The Prolate Spheroidal Phenomenon as a Consequence of Bispectrality

Transcription

1 Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume 37, 2004 The Prolate Spheroidal Phenomenon as a Consequence of Bispectrality F. Alberto Grünbaum and Milen Yakimov Abstract. In this paper we announce that a very large class of integral operators derived from bispectral algebras of rank 1 and 2 parametrized by Lagrangian Grassmannians of infinitely large size) posses commuting differential operators. The examples of Landau, Pollak, Slepian, and Tracy, Widom, used in time-band limiting and random matrix theory arise as special cases of this result. 1. Introduction It was discovered by Landau, Pollak, Slepian, and Tracy, Widom, that certain integral operators associated to the Airy and Bessel special functions possess commuting differential operators. They found important applications of this to time-band limiting, and to the study of asymptotics of Fredholm determinants, relevant to scaling limits of random matrix models. We call this phenomenon the prolate spheroidal phenomenon. On the other hand, the problem of bispectrality was posed [7] about 20 years ago by one of us A. G.) and J. J. Duistermaat as a tool to understand this prolate spheroidal property of integral operators. The aim was to extend it to larger classes and search for possible applications. Despite the dramatic recent developments in the areas of random matrices and bispectrality the two problems remained isolated except for several common examples, see [9 11]. In addition only a few integral operators possessing a commuting differential operator were found. Here we announce that any self-adjoint bispectral algebra of ordinary differential operators see Definition 3.3 and Definition 4.3 and Section 2 for general definitions) of rank 1 and 2 induces an integral operator possessing the prolate spheroidal property. The kernel of such an operator is of the form Kx, z) = Ψx, z)ψy, z) dz Γ 2 where Ψx, z) is the corresponding bispectral eigen)function and Γ 2 is a contour in the complex plane with 1 or 2 end points. It acts on the space L 2 Γ 1 ) again for 2000 Mathematics Subject Classification. Primary 47G10; Secondary 15A52, 37K10. This is the final form of the paper. 303 c 2004 American Mathematical Society

2 304 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV a contour with the same property. The main results are stated in Theorem 3.8 and Theorem 4.8. The integral operators of Landau, Pollak, Slepian, and Tracy, Widom correspond to the cases Ψx, z) = xzj ν+1/2 ixz) and Ψx, z) = Ax + z) in terms of the Bessel functions of first kind and the Airy function) which are known to be basic bispectral functions, in the sense that the other rank 1 and 2 bispectral functions are obtained from them by certain types of Darboux transformations see [3, 4, 7, 14, 23]). Although in these cases [15, 16, 19 21] the commuting differential operator is of order 2, in general it is of arbitrarily large order. In the rest of the introduction we describe our strategy for proving Theorem 3.8 and Theorem 4.8 which relies on a very interesting property of the size of bispectral algebras of rank 1 and 2. Consider, more generally, a holomorphic function Ψx, z) in some domain of C C which is not an eigenfunction of any differential operator in x or z. Denote by B Ψ the algebra of differential operators Rx, x ) with rational coefficients for which there exists an operator Sz, z ) with rational coefficients such that 1.1) Rx, x )Ψx, z) = Sz, z )Ψx, z). The algebra of all differential operators Sz, z ) obtained in this way will be denoted by C Ψ. The equality b Ψ Rx, x ) ) := Sz, z ) correctly defines an anti-isomorphism from B Ψ to C Ψ. Recall that such a function Ψx, z) is called bispectral if both algebras B Ψ and C Ψ contain rational functions. The subalgebra of B Ψ and C Ψ, consisting of differential operators in x and z for which Ψx, z) is an eigenfunction, are commutative. Algebras obtained in this way are called bispectral algebras. Their rank which is equal and is also called rank of the bispectral function Ψx, z)) is the greatest common divisor of the orders of the operators of these algebras. We derive our main result from the following remarkable property: Property. Consider the Z + Z + filtration of the algebra B Ψ given by B l1,l2 Ψ = {Rx, x ) B Ψ ord Rx, x ) 2l 1, ordb Ψ R)x, x ) 2l 2 }. If Ψx, z) is a bispectral function of rank r = 1 or 2 then the the size of the spaces B l1,l2 Ψ in this filtration is large in the sense dim B l1,l2 Ψ 2 r 2l 1l 2 + l 1 + l 2 ) const where the constant is independent of l 1 and l 2. For the basic bispectral functions expxz), Ax + z), and xzj ν+1/2 ixz), ν C\Z of rank 1 and 2, the dimension of these spaces is exactly equal to the righthand side with const = 1. This is remarkable since it shows that for all bispectral functions Ψx, z) of rank 1 and 2 the spaces of the above natural filtration of the algebra B Ψ are almost of the same dimension as the spaces corresponding to the basic bispectral functions. It is very interesting to understand the relation between our results and the approach of Adler, Shiota, and van Moerbeke [1, 2] to the Tracy Widom system of differential equations via Virasoro constraints. One possible way would be to

