The Spectral Theory of the X 1 -Laguerre Polynomials

Advances in Dynamical Systems and Applications ISSN 973-5321, Volume x, Number x, pp. 25 36 (2xx) http://campus.mst.edu/adsa The Spectral Theory of the X 1 -Laguerre Polynomials Mohamed J. Atia a, Lance L. Littlejohn b, and Jessica Stewart c a Faculté de Science, Université de Gabès, Tunisia b,c Department of Mathematics, Baylor University Waco, TX 76798-7328, USA Corresponding Author jalel.atia@gmail.com Lance Littlejohn@baylor.edu Jessica Stewart@baylor.edu Abstract In 29, Gómez-Ullate, Kamran, and Milson characterized all sequences of polynomials {p n } n=1, with deg p n = n 1, that are eigenfunctions of a secondorder 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 X 1 -Laguerre and the X 1 -Jacobi polynomials. In this paper, we discuss the self-adjoint operator, generated by the second-order X 1 -Laguerre differential expression, that has the X 1 -Laguerre polynomials as eigenfunctions. AMS Subject Classifications: 33C65, 34B2, 47B25. Keywords: X 1 -Laguerre polynomials, self-adjoint operator, Glazman-Krein-Naimark theory. 1 Introduction The classical orthogonal polynomials {p n } n= 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 {p n } n= satisfying any of the following four equivalent conditions: (a) the sequence of first derivatives {p n(x)} n=1 is also orthogonal with respect to a positive measure; Received, 2; Accepted, 2 Communicated by

26 M. J. Atia, L. L. Littlejohn, and J. Stewart (b) the polynomials {p n } n= satisfy a Rodrigues formula p n (x) = Kn 1 1 dn (w(x)) dx n (ρn (x)w(x)) (n N), where K n >, w > on some interval I = (a, b), and where ρ is a polynomial of degree 2; (c) the polynomials {p n } n= satisfy a differential recursion relation of the form π(x)p n(x) = (α n x + β n )p n (x) + γ n p n 1 (x) (n N, p 1 (x) = ) where π(x) is some polynomial and {α n } n=, {β n } n=, and {γ n } n= are real sequences; (d) there exists a second-order differential expression m[y](x) = b 2 (x)y (x) + b 1 (x)y (x) + b (x)y(x), and a sequence of complex numbers {µ n } n= such that y = p n (x) is a solution of m[y](x) = µ n y(x) (n N ). Three excellent sources for further information on the theory of orthogonal polynomials are the texts of Chihara [3], Ismail [15], and Szegö [26]. The characterization given in (d) above is generally attributed to S. Bochner [2] in 1929 and P. A. Lesky [2] 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. 59]. 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 29, Gómez-Ullate, Kamran, and Milson [7] (see also [8 13]) characterized all polynomial sequences {p n } n=1, with deg p n = n 1, which satisfy the following conditions: (i) there exists a second-order differential expression l[y](x) = a 2 (x)y (x) + a 1 (x)y (x) + a (x)y(x), and a sequence of complex numbers {λ n } n=1 such that y = p n (x) is a solution of l[y](x) = λ n y(x) (n N); each coefficient a i (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 X 1 -Laguerre Polynomials 27 (ii) if c is any non-zero constant, y(x) c is not a solution of l[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 p n (x)p m (x)w(x)dx = K n δ n,m, where K n > for each n N; I (iv) all moments {µ n } n= of w, defined by µ n = x n w(x)dx (n =, 1, 2,...), exist and are finite. I Up to a complex linear change of variable, the authors in [7] show that the only solutions to this classification problem are the exceptional X 1 -Laguerre and the X 1 -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 (p, q) = pqdµ. 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 impetus 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 X 1 -Laguerre and X 1 -Jacobi polynomials, as well as subsequent generalizations, are exceptional in the sense that they start at degree l (l 1) instead of degree, thus avoiding the restrictions of the Bochner classification but still satisfy second-order differential equations of spectral type. Reformulation within the framework of onedimensional quantum mechanics and shape invariant potentials followed their work by various other authors; for example, see [22] and [24]. Furthermore, the two secondorder differential equations that they discover in their X 1 classification are important R

