ( z ), itself expressible in terms of a Wronskian ˜ W ( f ( z ) , g ( z )) of some appropriatefunctions of z , as follows:( i ) V (+) = V l − , φ = φ I l − ,m , φ = φ I l − ,m , ≤ m < m , W ( φ , φ ) = ωx ( χ I l − ) g µ ( z ) ,g µ ( z ) = ˜ W ( L ( α − m ( − z ) , L ( α − m ( − z )) , µ = m + m −
1; (3.9)( ii ) V (+) = V l +2 , φ = φ II l +2 ,m , φ = φ II l +2 ,m , ≤ m < m < α + 2 , W ( φ , φ ) = ωx ( χ II l +2 ) g µ ( z ) ,g µ ( z ) = ˜ W ( L ( − α − m ( z ) , L ( − α − m ( z )) , µ = m + m −
1; (3.10)( iii ) V (+) = V l , φ = φ I l,m , φ = φ II l,m , ≤ m , ≤ m < α, W ( φ , φ ) = 2 x χ I l χ II l g µ ( z ) ,g µ ( z ) = z ˜ W ( L ( α ) m ( − z ) , L ( − α ) m ( z )) − ( z + α ) L ( α ) m ( − z ) L ( − α ) m ( z ) ,µ = m + m + 1 . (3.11)In all three cases, we can write (provided g µ does not have any zero on thehalf-line) V (1) = V l ′ −
12 ( E + E ) ,V (2) = V l − ω ( g µ g µ + 4 z " ¨ g µ g µ − (cid:18) ˙ g µ g µ (cid:19) −
12 ( E + E ) + C, (3.12)where a dot denotes a derivative with respect to z and C = − ω , 2 ω , or 0 in case( i ), ( ii ), or ( iii ), respectively. In the associated PSUSY approach, which we cannot6etail here due to space restrictions, the intermediate potential is some V l − , ext , V l +1 , ext , or V l +1 , ext potential. Note that it reduces to some bare radial oscillatorpotential in the special case where m = 0. Furthermore, if we adopt the reverseorder for the φ ’s, e.g. φ = φ II l,m , φ = φ I l,m in case ( iii ), the final potential will bethe same, but the intermediate one will be different, e.g. some V l − , ext potential incase ( iii ).Both V (1) and V (2) have the same bound-state energy spectrum, given by E (1) νl = E (2) νl = ω (2 ν + 2 l + m + m −
1) in case ( i ) ,ω (2 ν + 2 l − m − m + 5) in case ( ii ) ,ω (2 ν + 2 l + m − m + 2) in case ( iii ) , (3.13)where ν = 0, 1, 2, . . . . The bound-state wavefunctions ψ (2) ν ( x ) of V (2) can beobtained either by acting with A on those of V (1) , ψ (1) ν ( x ) ∝ η l ′ ( z ) L ( α ′ ) ν ( z ), or bydirectly inserting the expression ψ (2) ν ( x ) ∝ η l ( z ) g µ ( z ) y n ( z ) , n = µ + ν, ν = 0 , , , . . . , (3.14)in the Schr¨odinger equation for V (2) ( x ). As a result, we obtain the following differ-ential equation for y n ( z ), (cid:20) z d dz + (cid:18) α + 1 − z − z ˙ g µ g µ (cid:19) ddz + ( z − α ) ˙ g µ g µ + z ¨ g µ g µ (cid:21) y n ( z ) = ( µ − n ) y n ( z ) . (3.15)The orthonormality and completeness of ψ (2) ν ( x ), ν = 0, 1, 2, . . . , on the half-lineimply that for any n = µ + ν , ν = 0, 1, 2, . . . , the differential equation (3.15)admits a n th-degree polynomial solution and that when ν runs over 0, 1, 2, . . . ,such polynomials form an orthogonal and complete set with respect to the positive-definite measure z α e − z g − µ dz . We shall denote these polynomials by L I , I α,m ,m ,n ( z ), L II , II α,m ,m ,n ( z ), and L I , II α,m ,m ,n ( z ) in cases ( i ), ( ii ), and ( iii ), respectively.It is worth noting that in cases ( i ) and ( ii ), the differential equation (3.15) canbe rewritten in a slightly different form. The definitions of g µ ( z ) in (3.9) and (3.10),combined with Laguerre equation, indeed lead to z ¨ g µ = ( z ¯ g µ − ( α + z ) ˙ g µ + µg µ in case ( i ) , z ¯ g µ + ( α + z ) ˙ g µ − µg µ in case ( ii ) , (3.16)7here ¯ g µ = ˜ W (cid:0) ˙ L ( α − m ( − z ) , ˙ L ( α − m ( − z ) (cid:1) or ¯ g µ = ˜ W (cid:0) ˙ L ( − α − m ( z ) , ˙ L ( − α − m ( z ) (cid:1) , respec-tively. As a result, we get (cid:20) z d dz + (cid:18) α + 1 − z − z ˙ g µ g µ (cid:19) ddz − α ˙ g µ g µ + 2 z ¯ g µ g µ (cid:21) L I , I α,m ,m ,n ( z )= − nL I , I α,m ,m ,n ( z ) (3.17)and (cid:20) z d dz + (cid:18) α + 1 − z − z ˙ g µ g µ (cid:19) ddz + 2 zg µ ( ˙ g µ + ¯ g µ ) (cid:21) L II , II α,m ,m ,n ( z )= (2 µ − n ) L II , II α,m ,m ,n ( z ) . (3.18)The latter equation coincides with that obtained for ˆ L ( α,m ,m ) n ( z ) in Ref. 24. In cases ( i ) and ( ii ), the lowest-degree example for g µ corresponds to m = 0, m = 1, leading to g = 1 or g = − L I , I α, , ,n = L II , II α, , ,n = L ( α ) n . a On assuming m = 0, m = m + 1 ( m ≥ g m = L ( α − m ( − z ) or g m = − L ( − α − m ( z ), so that L I , I α, ,m +1 ,n = L I α,m,n and L II , II α, ,m +1 ,n = L II α,m,n ; hence