Exactly Solvable Quantum Mechanics and Infinite Families of Multi-indexed Orthogonal Polynomials

Infinite families of multi-indexed orthogonal polynomials are discovered as the solutions of exactly solvable one-dimensional quantum mechanical systems. The simplest examples, the one-indexed orthogonal polynomials, are the infinite families of the exceptional Laguerre and Jacobi polynomials of type I and II constructed by the present authors. The totality of the integer indices of the new polynomials are finite and they correspond to the degrees of the `virtual state wavefunctions' which are `deleted' by the generalisation of Crum-Adler theorem. Each polynomial has another integer n which counts the nodes.


Introduction
We will report on the discovery of infinite families of multi-indexed orthogonal polynomials, which form the main parts of the eigenfunctions of exactly solvable onedimensional quantum mechanical systems. This will add a huge number of new members to the inventory of exactly solvable quantum mechanics [1,2], which had seen rapid increase due to the recent discovery of various kinds of infinite families of exceptional orthogonal polynomials [3]- [8]. The method for deriving the multi-indexed orthogonal polynomials is simple generalisation of the wellknown method of Crum [9] and its modification by Krein-Adler [10]. Starting from an exactly solvable Hamiltonian system, Crum's method provides another exactly solvable system by successively deleting the lowest lying eigenstate. The modification due to Krein-Adler uses deletion of a finite number of higher eigenstates satisfying certain conditions. The present method is based on the same spirit of 'deleting' a finite number of virtual states instead of the eigenstates. Here the virtual state wavefunctions are the solutions of the Schrödinger equation of the original system and they have negative energies, with respect to the groundstate eigenvalue. The virtual state wavefunctions have no node in the interior and they are square-non-integrable. Their reciprocals are square-nonintegrable, too. We apply this method to the Hamiltonian systems of the radial oscillator potential and the Darboux-Pöschl-Teller potential, which are the best known examples of exactly solvable quantum mechanics [1,2]. Their eigenfunctions consist of the Laguerre and the Jacobi polynomials, respectively.
When only one virtual state of degree ℓ (≥ 1) is deleted, the resulting system is the exceptional X ℓ Laguerre or Jacobi polynomials of type I or II [3,5]. In general M distinct virtual states of degree v j (≥ 1) are deleted and the resulting eigenfunctions consist of orthogonal polynomials indexed by the integers (v 1 , . . . , v M ). The multi-indexed orthogonal polynomials are natural generalisation of the exceptional orthogonal polynomials. They start at degree ℓ ≥ 1 instead of a degree 0 constant term, thus avoiding the constraints by Bochner's theorem [11,12]. As solutions of exactly solvable quantum mechanical systems they constitute a complete system of orthogonal functions. The first examples of the exceptional Laguerre and Jacobi polynomials are constructed by Gómez-Ullate et al. [12] in the framework of Sturm-Liouville theory and by Quesne [13] within the shape invariant [14] and exactly solvable quantum mechanics. They are equivalent to the lowest members (ℓ = 1) of the X ℓ Laguerre and Jacobi polynomials derived in [3] by the present authors. The present method could be considered as the generalisation of the Darboux-Crum transformations [15] applied for the derivation of the exceptional orthogonal polynomials [16]- [20].
It is expected that these new polynomials would play important roles in physics, mathematics and related disciplines. The multi-indexed Jacobi polynomials provide infinitely many global solutions of second order Fuchsian differential equations with 3 + ℓ regular singularities [21]. Other theoretical developments are: non-linear identities underlying the shape invariance [22] are clarified, the structure of the extra zeros of the new orthogonal polynomials [23] are exemplified, two-step Darboux transformations [24] are explicitly constructed for the type II exceptional Laguerre polynomials, various use of nonphysical states [25,26] for generating the exceptional polynomials are suggested and several applications [27,28] are reported. In this connection, let us mention references of multi-step Darboux-Crum transformations, higher order susy transformations, N -fold susy transformations, etc [29]. Some of them utilise non-physical solutions.
This Letter is organised as follows. The general scheme of virtual states deletion is presented in section two. The explicit forms of the new polynomials are derived in section three, together with the forward-backward shift operators and the second order differential operators. The final section is for a summary and comments.