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 X 1 -Laguerre polynomials; the main objective will be to fill in the details of the analysis briefly outlined in [5] and [7]. 2 Summary of Properties of X 1 -Laguerre Polynomials We summarize some of the main properties of the X 1 -Laguerre polynomials {ˆL α n} n=1, as discussed in [7]. Unless otherwise indicated, we assume that the parameter α >. The X 1 -Laguerre polynomial y = ˆL α n(x) (n N) satisfies the second-order differential equation l α [y](x) = λ n y(x) ( < x < ), where and l α [y](x) := xy + (x α) (x + α + 1) y (x α) (x + α) (x + α) y (2.1) λ n = n 1 (n N). (2.2) These polynomials are orthogonal on (, ) with respect to the weight function w α (x) = xα e x (x + α) 2 (x (, )). (2.3) The X 1 -Laguerre polynomials form a complete orthogonal set in the Hilbert-Lebesgue space L 2 ((, ); w α ), defined by L 2 ((, ); w α ) := {f : (, ) C f is measurable and f 2 w α < }. More specifically, with ( 1/2 f α := f(x) 2 w α (x)dx), it is the case that ˆL α n 2 = α ( α + n ) Γ(α + n) α + n 1 (n 1)! (n N). The fact that {ˆL α n} n=1 forms a complete set L 2 ((, ); w α ) is remarkable, and certainly counterintuitive, since f(x) = 1 L 2 ((, ); w α ).

The Spectral Analysis of the X 1 -Laguerre Polynomials 29 The X 1 -Laguerre polynomials can be written in terms of the classical Laguerre polynomials {L α n(x)} n= ; specifically, ˆL α n(x) = (x + α + 1)L α n 1(x) + L α n 2(x) (n N), where L α 1(x) =. Moreover, the X 1 -Laguerre polynomials satisfy the (non-standard) three-term recurrence relation: = (n + 1) ( (x + α) 2 (n + α) α ) ˆLα n+2 (x) + (n + α) ( (x + α) 2 (x 2n α 1) + 2α ) ˆLα n+1 (x) + (n + α 1) ( (x + α) 2 (n + α + 1) α) ) ˆLα n (x). However in stark contrast with the classical Laguerre polynomials {L α n} n=, whose roots are all contained in the interval (, ) when α > 1, the X 1 -Laguerre polynomials {ˆL α n} n=1 have n 1 roots in (, ) and one zero in the interval (, α) whenever α >. 3 X 1 -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 X 1 -Laguerre differential expression (2.1) is given by l α [y](x) = 1 ( ( ) x α+1 e x ) w α (x) (x + α) 2 y (x) xα e x (x α) y(x) (x + α) 3 (x > ), (3.1) where w α (x) is the X 1 -Laguerre weight defined in (2.3). The maximal domain associated with l α [ ] in the Hilbert space L 2 ((, ); w α ) is defined to be := { f : (, ) C f, f AC loc (, ); f, l α [f] L 2 ((, ); w α ) }. The associated maximal operator is defined by T 1 : D(T 1 ) L 2 ((, ); w α ) L 2 ((, ); w α ), T 1 f = l α [f] (3.2) f D(T 1 ) : =.

3 M. J. Atia, L. L. Littlejohn, and J. Stewart For f, g, Green s formula can be written as l α [f](x)g(x)w α (x)dx = [f, g](x) x= x= + where [, ]( ) is the sesquilinear form defined by and where f(x)l α [g](x)w α (x)dx, [f, g](x) := xα+1 e x (x + α) 2 (f(x)g (x) f (x)g(x)) ( < x < ), (3.3) [f, g](x) x= x= := [f, g]( ) [f, g](). By definition of (and the classical Hölder s inequality), notice that the limits [f, g]() := lim x +[f, g](x) and [f, g]( ) := lim [f, g](x) x exist and are finite for each f, g. The term maximal comes from the fact that is the largest subspace of functions in L 2 ((, ); w α ) for which T 1 maps back into L 2 ((, ); w α ). It is not difficult to see that is dense in L 2 ((, ); w α ); consequently, the adjoint T of T 1 exists as a densely defined operator in L 2 ((, ); w α ). For obvious reasons, T is called the minimal operator associated with l α [ ]. From [1] or [21], this minimal operator T : D(T ) L 2 ((, ); w α ) L 2 ((, ); w α ) is defined by T f = l α [f] (3.4) f D(T ) : = {f [f, g] x= x= = for all g }. The minimal operator T is a closed, symmetric operator in L 2 ((, ); w α ); furthermore, because the coefficients of l[ ] are real, T necessarily has equal deficiency indices m, where m is an integer satisfying m 2. Therefore, from the general Stone-von Neumann [4] theory of self-adjoint extensions of symmetric operators, T has self-adjoint extensions. We seek to find the self-adjoint extension T in L 2 ((, ); w α ), generated by l[ ], which has the X 1 -Laguerre polynomials {ˆL α n} n=1 as eigenfunctions. In order to compute the deficiency indices, it is necessary to study the behavior of solutions of l[y] = near each of the singular endpoints x = and x =. 3.1 Endpoint Behavior Analysis The endpoint x = is, in the sense of Frobenius, a regular singular endpoint of the X 1 - Laguerre equation l α [y] = λy, for any value of λ C. The Frobenius indicial equation at x = is r(r + α) =.