3 THE PROLATE SPHEROIDAL PHENOMENON AS A CONSEQUENCE OF BISPECTRALITY305 incorporate the representation theoretic meaning of the Bessel tau functions from [5] in terms of representations of the W 1+ algebra, see [8]. Another important problem is to understand the relation of this work to the isomonodromic deformations approach to random matrices from the works of Palmer [18], Harnad, Tracy, and Widom [13,22], Its and Harnad [12], and Borodin and Deift [6]. The proofs of the results announced here will appear in a forthcoming publication. Acknowledgments. M. Y. would like to thank the organizers of the Workshop on Superintegrability, September 2002, CRM, Montreal and especially John Harnad and Pavel Winternitz for the opportunity to participate at the conference and for their interest in this work. 2. Bispectrality and commutativity Denote by W rat the algebra of differential operators in one variable with rational coefficients. Denote by a ) the formal adjoint of operators in W rat : n ) n 2.1) a b k x) x n = x ) n b k x). k=0 For any ξ C consider the map from the subalgebra of W rat consisting of operators regular at ξ to the set of bidifferential operators at ξ n ) n k 1 2.2) φ ξ b k x) x n = 1) i x k i 1 xb i k x). x=ξ k=0 k=0 i=0 For an oriented contour Γ in C we will denote its endpoints by eγ). For any ξ eγ) set πξ) = 1 or 0 depending on whether ξ is a left or right endpoint of Γ. Assume that n Dx, x ) = b k x) x n W rat k=0 is a differential operator which is regular along Γ. Let fx) and gx) be smooth functions on Γ that decrease rapidly when x i.e., lim x px)f k) x) = 0 for any polynomial px) and any integer k, similarly for gx)). By standard integration by parts one gets that 2.3) Dx, x )fx) ) gx) dx = 1) πξ) φ ξ Dx, x ) ) fx) gx) ) Γ ξ eγ) + fx) adx, x )gx) ) dx. Γ Let Ψx, z) be a holomorphic function in some domain of C C which is not an eigenfunction of a differential operator in x or z, as in the introduction. Recall the definition 1.1) of the algebras of differential operators B Ψ and C Ψ. Assume that B Ψ and C Ψ are two subalgebras of B Ψ and C Ψ, respectively, that are stable under the formal adjoint map 2.1) and such that k=0 b Ψ B Ψ ) = C Ψ.