Adler-Crum Scheme for Virtual States Deletion
Here we present the method of virtual states deletion in its generic and formal form. For rigorous treatments of possible singularities at the boundaries, we need to specify the details of the Hamiltonian systems, which will be done in the next section. Throughout this paper we choose all the solutions of the Schrödinger equations to be real. We start with an exactly solvable one dimensional quantum mechanical system defined in an interval For simplicity, we assume that it has discrete eigenvalues only with vanishing groundstate energy: 0 = E 0 < E 1 < E 2 < · · · , which allows us to express the potential in terms of the groundstate wavefunction φ 0 (x) which has no node: Let us suppose that the Schrödinger equation has a finite or infinite number of virtual state solutions with negative energies: which have no node in the interior x 1 < x < x 2 . Here V denotes the index set of the virtual states. We also require the infinite norm conditions, and this means that the virtual state wavefunctions vanish at one boundary and diverge at the other: where j = k ∈ {1, 2} and ∀v ∈ V. The set of eigenfunctions {φ n (x)} forms a complete set in an appropriate Hilbert space H, for example: ) That is any function in H which is orthogonal to all the eigenfunctions has a zero norm: After M -steps of deleting the lowest lying eigenstates by Crum's method [9], the resulting Hamiltonian system is iso-spectral to the original one given by ,k≤n is the Wronskian. These formulas have been repeatedly rediscovered under various names, higher derivative susy, N -fold susy, etc [29]. The corresponding Adler-Crum formulas for deleting the set of M distinct eigenstates specified by The set D must satisfy the conditions [10,30]. The exactly solvable Hamiltonian system obtained by deleting M distinct virtual states D = {d 1 , . . . , d M }, d j ∈ V looks almost the same: It should be stressed that the final results after M deletions are independent of the orders of deletions. The algebraic side of the derivation of (6)-(11) is basically the same as that for the Adler-Crum case. The essential difference is the verification of the inheritance at each step of the properties of the virtual states; no-nodeness and the proper boundary behaviours. This is guaranteed by choosing the parameters of the theory g and h large enough, see (23) and (24).
Here we show the process H → H [1] . It is elementary to show Sinceφ d (x) has no node, the operatorÂ d andÂ † d are regular in the interior. Next we define H [1] by interchanging the order of these two operators: It is straightforward to show that H and H [1] are isospectral Obviously the state d is deleted from the virtual state spectrum of H [1] , forÂ dφd (x) = 0. Since the zero-modes ofÂ d andÂ † d areφ d (x) and 1/φ d (x), which have infinite norms, the sets of eigenfunctions {φ n (x)} and {φ [1] n (x)} are isospectral and in one to one correspondence. Thus the new set {φ [1] n (x)} is complete. If not, there must exist a finite norm vector Ψ in the Hilbert space which is orthogonal to all the basis: (Ψ, φ [1] n ) = 0, ∀n ∈ Z ≥0 . This would lead to a contradiction (Â † d Ψ, φ n ) = 0, ∀n ∈ Z ≥0 , sinceÂ † d Ψ = 0. Next we show the logic for demonstrating that the virtual state solutions {φ [1] v (x)} have no node in the interior. By using the Schrödinger equations for them we obtain which has no node. Since we can verify that W in the interior follows. When the virtual states d and v belong to different types (I and II) the Wronskians might not vanish at both boundaries. In that case we can show that they have the same sign at both bound- show, for all the explicit examples in the next section that φ [1] v and 1/φ [1] v have infinite norms. The steps going from H [1] → H [2] and further are essentially the same.

Multi-indexed Laguerre and Jacobi Polynomials
Here we will apply the method of virtual states deletion to two well-known systems of exactly solvable quantum mechanics with the radial oscillator and Darboux-Pöschl-Teller potentials: in which L and J stand for the names of their eigenfunctions, the Laguerre and Jacobi polynomials. For the actual parameter ranges consistent with specific deletions, see (23)- (24). The eigenfunctions are factorised into the groundstate eigenfunction and the polynomial in the sinusoidal coordinate η = η(x) [31]: in which λ stands for the parameters, g for L and (g, h) for J. Their explicit forms are .