The Spectral Analysis of the X 1 -Laguerre Polynomials 31 Consequently, two linearly independent solutions of l α [y] = will behave asymptotically like z 1 (x) := 1 and z 2 (x) := x α ( < x 1) Now, for any α >, however, 1 1 z 1 (x) 2 w α (x)dx < ; z 2 (x) 2 w α (x)dx < only when < α < 1. In the vernacular of the Weyl limit-point/limit-circle analysis (see [14]), this Frobenius analysis shows that the X 1 -Laguerre differential expression is in the limit-circle case at x = when < α < 1 and in the limit-point case when α 1. The analysis at the endpoint x = is more complicated since x = is an irregular singular endpoint of the X 1 -Laguerre differential expression; consequently, another asymptotic method must be employed. Fortunately, we are able to explicitly solve the differential equation l α [y](x) = for a basis {y 1 (x), y 2 (x)} of solutions. Indeed, and, for fixed but arbitrary x >, y 1 (x) = x + α + 1 (x > ) x e t 2 (t + α) y 2 (x) = (x + α + 1) x t α+1 2 dt (x > ) (3.5) (t + α + 1) are independent solutions to l α [y](x) =. In fact, y 1 (x) = ˆL α 1 (x) is the X 1 -Laguerre polynomial of degree 1, while y 2 (x) is obtained by the well-known reduction of order method. It is straightforward to see that However, y 1 (x) 2 w α (x)dx <. (3.6) Lemma 3.1. For any α >, the solution y 2 (x), defined in (3.5), satisfies that is to say, y 2 / L 2 ((1, ); w α ). 1 y 2 (x) 2 w α (x)dx = ; (3.7)

32 M. J. Atia, L. L. Littlejohn, and J. Stewart Proof. To begin, we note that, for x > 1, x 2 e t (t + α) x t α+1 (t + α + 1) 2 dt (x + α) 2 (x + α + 1) 2 (x + α) 2 (x + α + 1) 2 x x x e t dt tα+1 e t/2 dt for large enough x > x Ae x/2 (x x 1 x ) for large enough x 1 x and where A is some positive constant. It follows that ( x y 2 (x) 2 = (x + α + 1) 2 e t 2 (t + α) x t α+1 (t + α + 1) 2 dt Hence, for any α >, we see that 1 A 2 (x + α + 1) 2 e x for x x 1. y 2 (x) 2 w α (x)dx x 1 y 2 (x) 2 w α (x)dx ) 2 A 2 x α (x + α + 1) 2 dx x 1 (x + α) 2 A 2 x α dx =. x 1 To summarize, Theorem 3.2. For α >, let l α [ ] be the X 1 -Laguerre differential expression, defined in (2.1) or (3.1), on the interval (, ). (a) l α [ ] is in the limit-point case at x = when α 1 and is in the limit-circle case at x = when < α < 1; (b) l α [ ] is in the limit-point case at x = for any choice of α >. Consequently, Theorem 3.3. Let T be the minimal operator in L 2 ((, ); w α ), defined in (3.4), generated by the X 1 -Laguerre differential expression l α [ ]. (a) If < α < 1, the deficiency index of T is (1, 1); (b) If α 1, the deficiency index of T is (, ).