the associated extended potentials coincide with thosealready obtained from first-order SUSYQM. The next values m = 1, m = 2 donot provide any new result either because g = L ( − α − ( z ) or g = − L ( α − ( − z ) andtherefore L I , I α, , ,n = L II α, ,n and L II , II α, , ,n = L I α, ,n . b Going to m = 1, m = 3 at lastgives rise to a new result (the same in the two cases), because g = [ z + 3 αz +3( α − α + 1) z + ( α − α ( α + 1)] / L ( α − ( − z ) and L ( − α − ( z )leading to L I α, ,n and L II α, ,n , respectively.In case ( iii ), considering either m ≥ m = 0 or m = 0, m ≥ L I , II α,m , ,n = L I α,m +1 ,n or L I , II α, ,m ,n = L II α,m +1 ,n , since g m +1 = − ( m + 1) L ( α − m +1 ( − z ) or g m +1 = ( m + 1) L ( − α − m +1 ( z ) as a consequence of known a These equalities and the following ones are of course dependent on the normalization chosenfor the EOP. We assume here that the latter can be appropriately adjusted. b It is worth observing here that the last reduction has not been noted in Ref. 24. m = m = 1, but the resulting g coincides with thatobtained for cases ( i ) and ( ii ).We conclude that SSUSY does not lead to any new extended potential of linearnor quadratic type, but gives rise to a new cubic one. Its explicit form can beobtained from V (2) for µ = 3, which we rewrite as V l, ext = V l + V l, rat , V l, rat = − ω ( g g + 4 z " ¨ g g − (cid:18) ˙ g g (cid:19) , (4.1)by dropping the additional constant. On combining Eq. (3.3) with the expressionof g ( z ) given above, we get V l, rat ( x ) = N ( x ) D ( x ) + N ( x ) D ( x ) , (4.2)where N ( x ) = 12 ω [ ω x − (2 l + 1) + 28] ,N ( x ) = − ω [3(2 l + 1) ω x + 4(2 l − l + 3) ωx + (2 l − l + 1)(2 l + 3)] ,D ( x ) = ( ωx + 2 l + 1) − ωx + 2 l + 1) . (4.3)This result may be compared with the two cubic-type extended potentials com-ing from first-order SUSYQM, which are given by Eq. (4.1) with either g ( z ) = L ( α − ( − z ) or g ( z ) = L ( − α − ( z ), and which can be written in the form (4.2) with N ( x ) = 12 ω [ ω x − (2 l − l + 5)] ,N ( x ) = − ω (2 l + 5)[(2 l + 9) ω x + 2(2 l + 3)(2 l + 5) ωx + (2 l + 1)(2 l + 3)(2 l + 5)] ,D ( x ) = ( ωx + 2 l + 5) − l + 5)(3 ωx + 6 l + 11) , (4.4)or N ( x ) = 12 ω [ ω x − (2 l − l + 11)] ,N ( x ) = 144 ω (2 l − l − ω x + 2(2 l − l − ωx + (2 l − l − l + 1)] ,D ( x ) = ( ωx + 2 l − + 2(2 l − ωx + 6 l − , (4.5)9espectively. Note that l is restricted to l > l > What has been done in detail in second-order SUSYQM can, in principle, be gen-eralized to higher order k [28, 29, 30, 31, 32]. The construction of the EOP andof the related extended potentials will then be governed by the choice of k seedsolutions φ ( x ), φ ( x ), . . . , φ k ( x ) of the initial Schr¨odinger equation and by theircorresponding Wronskian W ( φ , φ , . . . , φ k ).For k = 3, for instance, it is easy to see that the lowest-degree g µ corresponds to µ = 3, which may arise in the pure cases obtained from V (+) = V l − , φ i = φ I l − ,m i or V (+) = V l +3 , φ i = φ II l +3 ,m i , with i = 1, 2, 3, m < m < m , and µ = m + m + m − c Since g = ˜ W ( L ( α − ( − z ) , L ( α − ( − z ) , L ( α − ( − z )) = − L ( − α − ( z ) and g = ˜ W ( L ( − α − ( z ) , L ( − α − ( z ) , L ( − α − ( z )) = L ( α − ( − z ), it turns out, however,that we shall obtain the extended potentials associated with L II α, ,n ( z ) and L I α, ,n ( z ),respectively.We conclude that considering k th-order SUSYQM with k = 1, 2, 3, . . . leadsto exactly µ distinct extended radial oscillator potentials and corresponding EOPfamilies of µ th type for µ = 1, 2, and 3. Whether this result may be generalized tohigher values of µ is an interesting open conjecture.Other important points for future study are the construction of extended po-tentials of Morse or Coulomb type, also connected with Laguerre-type EOP (see,e.g., Refs. 16 and 18), as well as that of extended potentials related to Jacobi-typeEOP. A direct proof of the shape invariance of the new potentials remains to begiven. The existence of different intermediate Hamiltonians, as observed in thePSUSY approach of Sec. 3, is also worth analyzing along the lines of type A N -foldsupersymmetry [31, 33]. c Here we assume m >
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