They have two types of virtual states, which are again polynomial solutions. The total number is finite depending on the parameter values g and h, except for L1 which has infinite. The virtual states wavefunctions for L are: in which [a] ′ denotes the greatest integer less than a and δ I,II will be used later. For no-nodeness of ξ I,II v , see (2.39) of [22]. The virtual states wavefunctions for J are: The larger the parameter g and/or h become, the more the virtual states are 'created'. The L1 system is obtained from the J1 in the limit h → ∞ [5]. This explains the infinitely many virtual states of L1. Let us denote by V I,II the index sets of the virtual states of type I and II for L and J. Due to the parity property of the Jacobi polynomial P . For no-nodeness of ξ I,II v , see (2.40) of [22] and (3.2) of [6]. It is also obvious that the virtual state wavefunctionsφ v and their reciprocals 1/φ v are not square integrable in all four types. There is a third type of negative energy polynomial solutions for both L and J. For J, the degree n solution is (sin x) 1−g (cos x) 1−h P (−g+ 1 2 ,−h+ 1 2 ) n (η). Even when they have no node in the interior, their reciprocals are square integrable and they cannot be used for the virtual states deletion. The label 0 is special in that the wavefunctions Here the constant c F = 2 for L and c F = −4 for J. We will not use the label 0 states for deletion. In order to accommodate all these virtual states, the parameters g and h must be larger than certain bounds: See Figure 1 for the schematic structure of virtual states deletion. Like the original eigenfunctions (16) the n-th eigenfunction φ D,n (x; λ) ≡ φ [M,N ] n (x) after the deletion (9) can be clearly factorised into an x-dependent part, the common denominator polynomial Ξ D in η and the multi-indexed polynomial P D,n in η: in which φ 0 (x; λ) is the groundstate wavefunction (17) in which P n in (27) denotes the original polynomial, P n (η; g) for L and P n (η; g, h) for J. These polynomials depend on the order of deletions only through the sign of permutation. The multi-indexed polynomial P D,n is of degree ℓ + n and the denominator polynomial Ξ D is of degree ℓ in η, in which ℓ is given by Here the label n specifies the energy eigenvalue E n of φ D,n . Hence it also counts the nodes due to the oscillation theorem. Although they do not satisfy the three term recurrence relations, the multi-indexed polynomials {P D,n } form a complete set of orthogonal polynomials with the orthogonality relations: where the weight function of the original polynomials W (η; λ)dη = φ 0 (x; λ) 2 dx reads explicitly This is obtained by rewriting the orthogonality relations of the eigenfunctions (9) after the [M, N ] deletion.
In the rest of this section we explore various properties of the new multi-indexed polynomials {P D,n }. As for the exceptional orthogonal polynomials [3,4,5,21,20,7], the lowest degree polynomial P D,0 (η; λ) is related to the denominator polynomial Ξ D (η; λ) by the parameter shift λ → λ + δ (δ = 1 for L and δ = (1, 1) for J): The virtual state wavefunction with the index v ∈ D leading to the final [M, N ] deletion isφ It has the following form: Here the twist operator t acting on the parameters is defined as t II (g) = 1 − g for L, t I (g, h) = (g, 1 − h) and t II (g, h) = (1−g, h) for J. Note that the virtual states (20)- This has the same form as the various Hamiltonians of the exceptional orthogonal polynomials including those of the discrete quantum mechanics [3,4,5,7]. Reflecting the construction [19,6] it is shape invariant [14,3,22] This means that the operators A D (λ) and A D (λ) † relate the eigenfunctions of neighbouring degrees and parameters: in which the constants f n (λ) and b n−1 (λ) are the factors of the eigenvalue f n (λ)b n−1 (λ) = E n (λ): f n (λ) = −2 : L −2(n + g + h) : J , b n−1 (λ) = −2n : L, J.
