A Bochner type classification theorem for exceptional orthogonal ...

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Mar 14, 2016 - Ma ÁNGELES GARCÍA-FERRERO, DAVID GÓMEZ-ULLATE, AND ROBERT MILSON. Abstract. It was recently conjectured that every system of ...
arXiv:1603.04358v1 [math.CA] 14 Mar 2016

A BOCHNER TYPE CLASSIFICATION THEOREM FOR EXCEPTIONAL ORTHOGONAL POLYNOMIALS ´ ´ Ma ANGELES GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Abstract. It was recently conjectured that every system of exceptional orthogonal polynomials is related to classical orthogonal polynomials by a sequence of Darboux transformations. In this paper we prove this conjecture, which paves the road to a complete classification of all exceptional orthogonal polynomials. In some sense, this paper can be regarded as the extension of Bochner’s result for classical orthogonal polynomials to the exceptional class. As a supplementary result, we derive a canonical form for exceptional operators based on a bilinear formalism, and prove that every exceptional operator has trivial monodromy at all primary poles.

Contents 1. Introduction 2. Preliminaries 3. Rational Darboux transformations 4. Exceptional operators and characterization of U 5. Structure theorems for exceptional operators 6. Proof of the Theorem 7. Exceptional Orthogonal Polynomial Systems 8. Acknowledgements References

1 4 6 11 15 28 32 35 36

1. Introduction Exceptional orthogonal polynomials are complete systems of orthogonal polynomials that satisfy a Sturm-Liouville problem. They differ from the classical families of Hermite, Laguerre and Jacobi in that there are a finite number of exceptional degrees for which no polynomial eigenfunction exists. The total number of gaps in the degree sequence is the codimension of the exceptional family. As opposed to their classical counterparts [1, 2], the differential equation contains rational instead of polynomial coefficients, yet the eigenvalue problem has an infinite number of polynomial eigenfunctions that form the basis of a weighted Hilbert space. Because of the missing degrees, exceptional polynomials circumvent the strong limitations of Bochner’s classification theorem, which characterizes classical Sturm-Liouville orthogonal polynomial systems [3, 4]. The recent development of exceptional polynomial systems has received contributions both from the mathematics community working on orthogonal polynomials and special functions, 2010 Mathematics Subject Classification. 42C05, 33C45, 34M35. 1

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

and from mathematical physicists. Among the physical applications, exceptional polynomial systems appear mostly as solutions to exactly solvable quantum mechanical problems, describing both bound states [5–13] and scattering amplitudes [14–17]. But there are also connections with super-integrability [18, 19] and higher order symmetry algebras [20–22], diffusion equations and random processes [23–25], quantum information entropy [26], exact solutions to Dirac equation [27] and finite-gap potentials [28]. Some examples of exceptional polynomials were investigated back in the early 90s, [29] but their systematic study started a few years ago, where a full classification was given for codimension one, [30, 31]. Soon after that, Quesne recognised the role of Darboux transformations in the construction process and wrote the first codimension two examples, [32], and Odake & Sasaki showed families for arbitrary codimension, [10, 33]. The role of Darboux transformations was further clarified in a number of works, [11, 34, 35], and the next conceptual step involved the generation of exceptional families by multiple-step or higher order Darboux transformations, leading to exceptional families labelled by multi-indices, [36–38]. Other equivalent approaches to build exceptional polynomial systems have been developed in the physics literature, using the prepotential approach [39] or the symmetry group preserving the form of the Rayleigh-Schr¨ odinger equation [40], leading to rational extensions of the well known solvable potentials. In the mathematical literature, two main questions have centered the research activity in relation to exceptional polynomial systems: describing their mathematical properties and achieving a complete classification. Among the mathematical properties, the study of their zeros deserve particular attention. Zeros of exceptional polynomials are classified into two classes: regular zeros which lie in the interval of orthogonality and exceptional zeros, which lie outside this interval. Their interlacing, asymptotic behaviour, monotonicity as a function of parameters and electrostatic interpretation have been investigated in a number of works, [41–45], but there are still open problems in this direction. A fundamental object in the theory of orthogonal polynomials is the recurrence relation. Classical orthogonal polynomials have a three term recurrence relation, but exceptional polynomial systems have recurrence relations whose order is higher than three. There is a set of recurrence relations of order 2N + 3 where N is the number of Darboux steps [5, 46] with coefficients that are functions of x and n, and another set of recurrence relations whose coefficients are just functions of n (as in the classical case) and whose order is 2m + 3 where m is the codimension, [47–49]. While the former relations are generally of lower order and thus more convenient for an efficient computation, the latter are more amenable to a theoretical interpretation in terms of the usual theory of Jacobi matrices and bispectrality. The spectral theoretic aspects of exceptional differential operators were first addressed in [50, 51] and developed more recently in a series of papers [52–54]. The quest for a complete classification of exceptional polynomials has been fundamental problem that is now close to being solved, and the results in the present paper are a key step towards this goal. The first attempts to classify exceptional polynomial systems proceeded by increasing codimension. Codimension one systems were classified in [30] and they included just one X1 -Laguerre and one X1 -Jacobi family. The classification for codimension two was performed in [55], based on an exhaustive case-by-case enumeration of invariant flags under a given symmetry group. Due to the combinatorial growth of complexity with increasing codimension, this original approach proved to be unfeasible for the purpose of achieving a complete classification. However, a fundamental idea towards the full classification was also

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launched in [55], namely that every exceptional polynomial system can be obtained from a classical system by applying a finite number of Darboux transformations. More precisely, the following conjecture was formulated: Conjecture 1.1. [G´ omez-Ullate, Kamran, Milson 2012] Every exceptional orthogonal polynomial system of codimension m can be obtained by applying a sequence of at most m Darboux transformations to a classical orthogonal polynomial system. If the conjecture holds, then the program to classify exceptional polynomial systems becomes constructive: start from the three classical systems of Hermite, Laguerre and Jacobi and apply all possible Darboux transformations to describe the entire exceptional class. It should be stressed that only rational Darboux transformations need to be considered, i.e. those that map polynomial eigenfunctions into polynomial eigenfunctions, and this type of transformations are well understood and catalogued, and they are indexed by sequences of integers. This constructive approach has already been used to generate large classes of exceptional polynomial systems. The most general class obtained in this way can be labelled by two sets of indices or partitions (for the Laguerre and Jacobi classes) [56] or just one set (for the Hermite class) [5] which can be conveniently represented in a Maya diagram [57], a representation that takes naturally into account a number of equivalent sets of indices that lead to the same exceptional system, [58]. However, the question of whether this list contains all exceptional polynomials remained open. In all examples known so far, the weight for the exceptional system W (z) is a rational modification of a classical weight W0 (z) having the following form: (1)

W (z) =

W0 (z) , η(z)2

where η(z) is a polynomial in z (a Wronskian-like determinant) whose degree coincides with the codimension of the system. In this paper we prove that this is indeed the case for any possible exceptional polynomial system. One important point remains, namely that of ensuring that the transformed weight gives rise to a well defined spectral problem, which we shall refer to as the weight regularity problem. This means studying the sequence of Darboux transformations and the range of parameters for which: i) the weight has the right asymptotic behaviour at the endpoints ii) η(z) has no zeros inside the interval of orthogonality. The regularity problem has been solved for the exceptional Hermite class [5, 59] based on results by Krein [60], and Adler [61], and also for the Laguerre class, [62], using a remarkable correspondence between exceptional polynomials and discrete Krall type polynomials, [56]. The main result of this paper is the following theorem, which is essentially a proof of Conjecture 1.1, albeit without a bound on the number of Darboux steps. Theorem 1.2. Every exceptional orthogonal polynomial system can be obtained by applying a finite sequence of Darboux transformations to a classical orthogonal polynomial system. The essential consequence of this result is that it places on safe ground the constructive approach to the full classification described above. The strategy of the proof involves several steps.

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

First, we establish a number of factorization results for second-order order differential operators with rational coefficients. In particular, in Section 3 we show that every higherorder intertwiner can be factorized into a composition of first order operators, each of them corresponding to a one-step, rational Darboux transformation. We introduce exceptional operators in Section 4, and prove a fundamental theorem that relates the codimension to the sum of certain integer indices at the poles of the operator. Next, in section 5 we prove that every exceptional operator admits a canonical formulation as a bilinear relation between two polynomials. The key technical tools are some results on the local behaviour of solutions around the singular points of the differential equations corresponding to exceptional operators. A further key step is the demonstration that an exceptional operator has trivial monodromy at almost every point ζ ∈ C. This result was already known for the exceptional Hermite class [5,63], and we show that it can be extended to a general exceptional operator. The connection between trivial monodromy, bispectrality, Darboux transformations and the solvable character of Schr¨ odinger operators has been discussed in a number of papers (see for instance [63–68] and the references therein), and the results in this paper are one further piece of evidence of the close relationship among these concepts. In Section 6 we build on the structural properties of exceptional operators to prove the existence of a higher order intertwiner between any exceptional operator and a classical operator, extending the proof given by Oblomkov [63] for the rational extensions of the harmonic oscillator. Finally the proof of Theorem 1.2 is given in Section 7 making use of all the previous results. This section also contains Theorem 7.2 which states that the orthogonality weight for any exceptional polynomial system has the form (1).

2. Preliminaries In this preliminary section we introduce some key definitions and notation, and prove some essential results about second-order differential operators with rational coefficients. Let Q = C(z) denote the differential ring of univariate, complex-valued rational functions and P = C[z] the subring of polynomials. Let Pn ⊂ P, n ∈ N denote the vector space of polynomials of degree ≤ n, and Pn∗ ⊂ Pn the subset of polynomials whose degree is exactly equal to n. Let Diff(Q) = C(z)[Dz ] denote the ring of linear differential operators with rational coefficients and Diff(P) = C[z, Dz ] the subring of operators with polynomial coefficients. Alternatively, Diff(P) may be characterized as the subring of Diff(Q) that preserves P. When needed, will use RQ, RP, RPn to denote the corresponding real-valued subrings and subspaces, and Diff(RQ), Diff(RP) the corresponding rings of real-valued differential operators. For a sufficiently differentiable function y, we let Dzj y = y (j) (z) denote the j th derivative of y(z) with respect to z. The notation Dzz = Dz2 will also be employed. Let Diff ρ (Q) denote the set of ρth order differential operators; that is, operators of the form

(2)

L=

ρ X j=0

aj (z)Dzj ,

aj ∈ Q, aρ 6= 0,

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with action (3)

y 7→ L[y] =

ρ X

aj (z)y (j) (z),

y ∈ Q.

j=0

Definition 2.1. We say that a function φ(z) is quasi-rational if its log-derivative h i φ′ (z) Dz log φ(z) = φ(z) is a rational function of z.