4 306 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV Proposition 2.1. Let Γ 1 and Γ 2 be two contours in C such that Γ 1 Γ 2 is in the domain of Ψx, z) and Ψx, z) decreases rapidly when x or z go to along Γ 1 and Γ 2. If Dx, x ) B Ψ is such that 2.4) ab Ψ Dx, x ) ) = b Ψ a Dx, x ) ) and 2.5) φ ξ Dx, x )) = 0, ξ eγ 1 ); φ ξ bd)z, z )) = 0, ξ eγ 2 ) then the integral operator with kernel 2.6) Kx, y) = Ψx, z)ψy, z) dz Γ 2 on L 2 Γ 1 ) commutes with the differential operator Dx, x ) with domain all smooth functions on Γ 1 that decrease rapidly as x. Let us call a differential operator Dx, x ) W rat formally symmetric if and formally skewsymmetric if ad)x, x ) = Dx, x ) ad)x, x ) = Dx, x ). Lemma 2.2. i) A differential operator Dx, x ) W rat is formally symmetric if and only if it has the form n 2.7) Dx, x ) = x n c i x) x n i=0 for some integer n and some rational functions c i x). ii) The operator Dx, x ) given by 2.7) satisfies φ ξ Dx, x ) ) = 0 for some fixed ξ C if and only if i xc k )ξ) = 0 for k = 1,..., n, i = 0,..., k 1. For a given function Ψx, z) as before let B Ψ,sym be the subalgebra of B Ψ consisting of differential operators Rx, x ) for which both ) Rx, x ) and b Ψ R)z, z ) are formally symmetric. Set also C Ψ,sym := b Ψ BΨ,sym. Define the vector spaces 2.8) 2.9) B l1,l2 Ψ,sym = {Rx, x) B Ψ,sym ord Rx, x ) 2l 1 C l1,l2 Ψ,sym = {Sz, z) C Ψ,sym ordb 1 Ψ S)x, x) 2l 1 Clearly C l1,l2 Ψ,sym = b ΨB l1,l2 Ψ,sym ). From Proposition 2.1 and Lemma 2.2 we obtain: and ordb Ψ R)z, z ) 2l 2 }, and ord Sz, z ) 2l 2 }. Theorem 2.3. Assume the conditions from Proposition 2.1 for the the function Ψx, z) and the contours Γ 1 and Γ 2. If either of the following two conditions is satisfied then the integral operator with kernel 2.6) possesses a formally commuting symmetric differential operator of order less than or equal to 2l 1 and domain the space of rapidly decreasing smooth functions on Γ 1

5 THE PROLATE SPHEROIDAL PHENOMENON AS A CONSEQUENCE OF BISPECTRALITY307 Condition i) B l1,l2 Ψ,sym > l 1l 1 + 1)eΓ 1 )/2 + l 2 l 2 + 1)eΓ 2 )/2 + 1, Condition ii) eγ 1 ) = eγ 1 ), eγ 2 ) = eγ 2 ), all operators in B Ψ,sym are invariant under the transformation x x, and set B l1,l2 Ψ,sym > l 1l 1 + 1)eΓ 1 )/4 + l 2 l 2 + 1)eΓ 2 )/ Integral operators associated to self-adjoint Darboux transformations of Airy functions 3.1. The Airy bispectral function. Denote by Ax) the Airy function and 3.1) Ψ A x, z) = Ax + z). Recall that Ax) decreases rapidly when x in the sector π/3 < arg x < π/3. If L A x, x ) denotes the Airy differential operator then Ψ A x, z) satisfies L A x, x ) = 2 x x L A x, x )Ψ A x, z) = zψ A x, z), x Ψ A x, z) = z Ψ A x, z), xψ A x, z) = L A z, z )Ψ A x, z). For shortness denote the algebras B ΨA and C ΨA of differential operators with rational coefficients associated to the Airy function Ψ A x, z), recall 1.1), by B A and C A. It is straightforward to deduce: Lemma 3.1. The algebras B A and C A coincide with the Weyl algebra W poly of differential operators in one variable with polynomial coefficients. Moreover the anti-isomorphism b A associated to the Airy function Ψ A x, z), recall 3.1), is uniquely defined from the relations b A x) = L A z, z ) ), b A x ) = z, b A LA x, x ) ) = z Self-adjoint Darboux transformations from the Airy function. Note that C[x] = B A Cx) and C[L A x, x )] = b 1 A C A Cz)). The set of rational Darboux transformations D A from the Airy function was defined in [4] as the set of functions Ψx, z) for which there exist differential operators 3.2) P x, x ), Qx, x ) C[x]\0) 1 B A = W rat such that 3.3) 3.4) fl A x, x )) = Qx, x )P x, x ), Ψx, z) = 1 pz) P x, x)ψ A x, z),

