An integral formula for L 2 -eigenfunctions of a fourth order Bessel-type differential operator

Transcription

1 An integral formula for L -eigenfunctions of a fourth order Bessel-type differential operator Toshiyuki Kobayashi Graduate School of Mathematical Sciences The University of Tokyo Komaba, Meguro, Tokyo, , Japan. address: toshi@ms.u-tokyo.ac.p Jan Möllers Institut für Mathematik Universität Paderborn Warburger Str Paderborn, Germany address: moellers@math.uni-paderborn.de March 1, 1 Abstract We find an explicit integral formula for the eigenfunctions of a fourth order differential operator against the kernel involving two Bessel functions. Our formula establishes the relation between K-types in two different realizations of the minimal representation of the indefinite orthogonal group, namely the L -model and the conformal model. Mathematics Subect Classification: Primary 44A; Secondary E46, 33C45, 34A5. Key words and phrases: Fourth order differential equation, Laguerre polynomials, Bessel functions, minimal representations. Partially supported by Grant-in-Aid for Scientific Research B , Japan Society for the Promotion of Science, and the Alexander Humboldt Foundation. Partially supported by the International Research Training Group 1133 Geometry and Analysis of Symmetries, and the GCOE program of the University of Tokyo. 1

2 1 Introduction and statement of the results Let θ = x d dx be the one-dimensional Euler operator. We consider the following representation of the Bessel differential operator 1 x Q νθ = d dx + ν + 1 x d 1, ν C, dx d where Q ν is the quadratic transform of the Weyl algebra C[x, dx ] defined by Q ν P = P P + ν x, [ for P C x, ] d. dx Our obect of study is the L -eigenfunctions of the fourth order differential operator D µ,ν := 1 x Q νθ + µq ν θ = 1 x θ + µθ + µ + ν x θθ + ν x. Throughout this article we assume that the parameters µ and ν satisfy the following integrality condition: µ ν 1 are integers of the same parity, not both equal to We then have the following fact see [4, Theorem A]: Fact. The differential operator D µ,ν extends to a self-adoint operator on L R +, x µ+ν+1 dx with only discrete spectrum which is given by λ µ,ν := 4 + µ + 1, =, 1,,... The corresponding L -eigenspaces are one-dimensional. For instance, it is easily seen that the normalized K-Bessel function K ν z := z ν K ν z is an L -eigenfunction of D µ,ν for the eigenvalue λ µ,ν =. The purpose of this article is to establish the following integral formula for L -solutions of the differential equation D µ,ν u = λ µ,ν u. 1. Theorem A. Assume 1.1 and let u be an L -solution of the differential equation 1.. Then there exists a constant A µ,ν u such that for cos ϑ + cos ϕ > : ux J µ ax J ν bxxµ+ν+1 dx = A µ,ν u where we set a := cos ϑ + cos ϕ sin ϑ cos ϑ+cos ϕ and b := sin µ+ν+ C µ+1 cos ϑ ϕ cos ϑ+cos ϕ. ν+1 C + µ ν cos ϕ, 1.3

3 Here J α x = x α Jα x denotes the normalized J-Bessel function and C nx λ = ΓλCnx λ is the normalized Gegenbauer polynomial. The differential equation 1. has a regular singularity at x = with characteristic exponents, ν, µ and µ ν. Accordingly, the asymptotic behaviour of a non-zero L -solution u of 1. as x is of the following form see [4, Theorem 4. 1]: x ν + ox ν for ν >, ux B µ,ν u log x + olog x for ν =, o1 for ν = 1, with some non-zero constant B µ,ν u. The constant A µ,ν u in Theorem A is determined by B µ,ν u as follows: Theorem B. For any solution u of 1.: A µ,ν u B µ,ν u =!µ+ν µ+ µ ν + Γ 1 Γ Γ + µ ν+ Γ + µ ν + πγ + µ + 1 Γ ν for ν >, 1 for ν =, Γ ν for ν = 1. The proofs of Theorems A and B will be given in Sections and 3, respectively. In Section 4 we give some applications and discuss special values of Theorem A. One particularly interesting situation arises when both µ and ν are odd integers. In this case the solutions u of 1. can be expressed as { x ν e x M µ,ν x for ν 1, ux = const e x M µ,ν x for ν = 1, for some polynomial M µ,ν see [5]. For ν = ±1 these polynomials reduce to the classical Laguerre polynomials M µ,±1 x = L µ x. Hence, for ν = ±1 the integral formula in Theorem A collapses to integral formulas for the Laguerre polynomials. Even these we could not trace in the literature. Corollary C. Let µ 1 be an odd integer and cos ϑ + cos ϕ >. a := and b := sin. Then we have sin ϑ cos ϑ+cos ϕ ϕ cos ϑ+cos ϕ Set L µ x J µ ax cosbxxµ e x dx = 1 µ cos ϑ + cos ϕ π µ+1 cos + µ + 1 ϕ µ+1 C cos ϑ 3