For T ∈ Diff 2 (Q), write (4)

T = p(z)Dzz + q(z)Dz + r(z),

p, q, r ∈ Q

and define the quasi-rational functions (5a)

P (z) = exp

(5b)

W (z) =

(5c)

Z

z

 q(x) dx , p(x)

P (z) , p(z) R(z) = r(z)W (z).

Multiplying the eigenvalue relation T [y] = λy by W (z) gives an equivalent form as SturmLiouville type equation (6)

(P y ′ )′ + Ry = λW y.

Proposition 2.2. The operator T is formally symmetric with respect to W in the sense that Z z Z z (7) T [f ](x)g(x)W (x)dx − T [g](x)f (x)W (x)dx = P (z)(f ′ (z)g(z) − f (z)g′ (z)), where f, g are sufficiently differentiable functions. Proof. This follows by (6) and integration by parts.



Definition 2.3. We say that two rational operators T, Tˆ ∈ Diff 2 (Q) are gauge-equivalent if there exists a σ ∈ Q such that (8)

σT = Tˆσ.

We will refer to σ as the gauge-factor. Remark 2.4. Above we are using σ to denote both a rational function, and the multiplication operator y 7→ σy. The reason for the gauge-factor terminology is that the eigenvalue relation T [y] = λy is equivalent to the eigenvalue relation Tˆ[ˆ y ] = λˆ y , with yˆ = σy. Proposition 2.5. Let T, Tˆ ∈ Diff 2 (Q) be gauge equivalent by a gauge factor σ. Letting ˆ the corresponding weights p, q, r, pˆ, qˆ, rˆ be the coefficients of T and Tˆ as per (4), and W, W

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(5b) we have the following transformation laws p = pˆ 2σ ′ pˆ σ σ ′′ σ′ pˆ r = rˆ + qˆ + σ σ ˆ. W = σ2W q = qˆ +

(9)

3. Rational Darboux transformations The gauge-equivalence relation (8) is an intertwining relation of second-order operators by a zero-order multiplication operator. Consideration of higher-order intertwining relations leads naturally to the notion of a Darboux transformation. Definition 3.1. For T ∈ Diff 2 (Q) a rational factorization is a relation of the form (10)

T = BA + λ,

where A, B ∈ Diff 1 (Q) and λ ∈ C is a constant. Given a rational factorization, we call the operator Tˆ ∈ Diff 2 (Q) defined by (11) Tˆ := AB + λ the partner operator and say that T → 7 Tˆ is a rational Darboux transformation. Proposition 3.2. Suppose that T, Tˆ ∈ Diff 2 (Q) are related by a rational Darboux transformation. Then, the following intertwining relations hold (12) AT = TˆA, T B = B Tˆ. Proof. This is a direct consequence of (10) and (11).



Remark 3.3. The intertwining relation (12) implies that the eigenvalue relation T [y] = λy is formally equivalent to the eigenvalue relation Tˆ[ˆ y ] = λˆ y where yˆ = A[y]. Definition 3.4. For T ∈ Diff 2 (Q) and φ(z) quasi-rational, we will say that φ is a quasirational eigenfunction of T if (13)

T [φ] = λφ,

λ ∈ C.

Proposition 3.5. For T ∈ Diff 2 (Q), let φ(z) be a quasi-rational eigenfunction of T with eigenvalue λ, and let b(z) be an arbitrary, non-zero rational function. Define rational functions φ′ ˆb = p , , φ b and first order operators A, B ∈ Diff 1 (Q) by

(14)

(15)

w=

A = b(z)(Dz − w(z)),

w ˆ = −w −

q b′ + , p b

B = ˆb(z)(Dz − w(z)). ˆ

With A, B as above, the rational factorization relation (10) holds. Moreover, w is a solution of the Ricatti equation (16)

p(w′ + w2 ) + qw + r = λ.

Conversely, given a rational factorization (10), there exists a quasi-rational eigenfunction φ(z) with eigenvalue λ and a rational b(z) such that (14), (15), and (16) hold.

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Proof. By (13) we have pφ′′ qφ′ + + r = λ. φ φ The Ricatti relation (16) follows immediately. Applying (14), (15), and (16) we have (BA)[y] = B[by ′ − bwy] = ˆbby ′′ + (ˆbb′ − ˆbbw − bˆbw)y ˆ ′ + (wwb ˆ ˆb − ˆb(bw)′ )y  ′       pb q b′ q b′ b′ ′′ ′ ′ = py + +p − y + pw −w − + − pw − pw y b p b p b b ′′ ′ = py + qy + (r − λ)y. We now prove the converse. Suppose that (10) holds. Let b(z), w(z), ˆb(z), w(z) ˆ be rational functions dictated by the form (15). Define the quasi-rational function  Z z w(x)dx φ(z) = exp so that w = φ′ /φ. Then, (13) follows from (10). Expanding (BA)[y], as above shows that p = ˆbb, q = ˆbb′ − ˆbb(w + w), ˆ r − λ = wwb ˆ ˆb − ˆb(bw)′ .

From this (14) and (16) follow immediately.



Proposition 3.6. Suppose that T, Tˆ ∈ Diff 2 (Q) are related by a rational Darboux transformation. Then, T, Tˆ have the same second-order coefficients, while first- and zero-order ˆ (z), as defined by (5b), are coefficients q, qˆ, r, rˆ ∈ Q, and the quasi-rational weights W (z), W related by (17)

qˆ = q + p′ −

2b′ p, b

!  ′ 2 ′′  b′ b b rˆ = r + q + wp − (18) + 2w′ p q + p′ + 2 − b b b ˆ = p W, (19) W b2 where p(z) is the second-order coefficient of both T, Tˆ, and where b(z), w(z) ˆ are the rational functions defined in Proposition 3.5. ′



Proof. By (10)- (15), p(z) is the second-order coefficient of both T, Tˆ. Let qˆ(z) ∈ Q be the first-order coefficient of Tˆ. Relation (17) follows by (14) (15) and (11). Applying (5) and using (17) gives (19). Considering the hatted dual of (16) and applying (14) and (17) gives rˆ = λ − p(w ˆ′ + w ˆ2 ) − qˆw, ˆ     !    q b′ ′ q b′ 2 2pb′ q b′ ′ =λ−p −w − + + −w − + − q+p − −w − + . p b p b b p b The above simplifies to the expression shown in (18).



Next, we consider iterated rational Darboux transformations. In the context of Schr¨ odinger operators, these are known as higher-order Darboux or Darboux-Crum transformations. [69].

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

Definition 3.7. Let Tˆ, T ∈ Diff 2 (Q) be second-order operators with rational coefficients. We will say that Tˆ is Darboux connected to T if there exists an L ∈ Diff(Q) such that (20) TˆL = LT. Remark 3.8. Note that gauge-equivalent operators (8) are Darboux connected by definition, because they are related by a zeroth order intertwining relation. Definition 3.9. We will say that Tˆ, T ∈ Diff 2 (Q) are connected by a factorization chain if there exist second-order operators Ti ∈ Diff(Q), i = 0, 1, . . . , n with T0 = T and Tn = Tˆ; first-order operators Ai , Bi ∈ Diff 1 (Q), i = 0, 1, . . . , n − 1, and constants λi such that (21)

Ti = Bi Ai + λi ,

(22)

Ti+1 = Ai Bi + λi .

i = 0, 1, . . . , n − 1

It is trivial to show that two operators connected by a factorization chain are also Darboux connected. However, much less known is the fact that the converse is also true, as shown by the following theorem. Theorem 3.10. Two rational operators T, Tˆ ∈ Diff 2 (Q) are Darboux connected if and only if they are either gauge-equivalent, or they are connected by a factorization chain. Proof. Suppose that T and Tˆ are connected by a factorization chain. By assumption, Ti+1 Ai = Ai Bi Ai + λi Ai = Ai Ti ,

i = 0, 1, . . . , n − 1.

It follows by induction that Ti+1 Ai · · · A0 = Ai · · · A0 T0 . Therefore, (20) is satisfied with L = An−1 · · · A1 · A0 . The proof of the converse is a modification of an argument given in [63]. If ord L = 0, then T and Tˆ are gauge-equivalent. Thus, suppose that (20) holds and that ord L ≥ 1. Claim 1: no generality is lost if we assume that L does not have a right factor of the form T − λ. Indeed, suppose that ˜ − λ), λ ∈ C. L = L(T Since T commutes with T − λ, it follows that ˜ = LT ˜ Tˆ L is a lower order intertwining relation between Tˆ and T . Repeating this argument a finite number of times yields an intertwiner L with the desired property. Claim 2: T leaves ker L invariant. By relation (20), if y ∈ ker L, then     L T [y] = Tˆ L[y] = 0,

so T [y] ∈ ker L also. Claim 3: if T [y] = λy, then (23)

L[y] = F (z, λ)y + G(z, λ)y ′ ,

where F, G are polynomial in λ and rational in z. By assumption, T [y] = p(z)y ′′ + q(z)y ′ + r(z)y = λy

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where p(z), q(z), r(z) are rational in z. We have thus that y ′′ = −

q(z) ′ λ − r(z) y + y, p(z) p(z)

and hence a higher order derivative y (k) , k ≥ 2 can always be written as a linear combination of y and y ′ with coefficients that are polynomial in λ and rational in z. Since ker L is finite-dimensional and invariant with respect to T , let us choose an eigenvector φ ∈ ker L of T with eigenvalue λ0 . It follows that F (z, λ0 )φ + G(z, λ0 )φ′ = 0, with F, G defined above. Claim 4: G(z, λ0 ) is not identically zero. If ord L = 1 the claim is trivial. For ord L ≥ 2 we argue by contradiction and suppose that G(z, λ0 ) ≡ 0. Then, F (z, λ0 ) ≡ 0 also. Let ψ(z) be a second, independent solution of T [ψ] = λ0 ψ. By (23), L[ψ] = 0 also. Let ρ be the order of L. Choose polynomials π1 , . . . , πρ−1 so that π1 , . . . , πρ−1 , φ, ψ are linearly independent. Set fi = (T − λ0 )[πi ], Suppose that k1 , . . . , kρ−1

gi = L[πi ],

i = 1, . . . , ρ − 1. P ∈ C are constants such that i ki fi = 0. Then, # " X ki πi = 0. (T − λ0 ) i

Observe P that dim ker(T − λ0 ) = 2 because T is second-order. Hence {φ, ψ} span ker(T − λ0 ), and i ki πi must be a linear combination of φ and ψ. Since π1 , . . . , πρ−1 , φ, ψ are linearly independent, it follows that ki = 0. Hence, f1 , . . . , fρ−1 are linearly independent, and there exist rational functions a0 (z), a1 (z), . . . , aρ−2 (z) that satisfy the linear relations ρ−2 X

(j)

aj (z)fi (z) = gi (z),

i = 1, . . . , ρ − 1.

j=0

Define the operator ˜= L

ρ−2 X

aj (z)Dzj ,

j=0

and observe that by construction

˜ − λ0 )[πi ] = L[πi ], L(T ˜ − λ0 )[φ] = 0, L(T ˜ − λ0 )[ψ] = 0. L(T

i = 1, . . . , ρ − 1,

˜ − λ0 ) are differential operators of order ρ whose action on ρ + 1 We see thus that L and L(T ˜ −λ0 ), which violates the reducibility linearly independent functions coincides. Hence, L = L(T assumption established by Claim 1, and thereby establishes Claim 4.