6 308 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV for some polynomials ft) and pz). The polynomial pz) is included for normalization purposes only.) The quotient ring of B A by C[x]\{0} in 3.2) is well defined since C[x]\{0} is an Ore subset of B A, see [17]. It was also shown in [4, 14] and more conceptually proved in [3] that: Theorem 3.2. All rational Darboux transformations from the Airy function Ψx, z) are bispectral functions of rank 2. Definition 3.3. Define the set SD A of self-adjoint Darboux transformations from the Airy function Ψx, z) to consist of those functions Ψx, z) for which there exists a differential operator P x, x ) W rat such that 3.5) 3.6) gl A x, x )) 2 = ap )x, x )P x, x ), Ψx, z) = 1 gz) P x, x)ψ A x, z), for some polynomial gt). In fact SD A consists exactly of those Ψx, z) D A for which Qx, x ) = ap )x, x ) in 3.3) 3.4) with an appropriate normalization of the polynomial pz). One can show that as a consequence ft) is the square of some polynomial gt), compare to 3.5) 3.6) Size of the algebra B A relative to the anti-isomorphism b A. Consider the Z + Z + filtration of the algebra B A associated to the Airy function Ψ A x, z), defined analogously to 2.8) by 3.7) B l1,l2 A = {Rx, x ) B A ord Rx, x ) 2l 1, ordb A R)z, z ) 2l 2 }. Lemma 3.4. The vector space B l1,l2 A has a basis that consists of the differential operators { x m L A x, x ) ) n } { n l1, m l 2 x m x LA x, x ) ) n } n < l1, m < l 2. Note that the formal adjoint anti-involution a of W rat preserves the spaces B l1,l2 A. Since a2 = id the space B l1,l2 A is the direct sum of the eigenspaces of a with eigenvalues ±1. Denote the eigenvalue 1 subspace of B l1,l2 A by B l1,l2 A,sym. For the Airy function Ψ A x, z) one has ab A = b A a and thus: B l1,l2 A,sym = {Rx, x) B A ord Rx, x ) 2l 1, ordb A R)z, z ) 2l 2, arx, x ) = Rx, x ), ab A Rz, z )) = b A Rz, z )}. Lemma 3.5. The set of operators { 3.8) x m L A x, x ) ) n + LA x, x ) ) n x m } n l 1, m l 2 is a basis for the space B l1,l2 A,sym. In particular dim B l1,l2 A,sym = l 1 + 1)l 2 + 1).

7 THE PROLATE SPHEROIDAL PHENOMENON AS A CONSEQUENCE OF BISPECTRALITY Size of the algebra B Ψ,sym for a self-adjoint Darboux transformation from the Airy function, relative to the involution b Ψ. Fix an arbitrary self-adjoint Darboux transformation from the Airy function Ψ SD A given by 3.5) 3.6) for some P x, x ) W rat and gt) C[t]. Let 3.9) P x, x ) = 1 vx) Rx, x) for some Rx, x ) W poly = B A and vx) C[x]. Set 3.10) ord Rx, x ) = ρ 1 and ordb A R)x, x ) = ρ 2. Denote { } 1 S Ψ,1 = Span vx) Rx, 1 x)mx, x )ar)x, x ) vx) Mx, x) B l1 ρ1,l2 A,sym, S Ψ,2 = Span { } vx)mx, x )vx) Mx, x ) B l1,l2 ρ2 A,sym. Proposition 3.6. In the above setting: i) The spaces of differential operators S Ψ,1 and S Ψ,2 are subspaces of B l1,l2 Ψ,sym. ii) The dimension of the intersection S Ψ,1 S Ψ,2 is less than or equal to l 1 ρ 1 + 1)l 2 ρ 2 + 1). Theorem 4.2 of [3] shows that 3.5) 3.6) is equivalent to vl A z, z )) 2 1 = ab A R)z, z ) gz) 2 b AR)z, z ), Ψx, z) = 1 vx)gz) b AR)z, z )Ψ A x, z). The hard step is to prove the first equality.) One can show that ) 1 b Ψ vx) Rx, 1 x)mx, x )ar)x, x ) = gz)b A M)z, z )gz) vx) for all operators Mx, x ) B A. The fact that S Ψ,2 is a subspace of B l1,l2 Ψ,sym follows from this and the fact that S Ψ,1 B l1,l2 Ψ,sym by exchanging the roles of x and z. Theorem 3.7. For any self-adjoint Darboux transformation from the Airy function Ψx, z) SD A the dimension of the space of differential operators S Ψ,1 + S Ψ,2 is greater than or equal to l 1 + 1)l 2 + 1) ρ 1 ρ 2. In particular 3.11) dim B l1,l2 Ψ,sym l 1 + 1)l 2 + 1) ρ 1 ρ 2. Theorem 3.7 and Theorem 2.3 imply our final result for integral operators derived from Darboux transformations from the Airy function. Theorem 3.8. Let Ψx, z) SD A be a self-adjoint Darboux transformation from the Airy function, given by 3.5), 3.6), 3.9). Let Γ 1, Γ 2 be two connected contours in C that do not contain the roots of the polynomials vt) and gt) respectively and that begin at some finite points and go to infinity in the sector π/3 < arg x < π/3. Then the integral operator Kx, y) = Ψx, z)ψy, z)dz Γ 2

