The Spectral Theory of the X1-Laguerre Polynomials 1 Introduction

Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume x, Number x, pp. 25–36 (20xx) http://campus.mst.edu/adsa

The Spectral Theory of the X1-Laguerre Polynomials

Mohamed J. Atia a, Lance L. Littlejohn b,∗ and Jessica Stewart c a Faculte´ de Science, Universite´ de Gabes,` Tunisia b,c Department of Mathematics, Baylor University Waco, TX 76798-7328, USA ∗Corresponding Author [email protected] Lance [email protected] Jessica [email protected]

Abstract In 2009, Gomez-Ullate,´ Kamran, and Milson characterized all sequences of ∞ polynomials {pn}n=1, with deg pn = n ≥ 1, that are eigenfunctions of a second- order differential equation and are orthogonal with respect to a positive Borel measure on the real line having ﬁnite moments of all orders. Up to a complex linear change of variable, the only such sequences are the X1-Laguerre and the X1-Jacobi polynomials. In this paper, we discuss the self-adjoint operator, generated by the second-order X1-Laguerre differential expression, that has the X1-Laguerre polynomials as eigenfunctions.

AMS Subject Classiﬁcations: 33C65, 34B20, 47B25. Keywords: X1-Laguerre polynomials, self-adjoint operator, Glazman-Krein-Naimark theory.

1 Introduction

∞ The classical orthogonal polynomials {pn}n=0 of Jacobi, Laguerre, and Hermite can be characterized, up to a complex linear change of variable, in several ways. Indeed, ∞ they are the only positive-deﬁnite orthogonal polynomials {pn}n=0 satisfying any of the following four equivalent conditions:

0 ∞ (a) the sequence of ﬁrst derivatives {pn(x)}n=1 is also orthogonal with respect to a positive measure;

Received , 20; Accepted , 20 Communicated by 26 M. J. Atia, L. L. Littlejohn, and J. Stewart

∞ (b) the polynomials {pn}n=0 satisfy a Rodrigues formula dn p (x) = K−1(w(x))−1 (ρn(x)w(x)) (n ∈ ), n n dxn N

where Kn > 0, w > 0 on some interval I = (a, b), and where ρ is a polynomial of degree ≤ 2;

∞ (c) the polynomials {pn}n=0 satisfy a differential recursion relation of the form

0 π(x)pn(x) = (αnx + βn)pn(x) + γnpn−1(x)(n ∈ N, p−1(x) = 0)

∞ ∞ ∞ where π(x) is some polynomial and {αn}n=0, {βn}n=0, and {γn}n=0 are real sequences;

(d) there exists a second-order differential expression

00 0 m[y](x) = b2(x)y (x) + b1(x)y (x) + b0(x)y(x),

∞ and a sequence of complex numbers {µn}n=0 such that y = pn(x) is a solution of

m[y](x) = µny(x)(n ∈ N0).

Three excellent sources for further information on the theory of orthogonal polynomials are the texts of Chihara [3], Ismail [15], and Szego¨ [26]. The characterization given in (d) above is generally attributed to S. Bochner [2] in 1929 and P. A. Lesky [20] in 1962. However, in recent years, it has become clear that the question was addressed earlier by E. J. Routh [25] in 1885; see [15, p. 509]. Extensions of characterization (d) to a real linear change of variable was considered by Kwon and Littlejohn [17] in 1997 and to orthogonality with respect to a Sobolev inner product by Kwon and Littlejohn [18] in 1998. In 2009, Gomez-Ullate,´ Kamran, and Milson [7] (see also [8–13]) characterized all ∞ polynomial sequences {pn}n=1 , with deg pn = n ≥ 1, which satisfy the following conditions:

(i) there exists a second-order differential expression

00 0 `[y](x) = a2(x)y (x) + a1(x)y (x) + a0(x)y(x),

∞ and a sequence of complex numbers {λn}n=1 such that y = pn(x) is a solution of

`[y](x) = λny(x)(n ∈ N);

each coefﬁcient ai(x) is a function of the independent variable x and does not depend on the degree of the polynomial eigenfunctions; The Spectral Analysis of the X1-Laguerre Polynomials 27

(ii) if c is any non-zero constant, y(x) ≡ c is not a solution of `[y](x) = λy(x) for any λ ∈ C; (iii) there exists an interval I and a Lebesgue measurable function w(x)(x ∈ I) such that Z pn(x)pm(x)w(x)dx = Knδn,m, I where Kn > 0 for each n ∈ N; ∞ (iv) all moments {µn}n=0 of w, deﬁned by Z n µn = x w(x)dx (n = 0, 1, 2,...), I exist and are ﬁnite.