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

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Thus, G(z, λ0 ) is not identically zero, and therefore, φ′ (z) F (z, λ0 ) =− φ(z) G(z, λ0 ) is a rational function. Set T0 = T and L0 = L. By Proposition 3.5, there exists a rational factorization T = B0 A0 + λ0 with A0 [φ] = 0. Since φ ∈ ker L we also have a rational factorization L = L1 A0 ,

L1 ∈ Diff(Q).

Setting T1 = A0 B0 + λ0 we have

(TˆL1 − L1 T1 )A0 = 0

which implies that TˆL1 = L1 T1 . Claim 5: L1 has no right factors of the form T1 − λ. Suppose otherwise, so that ˜ 1 − λ). L1 = L(T ˜ = λ − λ0 we have Then, setting λ ˜ 0 = LA ˜ = LA ˜ 0 B0 − λ)A ˜ 0 (B0 A0 − λ) ˜ 0 (T − λ), L = L1 A0 = L(A which again violates the irreducibility assumption of Claim 1. Continuing by induction, we have TˆLi = Li Ti , i = 0, 1, . . . with Li reduced. Repeating the above argument, we construct rational factorizations Ti = Bi Ai + λi ,

Ti+1 = Ai Bi + λi ,

so that L = Li+1 Ai · · · A0 and

TˆLi+1 = Li+1 Ti+1 , and so that Li+1 is reduced as per Claim 1. This process terminates when Li is a firstorder operator, because then we can take Li = Ai , which gives Tˆ = Ti+1 , and completes the factorization chain that connects Tˆ and T .  Corollary 3.11. The property of being Darboux connected is an equivalence relation on Diff 2 (Q). Proof. Reflexivity of the relation is self-evident. We need to prove that the Darboux connected relation possesses both symmetry and transitivity. Suppose (20) holds. If L = µ is zero-order then, T µ−1 = µ−1 Tˆ, so that T is Darboux connected to Tˆ. If ord L ≥ 1 then Tˆ and T are related by a factorization chain. By inspection of the definition, the property of being connected by a factorization

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chain is symmetric; one simply switches the Ai and the Bi and reverses the order of the factorization chain. Next suppose that T1 L1 = L1 T2 , T2 L2 = L2 T3 , where T1 , T2 , T3 ∈ Diff 2 (Q) and L1 , L2 ∈ Diff(Q). Then, by associativity of operator composition, T1 L1 L2 = L1 T2 L2 = L1 L2 T3 , so that T1 is Darboux connected to T3 .  4. Exceptional operators and characterization of U Definition 4.1. A second-order operator T ∈ Diff 2 (Q) is exceptional if T has a polynomial eigenfunction for all but finitely many degrees. To be more precise, there exists a finite set of natural numbers {k1 , . . . , km } ⊂ N such that for all k ∈ / {k1 , . . . , km }, there exists a yk ∈ Pk∗ and a λk ∈ C such that T [yk ] = λk yk , and such that no such polynomial exists if k ∈ {k1 , . . . , km }. We will refer to k1 , . . . , km as the exceptional degrees. Remark 4.2. Note that in the above definition of an exceptional differential operator, m could be zero, i.e. exceptional operators include classical operators as a special case. In the recent literature on this subject, the adjective exceptional is usually reserved for the case m > 0 to differentiate them from the classical ones. However, for the purpose of this paper it is convenient to handle the general class. Thus, in order not to introduce further notation, we will stick to the term exceptional in this wider context, hoping that no confusion will arise. Remark 4.3. No generality is lost by assuming that an exceptional operator has rational coefficients. See Proposition 2.1 of [70] for more details. Observe that if T is an exceptional operator with eigenpolynomials yk ∈ Pk∗ and σ ∈ Pn∗ , n ≥ 1 a non-trivial polynomial, then the gauge-equivalent operator T˜ = σT σ −1 is also exceptional with eigenpolynomials y˜k+n = σyk . However, every equivalence class of gaugeequivalent exceptional operators admits a distinguished gauge, as per the following. Definition 4.4. We will say that an exceptional operator is in reduced gauge, if the corresponding eigenpolynomials do not all share a common root. For sake of brevity we will sometimes shorten the expression “an operator in reduced gauge” to the simpler “reduced operator”. Proposition 4.5. Every equivalence class of gauge-equivalent exceptional operators contains a reduced operator, which is unique up to a choice of scalar multiple. Proof. Suppose that T˜ ∈ Diff 2 (Q) is exceptional with eigenpolynomials y˜k ∈ Pk∗ . Let σ ∈ P be a maximal common polynomial factor of the y˜k . Then, the operator T = σ −1 T˜σ, ∗ admits polynomial eigenfunctions σ −1 y˜k ∈ Pk−deg σ which, by construction, do not possess a common root. 

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

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Example 4.6. Unreduced operators are, for all practical purposes, equivalent to their reduced counterpart. For example, consider the classical Hermite differential equation y ′′ − 2zy ′ + 2ny = 0,

n = 0, 1, 2, . . .

whose polynomial solutions are the classical Hermite polynomials y = Hn (z). One could ˆ n (z) = (1 + z 2 )Hn−2 (z), n ≥ 2. By construction, y = H ˆn instead consider the polynomials H is a solution of the differential equation     2z 8 2 ′ y ′′ − 2 z + − y + 4 + 2n + y = 0, 1 + z2 1 + z 2 (1 + z 2 )2 which is obtained by conjugating the classical Hermite operator by the multiplication operator 1 + z 2 . The ordinary hermite polynomials are orthogonal on (−∞, ∞) relative to the weight 2 ˆ n (z) are orthogonal relative to e−z , and hence by construction the modified polynomials H 2 ˆ n , n ≥ 2 constitute a family of exceptional orthogonal the weight e−z /(1 + z 2 )2 . Thus, H polynomials with 2 missing degrees. This type of construction is quite general, but does not produce genuinely new orthogonal polynomials. Definition 4.7. For an exceptional operator T ∈ Diff 2 (Q), let U ⊂ P denote the maximal invariant polynomial subspace, and ν the codimension of U in P. Proposition 4.8. An equivalent characterization of U is (24)

U = {y ∈ P : T j [y] ∈ P for all j ∈ N}.

Proof. Let U ′ denote the subspace defined by the right side of (24). For all y ∈ U ′ we have T [y] ∈ U ′ by definition. Hence, U ′ is T -invariant, and hence U ′ ⊂ U . On the other hand, if y ∈ U then T j [y] ∈ U ⊂ P for all j ∈ N. Therefore, U ⊂ U ′ , also.  Definition 4.9. Let T be an exceptional operator. We say that T is polynomially semi-simple if the action of T on every finite-dimensional, invariant polynomial subspace is diagonalizable. We will say that T is polynomially regular if there exists a positive-definite inner product on P relative to which the action of T is symmetric. Remark 4.10. By (24), U contains all eigenpolynomials of T , which means that ν ≤ m < ∞, where m is the number of exceptional degrees as per Definition 4.1. If T is also polynomially semi-simple, then U may be characterized as the span of the eigenpolynomials of T , in which case ν = m. However, in general U may contain polynomials that are not eigenvectors of T , in which case ν < m strictly. The polynomial semi-simplicity condition has not been considered previously in the literature. Rather in the context of orthogonal polynomial systems (see Section 7 for more details), the usual assumption is that T is related to a Sturm-Liouville operator with polynomial eigenfunctions, which under suitable assumptions (see Section 7) implies that T is polynomially regular. By the finite-dimensional Spectral Theorem, if T is polynomially regular, as per Definition 4.9, then the T -action on invariant, finite-dimensional, polynomial subspaces is diagonalizable. In other words, regularity implies semi-simplicity. To illustrate the above remark, consider the following example. Example 4.11. The operator T [y] = (1 − z 2 )y ′′ + 2(z − 2)y ′

13

is the α = 0, β = −4 instance of the classical Jacobi operator. This instance is degenerate, because the leading coefficient of the classical Jacobi polynomials is   α + β + 2n −n n α,β Pn (z) = 2 z + lower degree terms. n

Indeed, with the above choice of the α, β parameters, the third-degree Jacobi polynomial P3α,β degenerates to a constant. The constant y = 1 is an eigenfunction, but observe that T [z 3 + 6z 2 + 21z] = −72.

Hence, the vector space spanned by z 3 + 6z 2 + 21z and 1 is T -invariant, but the action is not diagonalizable. However, the Jacobi polynomials of all other degrees are eigenfunctions, so T does fit the definition of an exceptional operator. Regularity for Jacobi polynomials requires that α, β > −1. Since our example violates this assumption, there is no well-defined inner product. This lack of an inner-product permits an operator with an action that is not semi-simple. For the remainder of this section, T ∈ Diff 2 (Q) refers to an arbitrary exceptional operator with coefficients p, q, r ∈ Q as per (4). By a pole of T we mean a pole of one of these coefficients. Definition 4.12. For a meromorphic function f (z) and ζ ∈ C, we define ordζ f to be the degree of the leading term in the Laurent expansion of f ; i.e. k = ordζ f is the largest integer such that (z − ζ)−k f (z) is bounded near z = ζ. For example, ordζ f = 1 if f (z) is analytic with a simple root at z = ζ and ordζ f = −2 if z = ζ is a double pole of f (z). Definition 4.13. Let T ∈ Diff 2 (Q) be an exceptional operator and U ⊂ P its maximally invariant polynomial subspace. For a given ζ ∈ C, we define the order sequence of T at ζ as (25)

Iζ = {ordζ y : y ∈ U }.

Let νζ be the cardinality of N\Iζ ; that is, the number of gaps in the order sequence. Remark 4.14. An equivalent definition of the order sequence is that there exists a basis of U with polynomials (26)

yk (z) = (z − ζ)k + higher order terms,

k ∈ Iζ .