8 310 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV on L 2 Γ 1 ) commutes with a formally symmetric differential operator with rational coefficients of order less than or equal to 2ρ 1 ρ 2 +1) and domain all smooth functions on Γ 1 that decrease rapidly as x. 4. Integral operators associated to self-adjoint Darboux transformations of Bessel functions 4.1. The Bessel bispectral function. Denote by J ν the standard Bessel functions of first kind. By abuse of notation the functions 4.1) Ψ ν x, z) = xz) 1/2 J ν+1/2 ixz) will be also called Bessel functions. Consider the Euler operator and the operators D x = x x 4.2) L ν x, x ) = x 2 νν + 1) x 2 = 1 x 2 D x + ν)d x ν 1) to be called Bessel operators. The Bessel functions satisfy the equations 4.3) 4.4) 4.5) L ν x, x )Ψ ν x, z) = z 2 Ψ ν x, z), D x Ψ ν x, z) = D z Ψ ν x, z), x 2 Ψ ν x, z) = L ν z, z )Ψ ν x, z). For shortness the algebras B Ψν and C Ψν, associated to the Bessel function Ψ ν x, z), recall 1.1), will be denoted by B ν and C ν. The Bessel functions corresponding to ν 1, ν 2 C that differ by an integer can be obtained by a Darboux transformation from each other: Ψ ν+1 x, z) = 1 xz D x ν 1)Ψ ν x, z), Ψ ν x, z) = 1 xz D x + ν + 1)Ψ ν+1 x, z), which corresponds to the factorizations L ν = x 1 D x + ν + 1)x 1 D x ν 1), L ν+1 = xd x ν 1)x 1 D x + ν + 1). According to 4.3) 4.5) the algebras of differential operators B ν and C ν contain the operators L ν x, x ), D x, and x 2. Denote their subalgebras generated by those operators by B ν and C ν, respectively. Clearly 4.3) 4.5) imply b ν B ν ) = C ν. Similarly to Proposition 2.4 in [4] one shows: Lemma 4.1. i) For ν C\Z the algebras B ν and C ν are generated by the operators L ν x, x ), D x, and x 2, i.e., B ν = B ν, C ν = C ν. ii) For ν Z the subalgebras B ν and C ν of B ν and C ν consist of exactly those differential operators in B ν and C ν that are invariant under the transformation x x. Note that 4.6) ab ν P x, x ) ) = b ν a P x, x ) ), P x, x ) B ν.

9 THE PROLATE SPHEROIDAL PHENOMENON AS A CONSEQUENCE OF BISPECTRALITY Self-adjoint Darboux transformations from the Bessel functions. Similarly to the Airy case we have C[x 2 ] = B ν Cx), C[L ν x, x )] = b 1 ν Cν Cz) ). The set of rational Darboux transformations D ν from the Bessel function Ψ ν x, z) is defined to be the set of functions Ψx, z) for which there exist differential operators 4.7) P x, x ), Qx, x ) C[x 2 ]\{0}) 1 B A such that 4.8) 4.9) fl ν x, x )) = Qx, x )P x, x ), Ψx, z) = 1 pz) P x, x)ψ ν x, z), for some polynomials ft) and pz). Again C[x 2 ]\{0}) is an Ore subset of B A and quotient ring in 4.7) makes sense. The following theorem was proved in [23] for ν = 1 and in [3, 4] in the general case. Theorem 4.2. All rational Darboux transformations from the Bessel functions are bispectral of rank 2 if ν C\Z and of rank 1 if ν Z. Definition 4.3. Define the set of self-adjoint even, self-adjoint in the case ν Z) Darboux transformations SD ν from the Bessel functions Ψ ν x, z) to consist of all functions Ψx, z) for which there exists a differential operator such that 4.10) 4.11) P x, x ) C[x 2 ]\{0}) 1 B ν ) fl ν x, x )) = 1) m ap )x, x )P x, x ) Ψx, z) = for a polynomial ft) of the form 1 z m gz 2 ) P x, x)ψ ν x, z), 4.12) ft) = t 2m gt 2 ) 2, g0) 0; gt) C[t]. Here the term even reflects the fact that for ν Z the algebras B ν and C ν are bigger than B ν and C ν. The reason for this terminology is explained below. As in the Airy case the set SD ν consists of those rational Darboux transformations Ψx, z) from the Bessel function Ψ ν x, z) for which in the notation 4.8) 4.9) 4.13) Qx, x ) = 1) ord P ap )x, x ) with the additional property in the case ν Z 4.14) P x, x ) = P x, x ). As a consequence it is obtained that the polynomial ft) in 4.8) has the form 4.12) and an appropriate normalization of pz) is made. Note that Lemma 4.1 implies that in the rank 2 case ν C\Z the condition 4.14) is a consequence of 4.7). In the rank 1 case ν Z the term even reflects this extra condition. It is needed since in the case of Darboux transformations from the Bessel function the prolate spheroidal property will be deduced from the second condition in Theorem 2.3.

