Exceptional Laguerre and Jacobi polynomials and the corresponding potentials through Darboux-Crum Transformations

Here we construct in an elementary way the four sets of infinitely many shape invariant Hamiltonians and the corresponding exceptional (X ℓ ) polynomials [1,2]. The idea is quite simple. We start with a prepotential W ℓ (x; λ) and define a pair of factorised Hamiltonians H (±) ℓ (λ) which are intertwined by the the Darboux-Crum transformations [3,4] in terms of A ℓ (λ) and A ℓ (λ) † : A ℓ (λ)H (+) A ℓ (λ) Here λ stands for the set of parameters of the theory. The prepotential W ℓ (x; λ) is so chosen that H (+) ℓ (λ) is the well-known shape invariant [5] Hamiltonian of the radial oscillator [6,7] potential or the Darboux-Pöschl-Teller (DPT) potential [8] and H (-) ℓ (λ) is the Hamiltonian of the recently derived shape invariant potentials of Odake-Sasaki [1,2,9,10]. The essential property of the prepotential to achieve the above goal is that e ±W ℓ (x;λ) is not square integrable. We know that H (+) ℓ (λ) has a well defined groundstate. Thus these two functions e ±W ℓ (x;λ) cannot correspond to the groundstates of H ℓ (λ) are exactly iso-spectral including the groundstates: ℓ = 1, 2, . . . , n = 0, 1, 2, . . . , . By construction H (+) ℓ (λ) is shape invariant and exactly solvable. That is, the set of eigenvalues {E (+) ℓ,n (λ)} and the corresponding eigenfunctions {φ (+) ℓ,n (x; λ)} are exactly known. Throughout this paper we choose all the eigenfunctions to be real. They form a complete set of orthogonal functions: ℓ,m (x; λ)φ (+) ℓ,n (x; λ)dx = h ℓ,n (λ)δ m n , h ℓ,n (λ) > 0. (1.6) Thanks to the intertwining relations (1.2) the eigenfunctions {φ ℓ,n (x; λ), ℓ = 1, 2, . . . , n = 0, 1, 2, . . . , (1.7) and vice versa: ℓ,n (x; λ), ℓ = 1, 2, . . . , n = 0, 1, 2, . . . , . ( Of course {φ (1.9) These imply the completeness of the exceptional orthogonal polynomials and the above relationship (1.7) provides the formula relating the exceptional orthogonal polynomials to the classical orthogonal polynomials (i.e, the Laguerre or Jacobi polynomials) as shown in (2.1) and (2.3) of [10]. The orthogonality (1.9) corresponds to the integration formulas derived in §7 of [10]. These will be demonstrated in detail in subsequent sections. The above requirements lead to the following general form of the prepotential W ℓ (x; λ): W ℓ (x; λ) = w 0 (x; λ + ℓδ) + log ξ ℓ (η(x); λ), (1.10) in which w 0 (x; λ) is obtained by changing the sign of one term of the prepotential w 0 (x; λ) corresponding to the radial oscillator or the DPT potential. Here δ is the shift of the parameters. The change of the sign ensures the non-square integrability of e ±W ℓ (x;λ) . The additional term is the logarithm of the degree ℓ eigenpolynomial, the Laguerre or Jacobi polynomial with twisted parameters or arguments, introduced by Odake-Sasaki [1,2,10]. Before going to the details in the subsequent sections, let us make a few remarks about the background. This type of approach of deriving a new exactly solvable Hamiltonian from a known one in terms of Darboux-Crum [3,4] transformations has a long history and various aspects [11,12,13]. The method we are concerned in this paper is, in its essence, based on an alternative factorisation of an exactly solvable Hamiltonian (plus a constant), e.g., the radial oscillator and the DPT potential. Some refer to those newly found Hamiltonians as "conditionally exactly solvable". Junker and Roy [13] discussed an example of an alternative factorisation of the radial oscillator Hamiltonian by using the confluent hypergeometric function 1 F 1 , which could encompass the results of the L1 exceptional orthogonal polynomials [1] if the parameters and settings are properly chosen. After the introduction of the X 1 Laguerre polynomials by Gomez-Ullate et al [14] and Quesne [15,16] and the X ℓ (ℓ = 1, 2, . . .) polynomials by Odake-Sasaki [1,2], Roy and his collaborator [17] derived the X 1 and X 2 Laguerre polynomials of the L1 type in this way. This will be mentioned in a later section. A recent report by Gomez-Ullate eta al [18] has small overlap with the present work. So far as we are aware of, a "conditionally exactly solvable" treatment of the fully general (non-symmetric) DPT potential does not exist. Therefore, the present derivation of the J1 and J2 exceptional Jacobi polynomials from the classical Jacobi polynomials based on the Darboux-Crum transformation is new. The exceptional orthogonal polynomials were originally introduced in [14] by extending Bochner's theorem [19] for Sturm-Liouville problems. The characterisation of the exceptional orthogonal polynomials as polynomial solutions of Sturm-Liouville type equations under generalised Bochner problems will be discussed in section 4. We will show that the exceptional Laguerre and Jacobi polynomials have the bispectral property [20]. Here we will derive the L1 and L2 exceptional Laguerre polynomials as well as the corresponding Hamiltonians, that is the potentials. Let us start with the radial oscillator with λ = g > 0 and δ = 1: It is trivial to verify the shape invariance [5]: Its eigenvalues and eigenfunctions are φ n (x; g) = P n (η; g) e w 0 (x;g) , η ≡ η(x in which L (α) n (x) is the Laguerre polynomial satisfying the differential equation It should be stressed that the groundstate wavefunction φ 0 (x; g) = e w 0 (x;g) = e -x 2 /2 x g is square integrable and it provides the orthogonality measure of the Laguerre polynomials; ∞ 0 e 2w 0 (x;g) P m (x 2 ; g)P n (x 2 ; g)dx The radial oscillator Hamiltonian system is exactly solvable in the Heisenberg picture, too [23]. The exact annihilation/creation operators are obtained as the positive/negative energy parts of the Heisenberg operator solution (see, for example, (3.8) of [23]): x 2 4. (2.8) The action of these operators are )φ n-1 (x; g), a (+) φ n (x; g) = -(n + 1)φ n+1 (x; g). (2.9) Here we derive the L1 and L2 exceptional Laguerre polynomials. For each positive integer ℓ = 1, 2, . . ., let us consider the pair of Hamiltonians H (+) ℓ (g) and H (-) ℓ (g) corresponding to the following prepotentials (η ≡ η(x x 2 2 + (g + ℓ -1) log x + log ξ ℓ (η(x); g), g > 1/2, (2.10) L2: ) A ℓ (g) (2.15) By simple calculation using the differential equation for the Laguerre polynomial (2.6), the Hamiltonian H ℓ (g) is shown to be equal to the radial oscillator with g → g + ℓ -1 for L1 and with g → g + ℓ + 1 for L2 up to an additive constant: L2: The partner Hamiltonians are L1: L2: Up to additive constants, the above Hamiltonians (2.20) and (2.21) are equal to the Hamiltonians of the L1 and L2 exceptional orthogonal polynomials derived by Odake-Sasaki [1,10]: L1: L2: The definition of ξ ℓ (η; g) for the L1 and L2 Odake-Sasaki cases are the same as those given in (2.11) and (2.13). Let us note that e W ℓ (x;g) is not square integrable at x = ∞ and e -W ℓ (x;g) is not square integrable at x = 0 for the L1 case, whereas for the L2 case e W ℓ (x;g) is not square integrable at x = 0 and e -W ℓ (x;g) is not square integrable at x = ∞. In both cases the prepotential W ℓ (x; g) (2.10) and (2.12) are regular in the interval 0 < x < ∞. For this, it is enough to show that ξ ℓ (η(x); g) does not have a zero in 0 < x < ∞. In fact we have, as shown in (2.39) of [9]. Thus we find that e ±W ℓ (x;g) cannot be the groundstates of the Hamiltonians H (±) ℓ (g). However, we know quite well that H (+) ℓ (g), being the radial oscillator Hamiltonian, has a well-defined groundstate. This means that the partner Hamiltonian H (-) ℓ (g), thus the Hamiltonian of the L1 and L2 exceptional Laguerre polynomials, are exactly iso-spectral to the radial oscillator Hamiltonian H (+) ℓ (g), which have the following eigenvalues and the corresponding eigenfunctions: L2: (2.29) The intertwining relation (1.2) implies the simple expressions of the eigenfunctions of the partner Hamiltonians H (-) ℓ in terms of A ℓ (g): L1: (2.33) These are to be compared with the explicit expressions of the exceptional Laguerre polynomials P ℓ,n (η; g) derived in [10]: The final expressions of the eigenfunctions (2.31) and (2.33) are the same as (2.1) of [10] up to a multiplicative constant. Use is made of the identities of the Laguerre polynomials (E.11) and (E.12) of [10]. Thus we have derived the Hamiltonians as well as the eigenfunctions, that