We are now ready to state our first main result. Theorem 4.15. Let ζ1 , . . . , ζN ∈ C be the poles of an exceptional operator T , and let νi = νζi , i = 1, . . . , N be the cardinalities of the order gaps as per the above definition. Then, each νi is finite, and moreover the codimension of the maximal invariant polynomial subspace U is given by (27)

ν=

N X

νi .

i=1

The proof is based on a number of preliminary results, some of which will also be needed in subsequent sections. Lemma 4.16. For every ζ ∈ C, we have νζ ≤ ν.

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

Proof. By definition, it is not possible to choose more than ν linearly independent polynomials that are not in U . The polynomials {(z − ζ)n : n ∈ / Iζ } are linearly independent and not in U . Therefore, there are at most ν of them.  Definition 4.17. A differential functional of order k with support at ζ ∈ C is a linear map α : P → C of the form k X aj y (j) (ζ), aj ∈ C, ak 6= 0. α[y] = j=0

For k ∈ N and ζ ∈ C, define the linear functionals (k)

: y 7→ y (k) (ζ), y ∈ P.

δζ (0)

(k)

It is not hard to show that δζ , . . . , δζ true.

are linearly independent. Moreover, the following is

Proposition 4.18. A differential functional supported at ζ ∈ C cannot be given as a finite linear combination of differential functionals with support at other points. Proof. For i = 1, . . . , m, let αi : P → U be differential functionals of order ki with support at ζi ∈ C. Suppose that α is a differential functional of order k supported at ζ ∈ / {ζ1 , . . . , ζm }. Consider a polynomial of the form m Y y(z) = b(z) (z − ζi )ki +1 , b ∈ P, i=1

By construction,

αi [y] = 0, i = 1, . . . , m. It suffices then to show that α[y] 6= 0 for some choice of b. Suppose not and observe that  ker α = (ker α ∩ Pk ) ⊕ (z − ζ)k+1 P .

By the rank-nullity theorem, dim(ker α ∩ Pk ) = k. Since dim Pk = k + 1, there exists a b ∈ Pk such that the corresponding y(z) is divisible by (z − ζ)k+1 , which is impossible.  (k)

Lemma 4.19. If νζ > 0, then there exist νζ functionals αζ , with support at ζ and order k ∈ / Iζ that vanish on U . Moreover, if α is a differential functional with support at ζ that (k) vanishes on U , then necessarily it is a linear combination of these αζ , k ∈ / Iζ . Proof. Let n = 1 + max(N \ Iζ ). By (26), there exists a basis of U of the form X  (z − ζ)k (z − ζ)j (j) − ajk + O (z − ζ)n , j < n, j ∈ Iζ yζ (z) = j! k! k ∈I / ζ ,k>j

 (j) yζ (z) = (z − ζ)j + O (z − ζ)j+1 ,

j ≥ n.

For k ∈ / Iζ , set (28)

(k)

αζ [y] = y (k) (ζ) +

X

ajk y (j) (ζ).

j 0, then necessarily ζ is a pole of T . By Lemma 4.19 every αζi combination

(k) of the differential functionals αζi , k (k) only if it is annihilated by every αζi ,

is a linear

∈ / Iζi defined in Lemma 4.19. Therefore

i = 1, . . . , N, k ∈ / Iζi . These are linearly y ∈ U if and independent by Proposition 4.18, which establishes (27).  5. Structure theorems for exceptional operators Definition 5.1. Let T ∈ Diff 2 (Q) be a second-order rational operator, T = p(z)Dzz + q(z)Dz + r(z). We say that T admits a bilinear formulation, if p ∈ P2 and there exist polynomials s ∈ P1 and η, µ ∈ P such that the eigenvalue equation T [y] = λy is equivalent to the relation  1 (30) p(ηy ′′ − 2η ′ y ′ + η ′′ y) + p′ (ηy ′ + η ′ y) + s(ηy ′ − η ′ y) + 2p(log µ)′′ + p′ (log µ)′ ηy = λyη, 2 which, by inspection, is bilinear in y and η. Definition 5.2. A second order operator T is in the natural gauge, or more simply, T is a natural operator if it admits a bilinear formulation of the type (30) where µ is a constant, that is if the bilinear relation takes the particularly simple form 1 (31) p(ηy ′′ − 2η ′ y ′ + η ′′ y) + p′ (ηy ′ + η ′ y) + s(ηy ′ − η ′ y) = λyη. 2

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

16

Remark 5.3. Dividing (31) by η gives the form for the coefficients (4) of a natural operator, namely 2pη ′ p′ +s− 2 η  ′  ′ p η pη ′′ + −s . r= η 2 η q=

(32)

We are now ready to state the main result of this section, which is a structure theorem for the coefficients of an exceptional operator. Theorem 5.4. Every exceptional operator is gauge equivalent to a natural operator. Remark 5.5. Taken Theorem 5.4 as proven, we see that every class of gauge-equivalent exceptional operators has two distinguished gauges: the reduced gauge (Defintion 4.4) and the natural gauge (Defintion 5.2). Usually, these two choices of gauge are the same, but this is not always the case. Example 5.6 below illustrates one such instance where the natural and reduced gauges differ. It is probable that the discrepancy between the natural and reduced gauges is related to the confluence of zeros in the denominator polynomial η(z) of the exceptional weight [71, 72]. Example 5.6. The following example illustrates the difference between the natural and reduced gauge of an exceptional operator. The example is based on the following family of (α) two-step exceptional Laguerre polynomials [36]. Let Ln (z) denote the classical Laguerre polynomial of degree n. For n ≥ 2 set   (α) (α) (α) −z ˆ (α) L Wr Ln−2 (z), L1 (z), ez L2 (−z) . n (z) := e

(33)

ˆ (α) (z) = 0, and so we obtain a codimension-3 family of polynomials with By construction, L 3 degrees n = 2, 4, 5, 6, . . .. These polynomials can also be given using the following form introduced by Dur´ an [56]

(34)

(α)

(α) (α+1) (α+2) L (z) −L (z) L (z) n−2 n−3 n−4 (α) (α+1) ˆ (α) L , −L0 (z) 0 n (z) = L1 (z) (α) L (−z) L(α+1) (−z) L(α+2) (−z) 2 2 2

n = 2, 4, 5, 6, . . .

where Lj (z) is understood to be zero for j < 0. Let

 (α)  (α) η (α) (z) = e−z Wr L1 (z), ez L2 (−z) (α) −1 L1 (z) = (α) (α+1) L2 (−z) L2 (−z)  1 = − z 3 + (α + 4)z 2 − (α + 4)(α + 1)z − (α + 1)(α + 2)(α + 4) . 2

17 (α)

ˆ n (z), n = 2, 4, 5, . . . is exceptional and in the natural gauge, because The polynomial family L of the following bilinear relations:   1  (α) ˆ (α) ′ (α)′ ˆ (α) ′′ (α)′ ˆ (α) ′ (α)′′ ˆ (α) ˆ (α) (35) η L + η L + z η (α) L − 2η L + η L n n n n n 2    5 ˆ (α) η (α) = 0 ˆ (α) ′ − η (α)′ L ˆ (α) + (n − 3)L + −z + α + η (α) L n n n 2

It is easy to check that η (α) (z) 6= 0 for z ∈ [0, ∞) if and only if α ∈ (−∞, −4) ∪ (−2, −1). ˆ (α) Hence, for α ∈ (−2, −1) the polynomials L n (z) are orthogonal with respect to the inner product Z ∞ α+2 −z z e hf, gi = 2 f (z)g(z)dz. (α) 0 η (z) The discriminant of η (α) (z) is 18 (α+1)(α+4)2 (4α+7)2 . Hence, for α = − 74 the denominator polynomial has a multiple root. Indeed,   7 1 3 3 η (− 4 ) (z) = − ; z+ 2 4 there is a single root with a triple multiplicity. Moreover,    1 3 1 (− 7 ) L2 4 (−z) = z+ z− 2 4 4   7 3 (− ) . L1 4 (z) = − z + 4 Hence, ˆ (n L

− 74

)

3



(− 74 )



L (z) 1  1 3 Wr  n−2 3 , 1, ez z − 4 2 4 z+4          1 1 1 3 3 ( 41 ) 3 2 15 3 15 (− 3 ) (− 7 ) =− Ln−4 (z) − z+ z+ z+ Ln−34 (z) − z+ z+ Ln−24 (z) 2 4 2 4 4 2 4 4 

(z) = −e−z z +





has a root at z = − 34 for every n. Thus, for α = − 47 the natural gauge does not agree with reduced gauge. Let us therefore introduce the reduced family of polynomials   3 −1 ˆ (− 47 ) ˜ Ln+1 (z), n = 1, 3, 4, . . . . Ln (z) = z + 4 This family of polynomials is exceptional and reduced. The reduced inner product is Z ∞ 1 −z z4e hf, gi = 4 f (z)g(z)dz. 0 z + 43

To obtain the corresponding differential equation we conjugate (35) by z + 34 . Applying the transformation law described in Lemma 5.7, we obtain the differential equation   ˜′ ˜ 5 ′′ ˜ ′n + (n − 1)L ˜ n − 4z Ln + Ln = 0. ˜ −z L z Ln + 4 z + 34

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

In this way we recover the codimension 2 exceptional family first described in [36, Section 6.2.5]. This example also serves as an illustration of the principle that codimension very much depends on the choice of gauge. The generic family described above has codimension 3. However, for one particular value of the parameter, the “true” codimension, that is the codimension of the corresponding reduced family, is actually 2. The rest of this section is devoted to the proof of Theorem 5.4. We begin with some supplementary results. Lemma 5.7. Operator form (30) is gauge-equivalent to the natural form (31). Moreover, if the operator defined by (30) is exceptional, then so is the gauge-equivalent natural operator. Proof. Suppose that T admits a bilinear formulation (30). Set ηˆ = µη,

yˆ = µy,

Tˆ = µT µ−1 .