10 312 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV 4.3. Size of the algebras B ν relative to the anti-isomorphisms b ν. Consider the Z + Z + filtration of the algebras B ν 4.15) Bl 1,l 2 ν = {Rx, x ) B ν ord Rx, x ) 2l 1, ordb ν R)z, z ) 2l 2 } The formal adjoint involution a of W rat preserves the spaces Bν l1,l2. Similarly to Lemma 3.4 one shows: Lemma 4.4. The vector space Bν l1,l2 has a basis that consists of the differential operators { x 2m L ν x, x ) ) n } { n l1, m l 2 x 2m D x Lν x, x ) ) n } n < l1, m < l 2. B l1,l2 ν Since a 2 = id the space is the direct sum of the eigenspaces of a with l1,l2 l1,l2 eigenvalues ±1. The eigenvalue 1 subspace of B ν will be denoted by B ν,sym. The commutativity 4.6) of a and b ν on B l 1,l 2 ν implies: B l1,l2 ν,sym = {Rx, x ) B ν ord Rx, x ) 2l 1, ordb ν R)z, z ) 2l 2, arx, x ) = Rx, x ), ab ν Rx, x )) = b ν Rx, x )}. Lemma 4.5. The set of operators { 4.16) x 2m L ν x, x ) ) n + Lν x, x ) ) n x 2m } n l 1, m l 2 is a basis for the space B l1,l2 ν,sym. In particular dim B l1,l2 ν,sym = l 1 + 1)l 2 + 1) Size of the algebra B Ψ,sym for an even) self-adjoint Darboux transformation from a Bessel function. Fix an arbitrary self-adjoint and in addition even in the rank 1 case ν Z) Darboux transformation from a Bessel function Ψ ν x, z), Ψ SD ν given by 4.10) 4.11) for some P x, x ) C[x 2 ]\{0}) 1 Bν and ft) = t 2m gt 2 ) 2, gt) C[t]. Let 4.17) P x, x ) = 1 vx 2 ) Rx, x) for some Rx, x ) B ν and vx) C[x]. Set 4.18) ord Rx, x ) = ρ 1 and ordb ν R)x, x ) = ρ 2. Denote { 1 S Ψ,1 = Span vx 2 ) Rx, 1 x)mx, x )ar)x, x ) vx 2 ) Mx, x) S Ψ,2 = Span{vx 2 )Mx, x )vx 2 l1,l2 ρ2 ) Mx, x ) B ν,sym }. B l1 ρ1,l2 ν,sym Proposition 4.6. i) The spaces of differential operators S Ψ,1 and S Ψ,2 are subspaces of B l1,l2 Ψ,sym and are invariant under the transformation x x. ii) The dimension of the intersection S Ψ,1 S Ψ,2 is less than or equal to l 1 ρ 1 + 1)l 2 ρ 2 + 1). Finally we obtain the following theorem. },