is, the L1 and L2 exceptional Laguerre polynomials and the weight functions, from those of the radial oscillator by the Darboux-Crum transformations. Here we will derive the J1 and J2 exceptional Jacobi polynomials as well as the corresponding Hamiltonians, that is the potentials. The trigonometric DPT [8] potential has two parameters λ = (g, h), g > 0, h > 0 and δ = (1, 1), It is trivial to verify the shape invariance [5]: Its eigenvalues and eigenfunctions are: (3.5) in which P (α,β) n (x) is the Jacobi polynomial satisfying the second order differential equation The groundstate wavefunction φ 0 (x; g, h) = e w 0 (x;g,h) = (sin x) g (cos x) h is square integrable and it provides the orthogonality measure of the Jacobi polynomials: The trigonometric DPT potential is also exactly solvable in the Heisenberg picture [23]. The annihilation and creation operators are (see, for example, (3.28) of [23]): in which When applied to the eigenvector φ n (3.5) as E n (g, h) + (g + h) 2 = (2n + g + h) 2 , we obtain (see, for example, (3.29) and (3.30) of [23]): (3.12) Here we derive the J1 and J2 exceptional Jacobi polynomials. As explained in [9,10], the exceptional J1 and J2 orthogonal polynomials are 'mirror images' of each other, reflecting the parity property (x) of the Jacobi polynomial. Here we present both cases in parallel so that the structure of these polynomials can be better understood by comparison. For each positive integer ℓ = 1, 2, . . ., let us consider the pair of Hamiltonians H By simple calculation using the differential equation for the Jacobi polynomial (3.6), we obtain the trigonometric DPT potential for H (+) ℓ (g, h) up to an additive constant: J1: J2: The partner Hamiltonians are J1: J2: Up to additive constants, the above Hamiltonian (3.23) and (3.24) are equal to the Hamiltonian of the J1 and J2 exceptional Jacobi polynomials derived by Odake-Sasaki [1,10]: J1: J2: The definition of ξ ℓ (η; g) for the J1 and J2 Odake-Sasaki cases are the same as those given in (3.14) and (3.16). Again let us note that e W ℓ (x;g,h) is not square integrable at x = π/2 and e -W ℓ (x;g,h) is not square integrable at x = 0 for the J1 case, whereas for the J2 case e W ℓ (x;g,h) is not square integrable at x = 0 and e -W ℓ (x;g,h) is not square integrable at x = π/2. But the prepotentials W ℓ (x; g, h) (3.13) and (3.15) are regular in the interval 0 < x < π/2. For this, it is enough to show that ξ ℓ (η(x); g, h) does not have a zero in 0 < x < π/2. In fact we have, as shown in (2.40) of [9]. Thus we find that e ±W ℓ (x;g,h) cannot be the groundstates of the Hamiltonians H (±) ℓ (g, h). However, we know well that H (+) ℓ (g, h), being the trigonometric DPT Hamiltonian, has a well-defined groundstate. This means that the partner Hamiltonian H (-) ℓ (g, h), thus the Hamiltonian of the J1 and J2 exceptional Jacobi polynomials, are exactly iso-spectral to the trigonometric DPT Hamiltonians H (+) ℓ (g, h), which have the following eigenvalues and the corresponding eigenfunctions: J2: (3.34) The intertwining relation (1.2) implies the simple expressions of the eigenfunctions of the partner Hamiltonians H (-) ℓ in terms of A ℓ (g): J1: J2: )ξ ℓ (η; g + 1, h + 1)P (3.40) These are to be compared with the explicit expressions of the exceptional Jacobi polynomials P ℓ,n (η; g, h) derived in [10]: The final expressions (3.37) and (3.40) are the same as (2.3) of [10] up to a multiplicative constant. Use is made of the identity of the Jacobi polynomials (E.22) of [10]. Thus we have derived the Hamiltonians as well as the eigenfunctions of the J1 and J2 exceptional Jacobi polynomials from those of the trigonometric DPT by the Darboux-Crum transformations. The exceptional Laguerre and Jacobi polynomials satisfy a second order linear differential equation in the entire complex η plane: For later use we give the explicit form of the second order Fuchsian differential operator H O-S ℓ which was given in (3.5) of [10]: 4 Generalised Bochner problem: bispectral property The exceptional Laguerre polynomials L1 and L2 as well as the exceptional Jacobi polynomials J1 and J2 belong to complete orthogonal families of functions (the completeness follows from the well known properties of the Darboux process; indeed, the Darboux transformation which does not generate new eigenstates preserves the completeness of transformed system of eigenfunctions as solutions of the self-adjoint Schrödinger equation). These polynomials, however, do not belong to the ordinary families of orthogonal polynomials because polynomials of the first ℓ -1 degrees are absent in these systems [14]. Hence these polynomials do not satisfy 3-term recurrence relation which is a characteristic property of nondegenerate orthogonal polynomials (see, e.g. [24]). Nevertheless, as we will show, the exceptional polynomials J1, J2, L1 and L2 do satisfy 4ℓ + 1-term recurrence relations, i.e. they can be considered as eigenvectors of a semi-infinite matrix K having 4ℓ + 1 diagonals. In this sense the considered exceptional polynomials possess a very important bispectral property [20]: they are simultaneously eigenfunctions of a Sturm-Liouville operator and a matrix K. We consider exceptional orthogonal polynomials of the J1-type (the type J2 can be considered in the same manner). For simplicity of presentation we slightly change the previous notation, denoting the exceptional J1 polynomials as Pn (x) and introducing standard parameters We also use x for the sinusoidal coordinate η and denote π(x) = ξ ℓ (x; g, h) = P (a,-b) ℓ (x) which is a polynomial of degree ℓ. This polynomial will play a crucial role in the following. Then formula (O-S2.3) for the J1 polynomials can be presented in the form with . 1 where is the normalization constant. The exceptional J1 polynomials are orthogonal on the same interval ( Pn , Pm ) with the weight function and some nonzero normalization coefficients ĥn (in fact, these coefficients can easily be connected with the coefficients h n , however we need not their explicit expressions here). Using elementary properties of the Jacobi polynomials [21], or (2.24) of [10], we also have (x), (4.9) . Using the formula (4.9) with the repeated use of the three term recurrence relation This sum in general can be infinite, because the polynomials Pn (x) form a basis in a Hilbert space with the scalar product given by the formula (4.4) but due to existence of a "gap" in degrees of polynomials Pn (x) for a generic polynomial Q(x) we should have an expansion with infinitely many coefficients ζ ns . However, for some special choices of the polynomial Q(x) this expansion can contain only a finite number of terms. In order to find the coefficients η ns , let us multiply both sides of (4.11) by ŵ(x) Ps (x) ( ŵ(x) is given by (4.5)) and integrate over the interval [-1, 1]. Due to the orthogonality relation for polynomials Pn (x), in the rhs after integration we obtain the term ĥs η ns with the nonzero coefficient ĥs in (4.4). On the other hand, in lhs we have the integral Substituting expression (4.10) into (4.12) and using the orthogonality property of the Jacobi polynomials P (a+1,b-1) n (x) we have the relation Hence only 2ℓ + 1 coefficients η ns are nonzero. We thus have the expansion where the coefficients η ns are connected with ξ ns by the "mirror" relation (4.13). Formulas (4.10) and (4.14) can be considered as a generalization for generic ℓ of corresponding formulas obtained for ℓ = 1 in [14]. Eliminating the Jacobi polynomials P (a+1,b-1) n (x) from these formulas, we arrive at the recurrence relation with some real coefficients K ns . This recurrence relation belongs to the class of 4ℓ+1-diagonal relations. This means that in the operator form the relation (4.15) can be presented as where P is an infinite dimensional vector with components Pn (x), n = 0, 1, 2, . . . , and K is a matrix with entries K ns . This matrix has no more than 4ℓ + 1 nonzero diagonals. Corresponding polynomials Pn (x) satisfy 4ℓ+1-term recurrence relation (4.15). The ordinary orthogonal polynomials satisfy 3-term recurrence relation. Hence we have polynomials Pn (x) satisfying more general recurrence relation. Recurrence relations of such a type were studied e.g. by Durán and Van Assche [25] who showed that such polynomials should satisfy a matrix orthogonality