Observe that 2pµ′′ − 2p 2p(log µ) + p (log µ) = µ ′′







µ′ µ

2

+

p′ µ′ µ

and that ηˆ′ = µη ′ + µ′ η, ηˆ′′ = µη ′′ + 2µ′ η ′ + µ′′ η, with analogous relations for yˆ′ , yˆ′′ . Rewrite (31) but with yˆ, ηˆ in place of y, η, namely (36)

1 η yˆ′ + ηˆ′ yˆ) + s(ˆ η yˆ′ − ηˆ′ yˆ) = λˆ y ηˆ. p(ˆ η yˆ′′ − 2ˆ η ′ yˆ′ + ηˆ′′ yˆ) + p′ (ˆ 2

Substituting the preceding definitions into (36) and dividing by µ2 yields (30), and thereby establishes that Tˆ is a natural operator. Finally, by construction, if yk ∈ Pk∗ are eigenpolynomials of T , then yˆk+deg µ = µyk are eigenpolynomials of Tˆ.  Definition 5.8. Given a T ∈ Diff 2 (Q) , we define its Laurent decomposition at a given ζ ∈ C to be the sum X Tj , (37) T = j≥dζ

where the Tj are degree-homogeneous operators that map (z − ζ)k to a constant multiple of (z − ζ)j+k , where (38)

dζ = min{ordζ p − 2, ordζ q − 1, ordζ r},

and where p, q, r are the operator coefficients as per (4). Explicitly, (39)

Tj = pj+2 (z − ζ)j+2 Dzz + qj+1 (z − ζ)j+1 Dz + rj (z − ζ)j ,

19

where p(z) =

X

pj (z − ζ)j ,

pj ∈ C,

qj (z − ζ)j ,

qj ∈ C,

rj (z − ζ)j ,

rj ∈ C

j≥ordζ p

q(z) =

X

j≥ordζ q

r(z) =

X

j≥ordζ r

are the Laurent decompositions of p, q, r, respectively. We will refer to dζ as the leading order of the Laurent decomposition of T at ζ. Lemma 5.9. Given an exceptional T , the leading term, Tdζ , of its Laurent decomposition at a given ζ ∈ C preserves the space span{(z − ζ)k : k ∈ Iζ }. Proof. By Remark 4.14, there exists a basis of U of the form (26). Since U is T invariant and Tdζ is the smallest order term of T , the desired conclusion follows.  Lemma 5.10. Let T be an exceptional operator with leading order dζ < 0 at a given ζ ∈ C. Then, for every natural number j ∈ / Iζ , there exists a natural number nj > 0 such that i) j, j − dζ , . . . , j − (nj − 1)dζ ∈ / Iζ ; ii) j − dζ nj ∈ Iζ and Tdζ [(z − ζ)j−dζ nj ] = 0.   Proof. If j ∈ / Iζ then by Lemma 5.9 either j − dζ ∈ / Iζ or Tdζ (z − ζ)j−dζ = 0. Iterating this argument, and using Lemma 4.16 and the fact that the codimension is finite, we see that the first possibility can happen only a finite number of times.  Lemma 5.11. Given an exceptional operator T , for every ζ ∈ C we have dζ ≥ −2. Proof. Suppose that dζ < −2. For each j ∈ {0, 1, 2}, if j ∈ Iζ then Tdζ [(z − ζ)j ] = 0. If j ∈ / Iζ ,   then by Lemma 5.10 there exists an integer nj > 0 such that Tdζ (z − ζ)j−nj dζ = 0. In all cases, we see that Tdζ would be required to annihilate (z − ζ)k for three different integers k, and since it is a second order operator, this is impossible.  We are now able to derive some necessary conditions on the coefficients of T . Lemma 5.12. Given the coefficients p, q, r of an exceptional operator T as per (4), the following statements hold (i) p(z) is a polynomial; (ii) the poles of q(z) are simple; (iii) deg p ≤ 2, deg q ≤ 1, deg r ≤ 0. Moreover, if T is reduced, then (iv) the poles of r(z) are also simple; (v) every pole of r(z) is also a pole of q(z). Proof. Consider claim (i). If ζ ∈ C is a pole of p(z), then by (38) we would have dζ ≤ −3 which is forbidden by Lemma 5.11. Hence p(z) must be polynomial. To prove (ii) note that if ordζ q < −1, then dζ ≤ −3, which is again forbidden by Lemma 5.11.

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

In order to prove (iii), consider the polynomial division to write the following decompositions q(z) = qp (z) + qs (z), r(z) = rp (z) + rs (z), where qp , rp ∈ P and qs , rs ∈ Q with deg qs , deg rs < 0,

deg qp = deg q,

deg rp = deg r.

Next consider the decomposition T = Tp + Ts , where Tp = p(z)Dzz + qp (z)Dz + rp (z),

Ts = qs (z)Dz + rs (z).

By construction, deg Ts [y] < deg y,

y ∈ P.

Hence, since T preserves the degree of infinitely many eigenpolynomials, so does Tp . In Section 6 we consider the degree properties of operators with polynomial coefficients. Condition (iii) follows by Proposition 6.2 of that section. Next, suppose that T is reduced. If z = ζ is a pole of r(z), then as per (39), T−2 = p0 Dzz + q−1 (z − ζ)−1 Dz + r−2 (z − ζ)−2 , T−1 = p1 (z − ζ)Dzz + q0 Dz + r−1 (z − ζ)−1 , with rj = 0 for j < −2. Since T is reduced, 0 ∈ Iζ . By Lemma 5.9, we have Tdζ [1] = 0, which means that r−2 = 0. Claim (iv) has been established. By primitivity, there exists a y = 1 + a(z − ζ) + O((z − ζ)2 ) ∈ U . Observe that T [y] = (aq−1 + r−1 )(z − ζ)−1 + O(1). Hence, r−1 = −aq−1 , which implies that q−1 6= 0. This proves (v).



Lemma 5.13. Let T be a reduced exceptional operator and ζ ∈ C. If νζ > 0, then the gaps in the order sequence are the first νζ positive odd numbers; i.e., (40)

Iζ = {0, 2, 4, . . . , 2νζ } ∪ {2νζ + 1, 2νζ + 2, . . .}.

Moreover, p(ζ) 6= 0, with (41)

T−2

 = p(ζ) Dzz −

 2νζ Dz . (z − ζ)

Proof. Since νζ > 0 then ζ must be a pole of T . By Lemma 5.12, dζ = −2. By primitivity, 0 ∈ Iζ . We see that 1 ∈ / Iζ since otherwise T−2 would need to annihilate 1, (z − ζ) and (z − ζ)k for some k > 1, which is impossible. By Lemma 5.10, there exists an n ≥ 1 such that 1, 3, 5, . . . , 2n − 1 ∈ / Iζ , and T−2 [(z − ζ)2n+1 ] = 0. Since T−2 annihilates 1 and (z − ζ)1+2n , it cannot annihilate another monomial, which proves (40). By Lemma 5.12, (41) must hold with νζ = n. 

21

We now prove some supplementary results concerning the analytic properties of the series solutions to the eigenvalue equation T [y] = λy from the point of view of Frobenius’ method, [73]. For a second-order operator T = p(z)Dzz + q(z)Dz + r(z) we consider power series solutions of the second-order equation q(z) ′ r(z) − λ y + y = 0. (42) y ′′ + p(z) p(z) centered at a given ζ ∈ C. Definition 5.14. We say that ζ ∈ C is a regular point of the equation T [y] = λy if for every λ ∈ C, there exist two linearly independent formal series solutions to (42), in the usual sense of Frobenius’ method. Equivalently,     r−λ q ≥ −1, ordζ ≥ −2 ordζ p p for all λ ∈ C. Definition 5.15. We say that T has trivial monodromy at ζ ∈ C if T [y] = λy admits two linearly independent Laurent series solutions, i.e. if the general solution of (42) is meromorphic in a neighbourhood of ζ. Note that if these conditions holds for one λ, then they hold for all λ. The poles of q(z), r(z) and the zeros of p(z) thus have to be distinguished as points which may not be regular, and where the monodromy may be non-trivial. Definition 5.16. Let T ∈ Diff 2 (Q) be an exceptional operator and p, q, r its coefficients, as per (4). We say that z = ζ is i) a primary pole of T if it is a pole of q(z); ii) a secondary pole of T if it is a zero of p(z); iii) an ordinary point of T , if it is neither a pole of q(z) nor a zero of p(z). Proposition 5.17. A point ζ ∈ C cannot be both a primary and a secondary pole. Moreover, every pole of r(z) is either a primary or a secondary pole. Proof. If T is reduced, then the first assertion follows by Lemma 5.13 and the second by Lemma 5.12. For general exceptional operators, these assertions follow by Proposition 2.5.  Proposition 5.18. If T ∈ Diff 2 (Q) is an exceptional operator, then every primary pole and ordinary point is a regular point of the equation T [y] = λy. If p(z) has simple zeros, then every secondary pole is also a regular point. Proof. Repeat the argument from the proof of the preceding Proposition.



Consider the Laurent decomposition given in Definition 5.8. The indicial equation in Frobenius’ method is determined by the leading term of the operator via the condition Tdζ [(z − ζ)k ] = 0, which is equivalent to pdζ +2 k(k − 1) + qdζ +1 k + rdζ = 0. Note from Lemma 5.11 that dζ ≥ −2 at all finite points ζ ∈ C, so primary and secondary poles are both singular regular points of T .

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´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

If T is reduced, then at a primary pole, it can be seen from (41) that the two roots of the indicial equation are 0 and 2νζ + 1. Since they differ by an integer, there is a chance that one of the solutions has a logarithmic singularity. We show next that this is not the case. Proposition 5.19. An exceptional operator has trivial monodromy at ordinary points. Proof. It is well-known that a second-order, linear differential equation admits two linearly independent series solutions at an ordinary point.  Proposition 5.20. An exceptional operator T has trivial monodromy at all primary poles. Proof. Suppose that T is reduced with trivial monodromy at primary poles. Consider the general exceptional operator T˜ = σT σ −1 , σ ∈ P. The general solution of T˜[˜ y ] = λ˜ y has the form y˜ = σy, where y is the general solution of T [y] = λy. Thus, the primary poles of T˜ are either primary poles or ordinary points of T . By assumption and Proposition 5.19 it follows the non-reduced operator T˜ also has trivial monodromy at its primary poles. We therefore assume, without loss of generality, that T is reduced, that z = 0 is a primary pole of T , and that p(0) = 1. From Lemma 5.11 it follows that the order sequence of T at z = 0 is I0 = {0, 2, 4, . . . , 2ν0 } ∪ {2ν0 + 1, 2ν0 + 2, . . .}, ν0 > 0, and that the roots of the indicial equation are 0 and 2ν0 + 1. Hence T [y] = λy admits a unique holomorphic solution of the form ∞ X an (λ)z n , a2ν0 +1 = 1. y = a(z; λ) = n=2ν0 +1

By Frobenius’ method, T [y] = λy admits also a unique series solution of the form y = b(z; λ) =

∞ X

bj (λ)z j + c(λ) a(z; λ) log z,

b0 = 1, b2ν0 +1 = 0.

j=0

Our claim will be proven once we show that c ≡ 0; i.e. that the second independent solution b(z; λ) does not have a logarithmic singularity. Observe that   T−2 z 2ν0 +1 log z = (1 + 2ν0 )z 2ν0 −1 ,   T−2 z 2ν0 +1 = 0.

Hence, for 1 ≤ n ≤ 2ν0 we must have



(T − λ) 

n X j=0



bj (λ)z j  = O(z n−1 ).