Up to a complex linear change of variable, the authors in [7] show that the only solutions to this classification problem are the exceptional X1-Laguerre and the X1-Jacobi polynomials. Their results are spectacular and remarkable; indeed, it was believed, due to the ‘Bochner’ classification, that among the class of all orthogonal polynomials, only the Hermite, Laguerre, and Jacobi polynomials, satisfy second-order differential equations and are orthogonal with respect to a positive-definite inner product of the form Z (p, q) = pqdµ. R Even though the authors in [7] introduce the notion of exceptional polynomials via Sturm-Liouville theory, the path that they followed to their discovery was motivated by their interest in quantum mechanics, specifically with their intent to extend exactly solvable and quasi-exactly solvable potentials beyond the Lie algebraic setting. The im- petus that led to their work in [7] came from an earlier paper [16] of Kamran, Milson, and Olver on evolution equations reducible to finite-dimensional dynamical systems. It is important to note as well that the work in [7] was not originally motivated by orthogonal polynomials although they set out to construct potentials that would be solvable by polynomials which fall outside the realm of the classical theory of orthogonal polynomials. To further note, their work was inspired by the paper of Post and Turbiner [23] who formulated a generalized Bochner problem of classifying the linear differential operators in one variable leaving invariant a given vector space of polynomials. The X1-Laguerre and X1-Jacobi polynomials, as well as subsequent generalizations, are exceptional in the sense that they start at degree ` (` ≥ 1) instead of degree 0, thus avoiding the restrictions of the Bochner classification but still satisfy second-order differential equations of spectral type. Reformulation within the framework of one- dimensional quantum mechanics and shape invariant potentials followed their work by various other authors; for example, see [22] and [24]. Furthermore, the two second- order differential equations that they discover in their X1 classification are important 28 M. J. Atia, L. L. Littlejohn, and J. Stewart examples illustrating the Stone-von Neumann theory [4, Chapter 12] and the Glazman- Krein-Naimark theory [21, Section 18] of differential operators. In this paper, we will focus on the spectral theory associated with the X1-Laguerre polynomials; the main objective will be to fill in the details of the analysis briefly out- lined in [5] and [7].

2 Summary of Properties of X1-Laguerre Polynomials

ˆα ∞ We summarize some of the main properties of the X1-Laguerre polynomials {Ln}n=1, as discussed in [7]. Unless otherwise indicated, we assume that the parameter α > 0. ˆα The X1-Laguerre polynomial y = Ln(x) (n ∈ N) satisﬁes the second-order differential equation

`α[y](x) = λny(x) (0 < x < ∞), where (x − α)(x + α + 1) (x − α) ` [y](x) := −xy00 + y0 − y (2.1) α (x + α) (x + α) and λn = n − 1 (n ∈ N). (2.2) These polynomials are orthogonal on (0, ∞) with respect to the weight function

xαe−x wα (x) = (x ∈ (0, ∞)). (2.3) (x + α)2

The X1-Laguerre polynomials form a complete orthogonal set in the Hilbert-Lebesgue 2 space L ((0, ∞); wα), deﬁned by

Z ∞ 2 2 L ((0, ∞); wα) := {f : (0, ∞) → C |f is measurable and |f| wα < ∞}. 0 More speciﬁcally, with

Z ∞ 1/2 2 kfkα := |f(x)| wα(x)dx , 0 it is the case that

2 α + n Γ(α + n) ˆα Ln = (n ∈ N). α α + n − 1 (n − 1)!

ˆα ∞ 2 The fact that {Ln}n=1 forms a complete set L ((0, ∞); wα) is remarkable, and certainly 2 counterintuitive, since f(x) = 1 ∈ L ((0, ∞); wα). The Spectral Analysis of the X1-Laguerre Polynomials 29

The X1-Laguerre polynomials can be written in terms of the classical Laguerre poly- α ∞ nomials {Ln(x)}n=0; speciﬁcally,

ˆα α α Ln(x) = −(x + α + 1)Ln−1(x) + Ln−2(x)(n ∈ N),

α where L−1(x) = 0. Moreover, the X1-Laguerre polynomials satisfy the (non-standard) three-term recurrence relation:

2 ˆα 0 = (n + 1) (x + α) (n + α) − α Ln+2(x) 2 ˆα + (n + α) (x + α) (x − 2n − α − 1) + 2α Ln+1(x) 2 ˆα + (n + α − 1) (x + α) (n + α + 1) − α) Ln(x).