Since all the zeros of Ξ D (η; λ) are simple, (47) is a Fuchsian differential equation for the J case. The characteristic exponents at the zeros of Ξ D (η; λ) are the same everywhere, 0 and 3. The multi-indexed polynomials {P D,n (η; λ)} provide infinitely many global solutions of the above Fuchsian equation (47) with 3 + ℓ regular singularities for the J case [21]. The L case is obtained in a confluent limit. These situations are basically the same as those of the exceptional polynomials.
Although we have restricted d I,II j ≥ 1, there is no obstruction for deletion of d I,II j = 0. In terms of the multiindexed polynomial (27), the level 0 deletions imply the following: where the multiplicative factors A and B are From (33), Ξ D behaves similarly. Therefore including the level 0 deletion corresponds to M + N − 1 virtual states deletions. This is why we have restricted d I,II j ≥ 1. These relations (48)-(49) can be used for studying the equivalence of H D . It should be stressed that the same polynomials can have different but equivalent sets of multiindices D, which defines the denominator polynomial Ξ D . Since the overall scale of Ξ D is irrelevant (see (36)) for the multi-indexed polynomial {P D,n }, we consider the equivalence class of Ξ D . For example, it is straightforward to verify the following equivalence and their duals I ↔ II: The simplest, k = 1, of (50) corresponds to the well known fact that the X 1 type I and II polynomials are identical for L and J [5]. Deleting the first k consecutive excited states, like the l.h.s. of (50) has led to many interesting phenomena in the ordinary and discrete quantum mechanics [30,33,32]. Classification of the equivalent classes leading to the same polynomials is a challenging future problem.
In a recent paper [24] Gómez-Ullate et al discussed "twostep Darboux transformations," which corresponds to the examples of D = {m II 1 , m II 2 } for L in our scheme. This paper stimulated the present work.

Summary and Comments
This is a first short report on the discovery of infinite families of multi-indexed orthogonal polynomials. They are obtained as the main part of exactly solvable quantum mechanical systems, which are deformations of the radial oscillator and the Darboux-Pöschl-Teller potentials. Although these new polynomials start at degree ℓ (≥ 1), they form a complete set of orthogonal polynomials. The exactly solvable modified Hamiltonian and the eigenfunctions are obtained by applying the modified Crum's theorem due to Adler to the original system to delete a finitely many virtual state solutions of type I and II. The simplest case of one virtual state deletion reproduces the exceptional orthogonal polynomials [12,13,3,5,21]. For the Jacobi case, the new polynomials provide infinitely many examples of global solutions of second order Fuchsian differential equations with 3 + ℓ regular singularities [21]. Like the undeformed theory, the Laguerre results can be obtained from the Jacobi results in certain confluence limits. But we presented all results in parallel, for better understanding of the structure. The parameter ranges (23) and (24) are conservative sufficient conditions with which the proper boundary behaviours and the completeness are guaranteed.
As a first report, only the basic results are presented. Some important issues cannot be included due to space restrictions, for example, an alternative proof of shape invariance, the action of various intertwining operators, etc. Many important problems are remaining to be clarified. For example, the bispectrality [19], generating functions [21], properties of the zeros of the denominator polynomial Ξ D and the extra zeros of the multi-indexed polynomials {P D,n } [23], classification of the equivalent classes leading to the same new polynomials.
The exceptional Wilson, Askey-Wilson, Racah and q-Racah polynomials were constructed in the framework of discrete quantum mechanics [4,20,7,8], and the Crum-Adler theorem for discrete quantum mechanics were also presented [33,30,32]. The method of virtual states deletion presented in this Letter is applicable to discrete quantum mechanics and it is easy to write down general formulas like as (6)- (11). This gives multi-indexed orthogonal polynomials corresponding to the deformations of the Wilson, Askey-Wilson, Racah and q-Racah polynomials.
In concrete examples, more than two types of virtual state solutions are available and much richer structures are expected. We will report on these topics elsewhere.
Before closing this Letter, let us make two small comments. The original Schrödinger equation (1)   Among the multi-indices, we can include a finite number of continuous real parameters as labeling the type I virtual solutions as described in [6,16,17]. The resulting orthogonal functions are no longer polynomials, but most properties of the multi-indexed polynomials are shared by these new orthogonal functions.