Furthermore, a logarithmic singularity develops if and only if    2ν0 X ord0 (T − λ)  bj (λ)z j  = 2ν0 − 1. j=0

Therefore, to prove that c ≡ 0, it suffices to exhibit a polynomial β(z; λ) such that (T − λ)[β] = O(z 2ν0 ).

23

To construct this polynomial, we choose an order-adapted basis of U of the form yk (z) = z k + higher order terms,

k ∈ I0 ,

which is guaranteed to exist by Definition 4.13 and Remark 4.14. Since U is T -invariant, by (41), the action of T on y0 , y2 , . . . , y2ν0 can be expressed as T [y0 ] =

ν0 X

N0 X

B0k y2k +

k=0

A0k z k ,

k=2ν0 +1

T [y2j ] = 2j(2j − 1 − 2ν0 )y2j−2 +

ν0 X

Bjk y2k +

k=j

Nj X

Ajk z k ,

j = 1, 2, . . . , ν0 ,

k=2ν0 +1

for some Ajk , Bjk ∈ C and Nj ∈ N. Set β0 = 1, and define βj (λ), j = 1, . . . ν by the recurrence relations (43)

2j(2j − 1 − 2ν0 )βj + (Bj−1,j−1 − λ)βj−1 +

j−2 X

Bk,j−1 βk + B0,j−1 = 0;

k=1

Set β(z; λ) =

ν0 X

βj (λ)y2j (z).

j=0

By construction, (T − λ)[β] =

ν0 X

B0j y2j j=0 ν0 ν0 X X

+

+

ν0 X

2j(2j − 1 − 2ν0 )βj y2j−2

j=1

βj Bjk y2k −

j=1 k=j

=

νX 0 +1

ν0 X

βk Bk,j−1 y2j−2 −

νX 0 +1

B0,j−1 + 2j(2j − 1 − 2ν0 )βj +

j=1

= O(z

λβj−1 y2j−2 + O z 2ν0 +1

j=1

j=2 k=1

=

2j(2j − 1 − 2ν0 )βj y2j−2 +

j=1

j−1 νX 0 +1 X

ν0 X

λβj y2j + O(z 2ν0 +1 )

j=0

B0,j−1 y2j−2 +

j=1

+

ν0 X

j−2 X

βk Bk,j−1 + (Bj−1,j−1 − λ)βj−1

k=1

2ν0

),



!

y2j−2 + O(z 2ν0 )

as was to be shown. The key element in the proof is the fact that the invariance of U under T guarantees the absence of a logarithmic singularity.  Lemma 5.13 established the form of the T−2 term of a reduced, exceptional operator. The conclusion is that the Laurent expansion of q(z) at one of the primary poles has the form (i)

(44)

q(z) =

−2νi p0 (i) + q0 + O(z − ζi ) z − ζi

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

24

so that T−2 =

(i) p0

where



 2νi Dz , Dzz − z − ζi

(i)

p0 = p(ζi ) 6= 0. Using the trivial monodromy results we can now describe the T−1 term. Lemma 5.21. Let T ∈ Diff 2 (Q) be a reduced, exceptional operator, and (37) the Laurent decomposition at one of its primary poles ζi , i = 1, . . . , N . Then, !   1 νi (i) (i) 2 νi (3νi − 1) + q0 Dz − , (45) T−1 = p1 (z − ζi )Dzz − z − ζi z − ζi (i)

where p1 = p′ (ζi ). The proof is based on the following result characterizing monodromy-free Schr¨ odinger operators [64, Proposition 3.3]. Lemma 5.22 (Duistermaat-Gr¨ unbaum). Let U (x) be meromorphic in a neighborhood of x = 0 with Laurent expansion X U (x) = cj xj . j≥−2

Then all eigenfunctions of the Schr¨ odinger operator H = −Dxx + U (x) are single-valued around x = 0 if and only if c−2 = ν(ν + 1) for some integer ν ≥ 1, and c2j−1 = 0, 0 ≤ j ≤ ν. Proof of Lemma 5.21. Since p(ζi ) 6= 0 we can find an analytic change of variables z = ζ(x) that satisfies (46)

ζ ′ (x)2 = p(ζ(x)),

ζ(0) = ζi .

Explicitly, x= In this way

Z

z=ζ(x)

dz p . p(z)

1 Dxx = p(z)Dzz + p′ (z)Dz . 2

Set µ(z) = exp

1 2

Z

q(z) − 21 p′ (z) dz p(z)

!

.

Observe that µ(z) is analytic at z = ζi . A direct calculation shows that 1 µT µ−1 = p(z)Dzz + p′ (z)Dz + V (z), 2 where   q(z) − 21 p′ (z) q(z) − 32 p′ (z) p′′ (z) q ′ (z) − − + r(z). V (z) = 4 2 4p(z)

25

Set  H = −Dxx − V ζ(x) ,

so that T [y] = λy if and only if H[ψ] = −λψ, where   ψ(x) = µ ζ(x) y ζ(x) .

Hence, T has trivial monodromy at z = ζi if and only if H has trivial monodromy at x = 0. Using (44) and a direct calculation, gives (i)

(i)

(i)

(i) νi q0 + r−1 + p1 νi (νi − 1) νi (νi + 1)p0 V (z) = − + + O(1). (z − ζi )2 (z − ζi )

Relation (46) implies (ζ(x) − ζi )−1 = (ζ(x) − ζi )−2 = =

1 ζ ′ (0)

x−1 + O(1)

1 ζ ′′ (0) −1 −2 x − x + O(1) ζ ′ (0)2 ζ ′ (0)3 (i)

p1

1

x−2 − (i)

p0

(i) 2p0 ζ ′ (0)

x−1 + O(1).

Hence, U (x) = νi (νi + 1)x−2 −

1 (i) 1 (i) (i) (νi q0 + r−1 + p1 νi (3νi − 1))x−1 + O(1) ζ ′ (0) 2

By Lemma 5.22 the coefficient of x−1 must vanish. From this (45) follows directly.



Lemma 5.23. Every reduced, exceptional operator admits a bilinear formulation. Proof. Let T be a reduced, exceptional operator, ζi , i = 1, . . . , N the primary poles, and νi = νζi the gap cardinalities as per Definition 4.13. Set N Y (z − ζi )νi ; η(z) =

(47)

(48)

µ(z) =

i=1 N Y

(z − ζi )νi (νi −1)/2 .

i=1

Dividing (30) through by η(z) and letting p, q, r denote the operator coefficients, we have (49a) (49b)

1 η′ q = p′ + s − 2p 2 η r=p

η ′′ 2µ′′ + −2 η µ



µ′ µ

2 !

+p





η′ µ′ + 2η µ

We claim that every reduced, exceptional T has this form.





sη ′ . η

26

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

By (41) of Lemma 5.12 and (45) of Lemma 5.21, the Laurent expansions of p(z), q(z), r(z) at a primary pole z = ζi have the form (i)

(i)

(i)

(50)

p(z) = p0 + p1 (z − ζi ) + p2 (z − ζi )2

(51)

q(z) = −

(i)

(52)

r(z) =

2p0 νi (i) + q0 + O(z − ζi ) z − ζi

1 (i) 2 p1 νi (1

(i)

− 3νi ) − q0 νi + O(1) z − ζi

By Lemma 5.12, necessarily deg q ≤ 1 and deg r ≤ 0, which agrees with (49a) and (49b). Thus, it suffices to show that the Laurent expansions of (49a) and (49b) match (51) and (52), respectively. The definition of η(z) in (47) gives νi η ′ (z) = + τi + O(z − ζi ), η(z) z − ζi where τi =

X j6=i

Relation (49a) gives q(z) = −

νj . ζi − ζj

2p(ζi )νi (i) + q0 + O(z − ζi ), z − ζi

where (i)

(53)

(i)

q 0 = p1



1 − 2νi 2



(i)

(i)

+ s0 − 2p0 τi ,

(i)

s0 = s(ζi ).

We have νi (νi − 1) η ′′ (z) τ i νi = + + O(1), 2 η(z) (z − ζi ) z − ζi  ′ 2 1 νi (νi − 1) µ (z) µ′′ (z) =−2 − + O(1) µ(z) µ(z) (z − ζi )2 1 2 ν η′ µ′ + = 2 i + O(1). 2η µ z − ζi

From there, relation (49b) gives (i)

(i)

(i)

p0 τi νi + 12 p1 νi2 − s0 νi r(z) = + O(1). z − ζi On the other hand, (52) gives r(z) =

1 (i) 2 p1 νi (1

which agrees with the above.

 (i) − 3νi ) − p1

1 2

  (i) (i) − 2νi + s0 − 2p0 τi νi

z − ζi

+ O(1), 

27

Proof of Theorem 5.4. Since every exceptional operator is gauge equivalent to an operator in reduced gauge, it suffices to show that every reduced, exceptional operator admits a bilinear formulation. This is accomplished in Lemma 5.23. Finally, Lemma 5.7 shows that an exceptional operator in the reduced gauge is gauge-equivalent to an exceptional operator in the natural gauge.  Before moving onto the next section, we make a remark and state two corollaries of Theorem 5.4 that generalize results for exceptional Hermite polynomials previously established in [74]. These results are not used elsewhere in the paper, but they may have some significance for future research, in particular for the derivation of recurrence relations for exceptional polynomials. Remark 5.24. Since the roots of the indicial equation at a primary pole and at an ordinary point are non-negative, the general solution of T [y] = λy is not only meromorphic but holomorphic at such points. The only points at which the general solution of T [y] = λy might not be meromorphic are the secondary poles of T , i.e. the roots of p(z). In the case p(z) = 1 which corresponds to exceptional Hermite operators, the general solution is thus an entire function, as proved in [75]. Corollary 5.25. Let T ∈ Diff 2 (Q) be an exceptional operator in the natural gauge, and let (54)

η(z) =

N Y (z − ζi )νi , i=1

where νi = νζi are the gap cardinalties at the poles of the operator (see Definition 4.13 and Theorem 4.15). Let U ⊂ P be the maximal polynomial subspace. Then y ∈ U if and only if   1 ′ ′ ′ ′ ′ ′′ (55) 2pη y − pη + p η − sη y 2 is divisible by η. Proof. Let U ′ ⊂ P be the polynomial subspace consisting of those y ∈ P such that (55) is divisible by η. If y ∈ U , then T [y] ∈ P by Proposition 4.8. Decompose the operator in (32) as T = T0 + Ts where  ′  p T0 = pDzz + + s Dz 2  ′  ′ p η pη ′′ 2pη ′ Dz + + −s . Ts = − η η 2 η

Since T0 has polynomial coefficients, Ts [y] ∈ P. Hence, y ∈ U ′ , and therefore U ⊂ U ′ . To obtain equality, we use a codimension argument. For i = 1, . . . , N, j = 0, . . . , νi − 1, (j) define the differential functionals αi : P → C by     1 ′ j ′ ′ ′′ ′ ′ y 7→ Dz 2p(z)η (z)y (z) − p(z)η (z) + p (z)η (z) − s(z)η (z) y(z) . 2 z=ζi Observe that y ∈ P is divisible by η if and only if

y (j) (ζi ) = 0

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

28

(j)

for the range of i, j given above. Hence, U ′ is the joint kernel of the αi . By Proposition P 4.18, these functionals are linearly independent, and hence U ′ has codimension N i=1 νi in P. By Theorem 4.15, this is also the codimension of U in P. Therefore, U = U ′ .  Corollary 5.26. Let T ∈ Diff 2 (Q) be an exceptional operator in the natural gauge. Let U be the maximal invariant polynomial subspace, and let η be the polynomial defined in (54). Suppose that f ∈ P is such that f ′ is divisible by η. Then, multiplication by f preserves U ; i.e., f y ∈ U for every y ∈ U . Proof. Suppose that f ′ is divisible by η. Replacing y with f y in (55) yields       1 1 2pη ′ (f y)′ − pη ′′ + p′ η ′ − sη ′ f y = f 2pη ′ y ′ − pη ′′ + p′ η ′ − sη ′ y + 2pη ′ f ′ y. 2 2

By Corollary 5.25, if y ∈ U , then the above is divisible by η.