The Spectral Analysis of the X 1 -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, in particular the completeness of the X 1 -Laguerre polynomials in L 2 ((, ); w α ) and the fact that λ n as n, where λ n is defined in (2.2). Theorem 3.4. For α 1, the maximal and minimal operators coincide. In this case, T = T 1 is self-adjoint and is the only self-adjoint extension of the minimal operator T in L 2 ((, ); w α ). Furthermore, in this case, the X 1 -Laguerre polynomials {ˆL α n} n=1 are eigenfunctions of T ; the spectrum σ(t ) consists only of eigenvalues and is given by σ(t ) = N. Hence, for α 1, no boundary condition restrictions of the maximal domain are needed to generate a self-adjoint extension of the minimal operator T. 3.3 Spectral Analysis for < α < 1 From Theorem 3.3, the general Glazman-Krein-Naimark theory [21, Section 18] implies that one appropriate boundary condition at x = is needed in order to obtain a selfadjoint extension of the minimal operator T. Necessarily, this boundary condition takes the form [f, g ]() = (f ), for some appropriately chosen g D(T ), where [, ]( ) is the sesquilinear form defined in (3.3). In this section, we show that g (x) = 1 is such an appropriate function which generates the self-adjoint extension T having the X 1 -Laguerre polynomials {ˆL α n} n=1 as eigenfunctions. Notice that, for any α >, the function 1. The function y(x) = x α L 2 ((, ); w α ) if and only if < α < 1. Remarkably, a calculation shows that l α [x α ] = (α + 1)x α ; consequently, x α when < α < 1. Moreover, from (3.3), we find that [x α e x, 1]() = α lim x + (x + α) = 1 2 α. Hence 1 / D(T ). Therefore, from the Glazman-Krein-Naimark Theorem [21, Section 18, Theorem 4], we obtain the following result. Theorem 3.5. Suppose < α < 1. The operator T : D(T ) L 2 ((, ); w α ) L 2 ((, ); w α ), defined by T f = l α [f] f D(T ) : = {f [f, 1]() = },

34 M. J. Atia, L. L. Littlejohn, and J. Stewart is self-adjoint in L 2 ((, ); w α ) and has the X 1 -Laguerre polynomials {ˆL α n} n=1 as eigenfunctions. Moreover, the spectrum of T consists only of eigenvalues and is given by σ(t ) = N. 4 Final Remark: the Case α = The X 1 -Laguerre weight function w α (x), defined in (2.3), is defined for α >. When α =, this weight function reduces to w (x) := x 2 e x. Consequently, α = in the X 1 -Laguerre case corresponds to α = 2 in the ordinary Laguerre case. Furthermore, in this situation, the X 1 -Laguerre equation (2.1) reduces to the ordinary Laguerre equation l [y](x) := xy (x) + (x + 1)y (x) y(x) = λy(x) ( < x < ). (4.1) For each n N, the Laguerre polynomial y(x) = L 2 n (x) is a solution of l [y](x) = (n 1)y(x). The Laguerre polynomials {L 2 n } n=2 (of degree 2) form a complete orthogonal set in the Hilbert space L 2 ((, ); w ); moreover, for i =, 1, the polynomials L 2 i / L 2 ((, ); w ). The spectral theory for this Laguerre case, in L 2 ((, ); w ), was established in [6]; specifically, the authors in [6] develop the self-adjoint operator in L 2 ((, ); w ), generated by the Laguerre expression (4.1), which has the Laguerre polynomials {L 2 n } n=2 as eigenfunctions. These authors also discuss the spectral theory of (4.1) which has the entire sequence of Laguerre polynomials {L 2 n } n= as eigenfunctions. The analysis, in this case, is in the Hilbert-Sobolev space (S, (, )), where S = {f : [, ) C f, f AC loc [, ), f L 2 ((, ); e x )}, and where (, ) is the positive-definite inner product defined by (f, g) = 3f()g() f ()g() f()g () + f (x)g (x)e x dx. The authors find, and discuss, the self-adjoint operator in (S, (, )) having the Laguerre polynomials {L 2 n } n= as a complete orthogonal set of eigenfunctions. The key to establishing 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 212.

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