α ∞ However in stark contrast with the classical Laguerre polynomials {Ln}n=0, whose roots are all contained in the interval (0, ∞) when α > −1, the X1-Laguerre poly- ˆα ∞ nomials {Ln}n=1 have n − 1 roots in (0, ∞) and one zero in the interval (−∞, −α) whenever α > 0.

3 X1-Laguerre Spectral Analysis

The background material for this section can be found in the classic texts of Akhiezer and Glazman [1], Hellwig [14], and Naimark [21]. In Lagrangian symmetric form, the X1-Laguerre differential expression (2.1) is given by

α+1 −x 0 α −x 1 x e 0 x e (x − α) `α[y](x) = − 2 y (x) − 3 y(x) (x > 0), (3.1) wα(x) (x + α) (x + α) where wα(x) is the X1-Laguerre weight deﬁned in (2.3). 2 The maximal domain associated with `α[·] in the Hilbert space L ((0, ∞); wα) is deﬁned to be

0 2 ∆ := f : (0, ∞) → C | f, f ∈ ACloc(0, ∞); f, `α[f] ∈ L ((0, ∞); wα) .

The associated maximal operator

2 2 T1 : D(T1) ⊂ L ((0, ∞); wα) → L ((0, ∞); wα), is deﬁned by

T1f = `α[f] (3.2)

f ∈ D(T1) : = ∆. 30 M. J. Atia, L. L. Littlejohn, and J. Stewart

For f, g ∈ ∆, Green’s formula can be written as

Z ∞ Z ∞ x=∞ `α[f](x)g(x)wα(x)dx = [f, g](x) |x=0 + f(x)`α[g](x)wα(x)dx, 0 0 where [·, ·](·) is the sesquilinear form deﬁned by

xα+1e−x [f, g](x) := (f(x)g0(x) − f 0(x)g(x)) (0 < x < ∞), (3.3) (x + α)2 and where x=∞ [f, g](x) |x=0 := [f, g](∞) − [f, g](0). By deﬁnition of ∆ (and the classical Holder’s¨ inequality), notice that the limits

[f, g](0) := lim [f, g](x) and [f, g](∞) := lim [f, g](x) x→0+ x→∞ exist and are finite for each f, g ∈ ∆. The term ‘maximal’ comes from the fact that ∆ is the largest subspace of func- 2 2 tions in L ((0, ∞); wα) for which T1 maps back into L ((0, ∞); wα). It is not diffi- 2 cult to see that ∆ is dense in L ((0, ∞); wα); consequently, the adjoint T0 of T1 ex- 2 ists as a densely defined operator in L ((0, ∞); wα). For obvious reasons, T0 is called the minimal operator associated with `α[·]. From [1] or [21], this minimal operator 2 2 T0 : D(T0) ⊂ L ((0, ∞); wα) → L ((0, ∞); wα) is defined by

T0f = `α[f] (3.4) x=∞ f ∈ D(T0) : = {f ∈ ∆ | [f, g] |x=0 = 0 for all g ∈ ∆}.

2 The minimal operator T0 is a closed, symmetric operator in L ((0, ∞); wα); furthermore, because the coefficients of `[·] are real, T0 necessarily has equal deficiency indices m, where m is an integer satisfying 0 ≤ m ≤ 2. Therefore, from the general Stone-von Neumann [4] theory of self-adjoint extensions of symmetric operators, T0 has 2 self-adjoint extensions. We seek to find the self-adjoint extension T in L ((0, ∞); wα), ˆα ∞ generated by `[·], which has the X1-Laguerre polynomials {Ln}n=1 as eigenfunctions. In order to compute the deficiency indices, it is necessary to study the behavior of solutions of `[y] = 0 near each of the singular endpoints x = 0 and x = ∞.