The above Corollary allows to build recurrence relations for exceptional polynomials, where the traditional multiplication by x is substituted by multiplication by the polynomial f satisfying the above condition, [47, 48, 74, 76]. 6. Proof of the Theorem In this section we prove the previously conjectured result that every exceptional operator is Darboux connected to a classical operator. We begin with some preliminary definitions and results. Definition 6.1. For L ∈ Diff(Q) we define the degree of the operator as (56)

deg L = max{deg aj − j : j = 0, 1, . . . , ρ},

where the aj ∈ Q is the j th order coefficient as per (2), and where the degree of a rational function is the difference of the degrees of the numerator and denominator. The degree of an operator has an alternative, but equivalent characterization. Proposition 6.2. The degree of an operator L ∈ Diff(Q) is the smallest integer k such that (57)

deg L[y] ≤ k + n

for all y ∈ Pn∗ . The above inequality is strict for only finitely many values of n. Proof. By the defintion (56), inequality (57) holds for k = deg L. The second claim then implies that this is the smallest such integer. We therefore turn to the proof of the second claim. Set k = deg L and let aj ∈ Q, j = 0, 1, . . . , ρ be the coefficients of the operator. By assumption aj (z) = cj z j+k + bj (z), cj ∈ C, bj ∈ Q, where deg bj ≤ j + k − 1. Write L = L0 + L− , where L0 =

ρ X j=0

cj z k+j Dzj ,

29

is a homogeneous degree k operator and where L− =

ρ X

bj (z)Dzj

j=0

satisfies deg L− < k by construction. Next, observe that L0 [z n ] = σ(n)z n+k , where σ(n) =

ρ X

cj n(n − 1) · · · (n − j + 1).

j=0

Since σ(n) is a polynomial of degree ≤ ρ, we have σ(n) = 0 for at most ρ values of n. By construction, for y ∈ Pn , we have deg L− [y] < k + n. Therefore, if σ(n) 6= 0, then deg L[y] = k + n.  Definition 6.3. We say that T ∈ Diff 2 (P) is a Bochner operator (or classical operator) if deg T = 0. Before stating the main result of this section, we note the following. Proposition 6.4. Every Bochner operator is exceptional. Proof. By the proof of Proposition 6.2, T [z k ] = σ(k)z k + lower degree terms where σ(k) is a non-zero polynomial of degree ≤ 2. Hence, T − σ(k) maps Pk into Pk−1 for every k ∈ N. By the rank-nullity theorem, this linear map has a non-trivial kernel, which means that, for every k ∈ N, there exists a yk ∈ Pk such that T [yk ] = σ(k)yk . However deg yk may be strictly less than k, which means that yk may coincide with an eigenpolynomial of lower degree. However, this can happen only if σ(k) = σ(k′ ) for some k′ 6= k; i.e. if the eigenvalue is not simple. Since σ(k) is at most a quadratic function, and k is a positive integer, this can happen at most finitely many times. Therefore, a co-finite number of eigenvalues σ(k) are simple, which means that there are eigenpolynomials for a co-finite number of degrees k. Therefore, T is an exceptional operator according to Definition 4.1.  Remark 6.5. Note that Bochner operators need not have polynomial eigenfunctions for every degree k ∈ N. See for example Remark 4.10 and a counter-example in Example 4.11. The main result of this section is the following theorem.

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

30

Theorem 6.6. Every exceptional operator T ∈ Diff 2 (Q) is Darboux connected to a Bochner operator TB ∈ Diff 2 (P). Moreover, letting p ∈ P2 denote the second-order coefficient of T , and letting W, WB denote the respective weights, as defined by (5b), we have (58)

W (z) = WB (z)

χ(z) , η(z)2

where η ∈ P is a polynomial whose roots belong to the set of primary poles of T , and where χ ∈ Q is a rational function such that (59)

p(z)

χ′ (z) χ(z)

is a constant. The proof of Theorem 6.6 requires a number of preliminary results. Let T ∈ Diff 2 (Q) be an exceptional operator and consider the vector space L := {L ∈ Diff(P) : T k L ∈ Diff(P) for all k ∈ N}. The following is an equivalent characterization of L. Lemma 6.7. For L ∈ Diff(P), we have L ∈ L if and only if L[P] ⊂ U . Proof.  One  direction is trivial; we prove the converse. Suppose that L ∈ L so that we have T k L[y] ∈ P for all y ∈ P and all k ≥ 1. By Defintion 4.7, this implies that L[y] ∈ U , as was to be shown.  Next, define the subspace L(ρ) := {L ∈ L : ord L ≤ ρ, deg L ≤ 0} where it is clear that L(ρ1 ) ⊂ L(ρ2 ) for ρ1 < ρ2 . We will first show that at least one L(ρ) is non-trivial. Lemma 6.8. Let ζ1 , . . . , ζN be the primary poles of T , and ν1 , . . . , νN the corresponding gap cardinalities. Then, dim L(n) > 0 where n=

N X

2νi .

i=1

Proof. Set (60)

η(z) =

N Y (z − ζi )2νi . i=1

By construction, for every y ∈ P (k)

αζi [ηy] = 0,

i = 1, . . . , N, k ∈ / Iζi ,

(k)

where αζi are the linear functionals defined in (28). Hence, by Lemma 4.19, ηy ∈ U for all y ∈ P, and Lemma 6.7 implies that the differential operator L = η(z)Dzn , belongs to L. By Proposition 4.15 its degree is zero, so L ∈ L(n) as was to be proved.



31

Now, let ρmin be the minimum positive integer such that dim L(ρ) > 0, i.e. dim L(ρmin ) > 0 but dim L(ρ) = 0 for all ρ < ρmin . Lemma 6.9. For all non-zero L ∈ L(ρmin ) we have ord L = ρmin and deg L = 0 exactly. Proof. The order equality holds by the minimality assumption on ρmin . Similarly, suppose that there exists a non-zero L ∈ L(ρmin ) such that deg L = −d < 0. Since L has polynomial ˜ d , where L ˜ ∈ Diff(P). coefficients, such an operator would necessarily be of the form L = LD (ρ −d) ˜ ∈ L min , which would again contradict the minimality assumption This would imply that L for ρmin .  Lemma 6.10. dim L(ρmin ) ≤ ρmin + 1. Proof. Observe that ρmin + 1 is the dimension of the space of degree homogeneous differential operators of order ρmin . Hence if dim L(ρmin ) were to exceed this bound, we would be able to construct an operator L ∈ L(ρmin ) having strictly negative degree, which is impossible by Lemma 6.9.  Lemma 6.11. Let T be an exceptional operator. Then, there exist a decomposition T = T0 + Ts , where T0 ∈ Diff 2 (P) is a Bochner operator, and Ts ∈ Diff 1 (Q) has negative degree. Proof. Let p, q, r be the coefficients of T , as per (4). By Lemma 5.12, we must have p ∈ P2 , and polynomial division allows to write q = q1 + qs ,

r = r0 + rs ,

with q1 , r0 ∈ P, qs , rs ∈ Q, with deg q1 ≤ 1,

deg r0 = 0,

deg qs ≤ 0,

deg rs ≤ −1.

Taking T0 = pDzz + q1 Dz + r0 ,

Ts = qs Dz + rs

gives the desired decomposition.



Lemma 6.12. Let T be an exceptional operator and T0 , Ts its decomposition into Bochner and singular part according to Lemma 6.11 . If L ∈ L(ρmin ) is non-zero, then (61)

deg (T L − LT0 ) < 0.

Proof. By Lemma 6.9, deg L = 0. Hence, for y ∈ Pn∗ we have (62) (63) (64)

T [y] = σ1 (n) y + y1 , T0 [y] = σ1 (n) y + y2 , L[y] = σ2 (n) y + y3 ,

deg y1 < n, deg y2 < n, deg y3 < n,

where by Proposition 6.2, σ1 (n), σ2 (n) are polynomials in n. Hence, (65) (66)

(T L)[y] = T [σ2 (n) y + y3 ] = σ1 (n)σ2 (n)y + y4 , (LT0 ) [y] = L[σ1 (n) y + y2 ] = σ2 (n)σ1 (n)y + y5 ,

which establishes (61).

deg y4 < n deg y5 < n, 

32

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

Lemma 6.13. Let T be an exceptional operator and T0 , Ts its decomposition according to Lemma 6.11. Then, there exists a linear transformation A : L(ρmin ) → L(ρmin ) such that A(L)D = T L − LT0 ,

L ∈ L(ρmin ) .

Proof. Since T is second-order, ord(T L − LT ) ≤ ord L + 1,

L ∈ Diff(P).