11 THE PROLATE SPHEROIDAL PHENOMENON AS A CONSEQUENCE OF BISPECTRALITY313 Theorem 4.7. For any self-adjoint and even, self-adjoint in the case ν Z) Darboux transformation from a Bessel function Ψ ν x, z), Ψx, z) SD ν the space S Ψ,1 + S Ψ,2 consists of differential operators invariant under the transformation x x dims Ψ,1 + S Ψ,2 ) l 1 + 1)l 2 + 1) ρ 1 ρ 2. In particular the dimension of the subspace of B l1,l2 Ψ,sym of differential operators invariant under x x is greater than or equal to l 1 + 1)l 2 + 1) ρ 1 ρ 2. Theorems 4.7 and 2.3 imply our final result for Darboux transformations from the Bessel functions: Theorem 4.8. Let Ψx, z) SD ν be a self-adjoint and in addition even if ν Z) Darboux transformation from the Bessel function Ψ ν x, z), given by 4.10), 4.11), 4.12), 4.17). Let Γ 1, Γ 2 be two connected finite contours that do not contain the roots of the polynomials vt) and gt), respectively and such that eγ i ) = eγ i ). Then the integral operator with kernel Kx, y) = Ψx, z)ψy, z)dz Γ 2 on L 2 Γ 1 ) commutes with a formally symmetric differential operator with rational coefficients of order less than or equal to 2ρ 1 ρ 2 +1) and domain all smooth functions on Γ 1. References 1. M. Adler, T. Shiota, and P. van Moerbeke, Random matrices, Virasoro algebra, and noncommutative KP, Duke Math. J ), no. 2, M. Adler and P. van Moerbeke, The spectrum of coupled random matrices, Ann. Math ), no. 3, B. Bakalov, E Horozov, and M. Yakimov, General methods for constructing bispectral operators, Phys. Lett A ), no. 1-2, , Bispectral algebras of commuting ordinary differential operators, Comm. Math. Phys ), no. 2, , Highest weight modules over the W 1+ algebra and the bispectral problem, Duke Math. J ), no. 1, A. Borodin and P. Deift, Fredholm determinants, Jimbo Miwa Ueno τ-functions, and representation theory, Comm. Pure Appl. Math ), no. 9, J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys ), no. 2, E. Frenkel, V. Kac, A. Radul, and W. Wang, W 1+ and W gl N ) algebra with central charge N, Comm. Math. Phys ), no. 2, F. A. Grünbaum, Time-band limiting and the bispectral problem, Comm. Pure Appl. Math ), no. 3, , Band-time-band limiting integral operators and commuting differential operators, Algebra i Analiz ), no. 1, Russian); English transl., St. Petersburg Math. J ), no. 1, , Some bispectral musing, The Bispectral Problem Montréal, 1997), CRM Proc. Lecture Notes, vol. 14, Amer. Math. Soc., Providence, RI, 1998, pp J. Harnad and A. R. Its, Integrable Fredholm operators and dual isomonodromic deformations, Comm. Math. Phys ), no. 3, J. Harnad, C. A. Tracy, and H. Widom, Hamiltonian structure of equations appearing in random matrices, Low-Dimensional Topology and Quantum Field Theory Cambridge, 1992), NATO Adv. Sci. Inst. Ser. B Phys., vol. 315, Plenum, New York, 1993, pp A. Kasman and M. Rothstein, Bispectral Darboux transformations: the generalized Airy case, Phys. D ), no. 3-4,

12 314 F. ALBERTO GRÜNBAUM AND MILEN YAKIMOV 15. H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40, 1961), , Prolate spheroidal wave functions, Fourier analysis and uncertainty. III, Bell System Tech. J. 41, 1961), J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure Appl. Math., Wiley, Chichester, J. Palmer, Deformation analysis of matrix models, Phys. D ), no. 3-4, D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System Tech. J. 40, 1961), C. A. Tracy and H. Widom, Level spacing distributions and the Airy kernel, Comm. Math. Phys ), no. 1, , Level spacing distributions and the Airy kernel, Comm. Math. Phys ), no. 2, , Fredholm determinants, differential equations and matrix models, Comm. Math. Phys ), no. 1, G. Wilson, Bispectral algebras of commuting ordinary differential operators, J. Reine Angew. Math ), Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA address: grunbaum@math.berkeley.edu Department of Mathematics, University of California at Santa Barbara, Santa Barbara, CA 93106, USA address: yakimov@math.ucsb.edu