3.1 Endpoint Behavior Analysis

The endpoint x = 0 is, in the sense of Frobenius, a regular singular endpoint of the X1- Laguerre equation `α[y] = λy, for any value of λ ∈ C. The Frobenius indicial equation at x = 0 is r(r + α) = 0. The Spectral Analysis of the X1-Laguerre Polynomials 31

Consequently, two linearly independent solutions of `α[y] = 0 will behave asymptoti- cally like −α z1(x) := 1 and z2(x) := x (0 < x ≤ 1) Now, for any α > 0, Z 1 2 |z1(x)| wα(x)dx < ∞; 0 however, Z 1 2 |z2(x)| wα(x)dx < ∞ 0 only when 0 < α < 1. In the vernacular of the Weyl limit-point/limit-circle analysis (see [14]), this Frobenius analysis shows that the X1-Laguerre differential expression is in the limit-circle case at x = 0 when 0 < α < 1 and in the limit-point case when α ≥ 1. The analysis at the endpoint x = ∞ is more complicated since x = ∞ is an irregu- lar singular endpoint of the X1-Laguerre differential expression; consequently, another asymptotic method must be employed. Fortunately, we are able to explicitly solve the differential equation `α[y](x) = 0 for a basis {y1(x), y2(x)} of solutions. Indeed,

y1(x) = x + α + 1 (x > 0) and, for ﬁxed but arbitrary x0 > 0,

Z x et (t + α)2 y2(x) = (x + α + 1) dt (x > 0) (3.5) α+1 2 x0 t (t + α + 1)

ˆα are independent solutions to `α[y](x) = 0. In fact, y1(x) = L1 (x) is the X1-Laguerre polynomial of degree 1, while y2(x) is obtained by the well-known reduction of order method. It is straightforward to see that

Z ∞ 2 |y1(x)| wα(x)dx < ∞. (3.6) 0

However,

Lemma 3.1. For any α > 0, the solution y2(x), deﬁned in (3.5), satisﬁes

Z ∞ 2 |y2(x)| wα(x)dx = ∞; (3.7) 1

2 that is to say, y2 ∈/ L ((1, ∞); wα). 32 M. J. Atia, L. L. Littlejohn, and J. Stewart

Proof. To begin, we note that, for x0 > 1,

Z x et (t + α)2 (x + α)2 Z x et dt ≥ 0 dt α+1 2 2 α+1 x0 t (t + α + 1) (x0 + α + 1) x0 t 2 Z x (x0 + α) t/2 ≥ 2 e dt for large enough x0 > 0 (x0 + α + 1) x0 x/2 ≥ Ae (x ≥ x1 ≥ x0) for large enough x1 ≥ x0 and where A is some positive constant. It follows that

2 Z x t 2 ! 2 2 e (t + α) |y2(x)| = (x + α + 1) dt α+1 2 x0 t (t + α + 1) 2 2 x ≥ A (x + α + 1) e for x ≥ x1.

Hence, for any α > 0, we see that Z ∞ Z ∞ 2 2 |y2(x)| wα(x)dx ≥ |y2(x)| wα(x)dx 1 x1 Z ∞ α 2 2 x (x + α + 1) ≥ A 2 dx x1 (x + α) Z ∞ ≥ A2 xαdx = ∞. x1

To summarize,

Theorem 3.2. For α > 0, let `α[·] be the X1-Laguerre differential expression, deﬁned in (2.1) or (3.1), on the interval (0, ∞).

(a) `α[·] is in the limit-point case at x = 0 when α ≥ 1 and is in the limit-circle case at x = 0 when 0 < α < 1;

(b) `α[·] is in the limit-point case at x = ∞ for any choice of α > 0.

Consequently,

2 Theorem 3.3. Let T0 be the minimal operator in L ((0, ∞); wα), deﬁned in (3.4), generated by the X1-Laguerre differential expression `α[·].

(a) If 0 < α < 1, the deﬁciency index of T0 is (1, 1);

(b) If α ≥ 1, the deﬁciency index of T0 is (0, 0). The Spectral Analysis of the X1-Laguerre Polynomials 33

3.2 Spectral Analysis for α ≥ 1 From Theorem 3.3(b), the next result follows from the general theory of self-adjoint extensions of symmetric operators [4, Chapter 12, Section 4] and results from Section 2 2, in particular the completeness of the X1-Laguerre polynomials in L ((0, ∞); wα) and the fact that λn → ∞ as n → ∞, where λn is deﬁned in (2.2). Theorem 3.4. For α ≥ 1, the maximal and minimal operators coincide. In this case, T0 = T1 is self-adjoint and is the only self-adjoint extension of the minimal operator T0 2 ˆα ∞ in L ((0, ∞); wα). Furthermore, in this case, the X1-Laguerre polynomials {Ln}n=1 are eigenfunctions of T0; the spectrum σ(T0) consists only of eigenvalues and is given by σ(T0) = N0. Hence, for α ≥ 1, no boundary condition restrictions of the maximal domain are needed to generate a self-adjoint extension of the minimal operator T0.