By construction, T − T0 is a first-order operator, and hence ord(T L − LT0 ) ≤ ord L + 1,

L ∈ Diff(P)

also. By Lemma 6.12, if L ∈ L(ρmin ) , then ˜ T L − LT0 = LD ˜ ∈ Diff(P). By construction, (LD)[P] ˜ for some unique operator L ⊂ U which means that that ˜ ˜ ∈ L. If L ∈ L(ρmin ) then by the above L[P] ⊂ U as well. Hence, Lemma 6.7 implies that L ˜ ≤ ord L = ρmin and deg L ˜ ≤ 0. This implies that L ˜ ∈ L(ρmin ) . Our results we see that ord L ˜ claim is established once we set A(L) := L.  Proof of Theorem 6.6. Let T, T0 be as in the preceding Lemma. By Lemma 6.10, L(ρmin ) is finite dimensional. Hence, there exists an eigenvector L ∈ L(ρmin ) with eigenvalue γ of the linear transformation A defined in Lemma 6.13. This means that L ∈ Diff(P) and A(L) = γL so that T L = L(T0 + γD). Therefore TB = T0 + γD is the desired Bochner operator. By Theorem 5.4 and Lemma 5.7, the eigenvalue relation T [y] = λy is equivalent to (30), where η has zeros at the primary poles of T . Hence, Z z  ax + b W (z) = exp dx η(z)−2 , p(x) where a, b are constants. Let WB (z) be the weight associated to TB . By definition of T0 , Z z  ax + c WB (z) = exp dx , p(x)

where c is also a constant. By Proposition 3.6, W (z) is WB (z) times a rational factor, which implies that Z z  b−c 2 W (z) = exp dx χ(z) = η(z) WB (z) p(x) is a rational function. The desired conclusion follows immediately.  7. Exceptional Orthogonal Polynomial Systems In all of the previous sections the differential operator T was treated at a purely formal level, the emphasis being on the algebraic conditions leading to the existence of an infinite number of polynomial eigenfunctions. In this section, analytic conditions will be further imposed, in order to select those operators that have a self-adjoint action on a suitably defined Hilbert space. This observation motivates the following. / Definition 7.1. We say that a co-finite, real-valued polynomial sequence yk ∈ RPk∗ , k ∈ {k1 , . . . , km } forms a Sturm-Liouville orthogonal polynomial system (SL-OPS) provided

33

(i) the yk are the eigenpolynomials of an operator T ∈ Diff 2 (RQ) , (ii) there is an open interval I ⊂ R such that (ii-a) the associated weight function W (z), as defined by (5b), is single valued, and integrable on I, and moreover, (ii-b) all moments are finite, i.e. Z z j W (z)dz < ∞, j ∈ N; I

(ii-c) y(z)p(z)W (z) → 0 at the endpoints of I for every polynomial y ∈ P . (iii) the vector space span{yk : k ∈ / {k1 , . . . , km }} is dense in the weighted Hilbert space L2 (W (z)dz, I).

Assumption (i) means that T is an exceptional operator. By Proposition 2.5 and Theorem 5.4, no generality is lost if we assume that T is in the natural gauge; i.e., the eigenvalue relation takes the form (31), where η is given by (54). Proposition 2.2 and (ii-c) ensures that T is polinomially regular and that yk are orthogonal Z W (z)yi (z)yj (z)dz = ci δij , i, j ∈ / {k1 , . . . , km }, ci > 0. I

As it was already mentioned in Remark 4.10, regularity implies semi-simplicity, which means that U , the maximal invariant polynomial subspace, coincides with the span of the eigenpolynomials yk , k ∈ / {k1 , . . . , km }, and ν = m. Therefore, by assumption (iii), operator T is essentially self-adjoint on U . It has already been noted in all examples of exceptional orthogonal polynomials published in the literature, that the orthogonality weight for the exceptional OPS is a classical weight multiplied by a rational function. This can now be considered as a result. Proposition 7.2. The orthogonality weight W (z) of a SL-OPS has the form (67) where

W (z) =

WB (z) η(z)2

z

 ax + b dx , p ∈ RP2 , a, b ∈ R WB (z) = exp p(x) ∗. is the weight of a classical OPS, and where η ∈ RPm Z

Proof. Expression (67) follows by (32) and (5b). The condition on η is ensured by its definition (54), Theorem 4.15 and the fact already mentioned that ν = m.  Notice that η has zeros at the primary poles, which lie outside I by assumption (ii). Remark 7.3. If an SL-OPS has polynomial eigenfunctions for all degrees, i.e. m = 0 in Definition 7.1, then it defines a classical orthogonal polynomial system, which up to an affine transformation must be Hermite, Laguerre or Jacobi [3, 4]. Otherwise we have a genuinely exceptional orthogonal polynomial system whose operator T has one or more primary poles. Since every SL-OPS has an associated exceptional operator T , the notion of Darboux connectedness for operators can be naturally extended to SL-OPS. Definition 7.4. We say that two SL-OPS are Darboux connected if their associated exceptional operators are Darboux connected as per Definition 3.7.

34

´ ´ GARC´IA-FERRERO, DAVID GOMEZ-ULLATE, AND ROBERT MILSON Ma ANGELES

The weights associated with a SL-OPS fall into the same three broad categories as do classical orthogonal polynomials. Definition 7.5. We say that a SL-OPS is of, respectively, Hermite, Laguerre, and Jacobi type if the corresponding interval I = (a, b) and weight W (z), z ∈ I have the form 2

(68a)

I = (−∞, ∞),

(68b)

I = (0, ∞)

(68c)

I = (−1, 1)

e−z WH (z) = , η(z)2 z α e−z , WL (z) = η(z)2 WJ (z) =

α > −1,

(1 − z)α (1 + z)β , η(z)2

α, β > −1,

where η ∈ RP is a real-valued polynomial which is non-vanishing on I. Proposition 7.6. Up to an affine transformation of the independent variable, every SL-OPS belongs to one of the three types shown above. Proof. Up to an affine change of variable, the second-order coefficient of an exceptional operator takes one of the following forms: 1, z, z 2 , 1 + z 2 , 1 − z 2 . Applying (5b) and (32), we see that cases 1,2, and 5 correspond to weights of Hermite, Laguerre, and Jacobi type, respectively. It therefore suffices to rule out the remaining possibilities. These correspond to, respectively, weights of the following form: b

zae z , W (z) = η(z)2 W (z) =

ea arctan(z) (1 + z 2 )b , η(z)2

where a, b ∈ R are real constants. By inspection, there does not exist a choice of constants or an interval I ⊂ R such that of these forms can satisfy requirement (ii) in the definition of a SL-OPS.  The analysis of the regularity of the exceptional weight amounts to studying the range of parameters and the combination of Darboux transformations such that η(z) has no zeros on I, and such that the classical portion of the weight is integrable on I. For the case of exceptional Hermite polynomials, this was done in [59,75], for exceptional Laguerre polynomials in [56,62], and for exceptional Jacobi polynomials in [77]. Applying (31) with p(z) = 1, z, 1 − z 2 , respectively, we arrive at the following bilinear relations for the exceptional polynomials associated to the above 3 classes of SL-OPS: (69)

(ηHk′′ − 2η ′ Hk′ + η ′′ Hk ) − 2z(ηHk′ − η ′ Hk ) + 2k ηHk = 0

(70)

z(ηL′′k − 2η ′ L′k + η ′′ Lk ) + (1 + α − z)ηL′k + (z − α)η ′ Lk + (k − m) ηLk = 0,

(71)

(1 − z 2 )(ηPk′′ − 2η ′ Pk′ + η ′′ Pk ) + (−(2 + α + β)z + β − α)ηPk′ + + ((α + β)z − β + α)η ′ Pk + ((α + β)(n − m) + (n − 2m + 1)n) ηPk = 0,

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Here, Hk (z), Lk (z), Pk (z) denote, respectively, exceptional Hermite, Laguerre, and Jacobi ∗ , and valid for all polynomials of degree k corresponding to a particular choice of η(z) ∈ Pm k∈ / {k1 , . . . , km }. Setting m = 0 in the above equations recovers the usual Hermite, Laguerre, and Jacobi differential equations. It therefore makes sense to regard (69) (70) and (71) as the exceptional generalizations of these 3 classical equations. We are now ready to give the proof of the main theorem. Proof of Theorem 1.2. Let T ∈ R Diff(Q) be the exceptional operator associated with a SLOPS. By Theorem 6.6, T is Darboux connected to a Bochner operator TB with the corresponding weights related by (58). Since p ∈ RP2 is the same for both operators, both the W and WB belong to the same class of weights. In Proposition 7.2, we established that the polynomial η(z) is real-valued. Hence, the rational factor χ(z) in (58) must also be real-valued, by (59). Hence, TB has real coefficients. It remains to show that the weight parameters in WB satisfy the conditions in (68), so that the resulting measure has finite moments. We do not claim that TB is regular, but rather that, TB is Darboux connected to another Bochner operator that is regular. For the Hermite class, there is nothing to prove, because p(z) = 1, and hence χ(z) in (58) must be a constant. Let us consider the Laguerre class next. Write Tα = zDzz + (1 + α − z)Dz = (zDz + 1 + α − z) ◦ Dz . The corresponding weight is z α e−z . Performing a Darboux transformation gives Tα 7→ Dz ◦ (zDz + 1 + α − z) = Tα+1 − 1. Therefore, Tα is Darboux connected to Tα+1 − 1, and more generally to Tα+n − n, where n is an arbitrary integer. Hence, even though the TB produced by Theorem 6.6 may not be regular, it is Darboux connected to a regular Bochner operator, and hence so is T . Finally, let us consider the Jacobi class. Write  Tα,β = (1 − z 2 )Dzz + (−(2 + α + β)z + β − α)Dz = (1 − z 2 )Dz − (2 + α + β)z + β − α ◦ Dz . Performing a Darboux transformation gives

 Tα,β 7→ Dz ◦ (1 − z 2 )Dz − (2 + α + β)z + β − α = Tα+1,β+1 − 2 − α − β.

Therefore, Tα,β is Darboux connected to Tα+n,β+n − (2 + α + β)n for every integer n. By taking n sufficiently large, we can ensure that Tα+n,β+n is regular.  8. Acknowledgements M.A.G.F. acknowledges the financial support of the Spanish MINECO through a Severo Ochoa FPI scholarship. The work of M.A.G.F. is supported in part by the ERC Starting Grant 633152 and the ICMAT-Severo Ochoa project SEV-2015-0554. The research of D.G.U. has been supported in part by Spanish MINECO-FEDER Grants MTM2012-31714 and MTM2015-65888-C4-3 and by the ICMAT-Severo Ochoa project SEV-2015-0554. The research of the third author (RM) was supported in part by NSERC grant RGPIN-2280572009. D.G.U. would like to thank Dalhousie University for their hospitality during his visit in the Spring semester of 2014 where many of the results in this paper where obtained.

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´ ticas (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera 15, 28049 Instituto de Ciencias Matema Madrid, Spain. ´ rica II, Universidad Complutense de Madrid, 28040 Madrid, Spain. Departamento de F´ısica Teo ´ ticas (CSIC-UAM-UC3M-UCM), C/ Nicolas Cabrera 15, 28049 Instituto de Ciencias Matema Madrid, Spain. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada. E-mail address: [email protected], [email protected], [email protected]