relation. Thus the exceptional Jacobi polynomials belong to the class of polynomials satisfying higher-order recurrence relations. Note however, that in the approach of [25] it is required that polynomials Pn (x) have exactly degree n = 0, 1, 2, . . . . In our case the polynomial Pn (x) has degree n + ℓ, n = 0, 1, 2, . . . . This means that the methods of [25] should be modified if applied to the case of the exceptional polynomials. Formulas (4.10) and (4.14) admit an interesting algebraic interpretation. Introduce semiinfinite matrices Ξ and H by their entries ξ ns and η ns . Then we have where From the relation (4.17) it follows that where J is a Jacobi (3-diagonal) matrix corresponding to the Jacobi polynomials P x P (x) = J P (x). Thus the matrices H and Ξ appear under factorization of the 4ℓ + 1-diagonal matrix π 2 (J). The exceptional polynomials Pn (x) satisfy recurrence relation (4.18) which appear after refactorization (permutation) of the factors H and Ξ. Such permutation of the matrix factors is known as the Darboux transformation of the matrix π 2 (J). Similar Darboux transformations were already studied in [20] where refactorization of the quadratic polynomials in J corresponding to the Jacobi and Laguerre polynomials was considered. Clearly, under such refactorization one obtains new polynomials satisfying a 5-term recurrence relation. These polynomials are not classical orthogonal polynomials. Nevertheless, the authors of [20] showed that these new polynomials are eigenfunctions of a linear fourth-order differential operator. In our case we have almost the same construction as in [20] but the resulting polynomials are exceptional (i.e. some degrees are absent) and satisfy a second-order differential equation. It is also interesting to note that the weight function ŵ(x) given by (4.5), differs from the "classical" weight function (1 -x) a+1 (1 + x) b-1 for the Jacobi polynomials by the factor π -2 (x). For the ordinary orthogonal polynomials it is known that a rational modification of the weight function w(x) → w(x)U(x) -1 with some polynomial U(x) of degree M is equivalent to the application of M Geronimus transforms [22]. If P n (x) are orthogonal polynomials corresponding to the weight function w(x) then orthogonal polynomials Pn (x) corresponding to the weight function w(x)U(x) -1 are given by the linear combination [22] Pn (x) = A (0) n P n (x) + A (1) with some coefficients A (i) n , i = 0, 1, 2, . . . M. The formula (4.20) describes a special case of the M-time Geronimus transform (general case of the Geronimus transform allows an adding of M concentrated masses located on the roots of the polynomial U(x) [22]). If we now compare formulas (4.10) and (4.20) we see that the exceptional J1 Jacobi polynomials Pn (x) are connected with the ordinary Jacobi polynomials P However, this transformation can be considered as a degeneration of the ordinary Geronimus transform in a sense that the first ℓ polynomials Pi (x), i = 0, 1, . . . , ℓ -1 become zero. The case of the J2 exceptional Jacobi polynomials can be considered in the same manner leading to a recurrence relation of the same type (4.15). Consider the case of the exceptional L1 Laguerre polynomials. Here we list some well known formulas for the ordinary Laguerre polynomials (see, e.g. [21]): Recurrence relation: Differentiation formula: Geronimus transformation: n-1 (x). We already have that ξ ns = 0 if |n -s| > ℓ. Hence η ns = 0 if |n -s| > ℓ and we have the expansion with 2ℓ + 1 terms where the coefficients η ns are connected with ξ ns by (4.31). Similarly to the case of the J1 exceptional Jacobi polynomials we obtain a 4ℓ + 1-term recurrence relation K ns Ls (x), Ls (x) = 0, if s < 0, (4.33) The case of the L2 exceptional Laguerre polynomials can be considered in a similar manner leading to a recurrence relation of the same type (4.33). One can formulate a natural conjecture that all the exceptional orthogonal polynomials (i.e. polynomials satisfying a linear second-order Sturm-Liouville equation and orthogonal to one another, see the detailed description of this problem in [14]) satisfy a recurrence relation of the type (4.15) or (4.33) with an appropriately defined polynomial π(x). It is worthwhile to stress that the function π(x) 2 plays the important role of the eigenvalue of the recurrence relation (4.15) or (4.33). In the ordinary three term recurrence relation (4.2) or (4.21), the same role is played by the (sinusoidal) coordinate x itself. Here we will show that the bispectral property of the exceptional Laguerre and Jacobi polyno- However, it is easy to see that In other words, π(x) 2 V n belongs to the degree n + 2ℓ invariant polynomial subspace of the operator H O-S ℓ , see §6 of [10]. Therefore, π(x) 2 Pn (x) and π(x) 2 Ln (x) also belong to the degree n + 3ℓ invariant polynomial subspace and they can be expressed as a linear combination of This is essentially the same argument for demonstrating the three term recurrence relation. A few remarks are in order. Dutta and Roy [17] derived the first two members of the L1 exceptional Laguerre polynomials in a way shown in section 2. In our language, they used a prepotential (5.2) But erroneously they insist that the partner Hamiltonian H (-) ℓ (g) is not shape invariant on account of the fact that its partner is the radial oscillator Hamiltonian. Let us emphasise that the shape invariance is the intrinsic property of the Hamiltonian, or the potential, determined by the unique factorisation in terms of the groundstate wavefunction. Allowing for singularities, a generic quantum mechanical Hamiltonian H has infinitely many different factorisations. For example, if then it is trivial to show the following factorisation: The right hand side as a whole is non-singular, but each factor is singular, since φ n (x) has n zeros. Usually the groundstate n = 0 provides the unique non-singular factorisation. The results of the present paper assert that the radial oscillator (DPT) potential admits two families, L1 and L2 (J1 and J2), of infinitely many non-singular factorisations. Perhaps it would be worthwhile to reflect upon the significance of the infinitely many nonsingular factorisations in historical perspective. More than three decades ago, Miller [26] embarked on the program to classify and exhaust factorisations of shape invariant Hamiltonians (H = -d 2 /dx 2 + V (x)). Although he did not employ the word 'shape invariance', the essential ingredients were there. He considers the cases in which the Hamiltonian depends on one parameter only, say, g. Based on an assumption, rephrased in our notation, that the derivative of the prepotential has a finite power dependence on g dw(x; g) dx = M j=-N α j (x)g j , N, M ∈ Z + , (5.5) he came to the conclusion that the allowed factorisation types were the same as those listed by Infeld-Hull [6]. The present cases of the radial oscillator, for example the L2 (2.12) L2: dW ℓ (x; g) dx = -x -g + ℓ x + ∂ x ξ ℓ (η; g) ξ ℓ (η; g) (5.6) are not covered by his assumption. It should be remarked that the present method provides an alternative proof of shape invariance of the Odake-Sasaki Hamiltonians [1,2,9,10]. Here we will show the shape invariance of the Hamiltonians of the exceptional Laguerre polynomials. Starting from the prepotential W ℓ (x; g), (2.10) for the L1 and (2.12) for the L2, we obtain the Hamiltonian ℓ (g) by using the factorisation in terms of the groundstate wavefunction e -x 2 /2 x g+ℓ-1 for the L1 and e -x 2 /2 x g+ℓ+1 for the L2. By the shape invariance of the radial oscillator Hamiltonian, the result is simply H (+) ℓ (g + 1) + 4. Thus it is also obtained by the shifted prepotential W ℓ (x; g + 1), which produces the partner Hamiltonian H The direct proof of shape invariance of H (-) ℓ (g) by using the groundstate wavefunction was given in [9]. Obviously the above proof of shape invariance is also valid for the Hamiltonians of the J1, J2 exceptional Jacobi polynomials, discussed in section 3. Dutta-Roy paper [17] for the ℓ = 1 and 2 cases. The three term recurrence relations for the Laguerre and Jacobi polynomials are mapped to those of the exceptional orthogonal polynomials simply by A ℓ . The explicit forms are given in [10]. This is definitely different from the bispectral property discussed in the previous section 4.