3.3 Spectral Analysis for 0 < α < 1 From Theorem 3.3, the general Glazman-Krein-Naimark theory [21, Section 18] implies that one appropriate boundary condition at x = 0 is needed in order to obtain a self- adjoint extension of the minimal operator T0. Necessarily, this boundary condition takes the form [f, g0](0) = 0 (f ∈ ∆), for some appropriately chosen g0 ∈ ∆ D(T0), where [·, ·](·) is the sesquilinear form deﬁned in (3.3). In this section, we show that g0(x) = 1 is such an appropriate function which generates the self-adjoint extension T having the X1-Laguerre polynomials ˆα ∞ {Ln}n=1 as eigenfunctions. Notice that, for any α > 0, the function 1 ∈ ∆. −α 2 The function y(x) = x ∈ L ((0, ∞); wα) if and only if 0 < α < 1. Remarkably, a calculation shows that −α −α `α[x ] = −(α + 1)x ; consequently, x−α ∈ ∆ when 0 < α < 1. Moreover, from (3.3), we ﬁnd that e−x 1 [x−α, 1](0) = α lim = 6= 0. x→0+ (x + α)2 α

Hence 1 ∈/ D(T0). Therefore, from the Glazman-Krein-Naimark Theorem [21, Section 18, Theorem 4], we obtain the following result.

2 Theorem 3.5. Suppose 0 < α < 1. The operator T : D(T ) ⊂ L ((0, ∞); wα) → 2 L ((0, ∞); wα), deﬁned by

T f = `α[f] f ∈ D(T ) : = {f ∈ ∆ | [f, 1](0) = 0} , 34 M. J. Atia, L. L. Littlejohn, and J. Stewart

2 ˆα ∞ is self-adjoint in L ((0, ∞); wα) and has the X1-Laguerre polynomials {Ln}n=1 as eigenfunctions. Moreover, the spectrum of T consists only of eigenvalues and is given by

σ(T ) = N0.

4 Final Remark: the Case α = 0

The X1-Laguerre weight function wα(x), deﬁned in (2.3), is deﬁned for α > 0. When −2 −x α = 0, this weight function reduces to w0(x) := x e . Consequently, α = 0 in the X1-Laguerre case corresponds to α = −2 in the ordinary Laguerre case. Further- more, in this situation, the X1-Laguerre equation (2.1) reduces to the ordinary Laguerre equation

00 0 `0[y](x) := −xy (x) + (x + 1)y (x) − y(x) = λy(x) (0 < x < ∞). (4.1)

−2 For each n ∈ N0, the Laguerre polynomial y(x) = Ln (x) is a solution of `0[y](x) = −2 ∞ (n − 1)y(x). The Laguerre polynomials {Ln }n=2 (of degree ≥ 2) form a complete or- 2 thogonal set in the Hilbert space L ((0, ∞); w0); moreover, for i = 0, 1, the polynomials −2 2 2 Li ∈/ L ((0, ∞); w0). The spectral theory for this Laguerre case, in L ((0, ∞); w0), was established in [6]; speciﬁcally, the authors in [6] develop the self-adjoint operator 2 in L ((0, ∞); w0), generated by the Laguerre expression (4.1), which has the Laguerre −2 ∞ polynomials {Ln }n=2 as eigenfunctions. These authors also discuss the spectral theory −2 ∞ of (4.1) which has the entire sequence of Laguerre polynomials {Ln }n=0 as eigenfunctions. The analysis, in this case, is in the Hilbert-Sobolev space (S, (·, ·)), where

0 00 2 −x S = {f : [0, ∞) → C | f, f ∈ ACloc[0, ∞), f ∈ L ((0, ∞); e )}, and where (·, ·) is the positive-deﬁnite inner product deﬁned by

Z ∞ (f, g) = 3f(0)g(0) − f 0(0)g(0) − f(0)g0(0) + f 00(x)g00(x)e−xdx. 0

The authors ﬁnd, and discuss, the self-adjoint operator in (S, (·, ·)) having the Laguerre −2 ∞ polynomials {Ln }n=0 as a complete orthogonal set of eigenfunctions. The key to es- tablishing this analysis of (4.1) in (S, (·, ·)) is the general left-deﬁnite spectral theory developed by Littlejohn and Wellman in [19]. Acknowledgement. One of the authors, Mohamed J. Atia, gratefully acknowledges support from the Fulbright Program, sponsored by the United States Department of State and the Bureau of Educational and Cultural Affairs, during his visit to Baylor University in the fall semester of 2012. The Spectral Analysis of the X1-Laguerre Polynomials 35

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