4 and L µ x J µ ax sinbxxµ e x dx = 1 µ cos ϑ + cos ϕ π µ+1 sin + µ + 1 ϕ µ+1 C cos ϑ. Note that there appear 5 parameters in the integral formula in Theorem A, namely µ, ν, ϑ, ϕ and. Our scheme specialization of parameters, relation to representation theory is summarized in the following diagram: Special Functions Representation Theory K-Bessel functions Section 4.3 = Gegenbauer polynomials Theorem A Λ µ,ν x : L -solution to 1. Laguerre polynomials Corollary C ν Z+1 ν=1 Notation: N = {, 1,,...}, R + = {x R : x > }. Conformal model T L -model Polynomials x ν e x M µ,ν ν 3 x see [5]. Two models for the minimal representation of the indefinite orthogonal group In the proof of Theorem A we will use representation theory, namely two different models for the minimal representation of the indefinite orthogonal group G = Op, q where p q and p + q 6 is even. These two models were constructed by T. Kobayashi and B. Ørsted [7, 8] and investigated 4

5 further by T. Kobayashi and G. Mano [6]. This unitary representation is irreducible and attains the minimum Gelfand Kirillov dimension among all irreducible unitary representations of G. In physics the minimal representation of O4, appears as the bound states of the Hydrogen atom, and incidentally as the quantum Kepler problem..1 The conformal model We begin with a quick review of the conformal model for the minimal representation of G = Op, q from [7]. We equip M := S p 1 S q 1 with the standard indefinite Riemannian metric of signature p 1, q 1 by letting the second factor be negative definite. Then G acts on M by conformal transformations. The solution space Sol { M := f C M : } M f = of the Yamabe operator M = S p 1 S q 1 p + q is infinitedimensional. Further, it is invariant under the twisted action ϖ of G and hence defines a representation. The minimal representation of G is realized on the Hilbert completion H := Sol M of Sol M with respect to a certain G-invariant inner product. The maximal compact subgroup K = Op Oq of G acts on M as isometries, and the restriction of ϖ to K is given ust by rotations. To see the K-types we recall the space of spherical harmonics H k R n := { ϕ C S n 1 : S n 1ϕ = kk + n ϕ }, or equivalently, the space of restrictions of harmonic homogeneous polynomials on R n of degree k to the sphere S n 1. The orthogonal group On acts irreducibly on H k R n for any k by rotations in the argument. Then clearly and we put H R p H k R q Sol M if and only if k = + p q, V := H R p p q + H R q, =, 1,,..., on which K acts irreducibly. The multiplicity-free sum = V of irreducible representations of K is dense in the Hilbert space H. Let K be the isotropy group of K at 1,,...,,,...,, 1 S p 1 S q 1. Then K = Op 1 Oq 1. We write H K for the space of K -fixed vectors. 5

6 Lemma.1. In each K-type V N the subspace V H K is onedimensional and spanned by the functions ψ : S p 1 S q 1 C, v, v, v, v p+q 1 p C v q C + p q v p+q 1..1 Proof. It is well-known that any On 1-invariant spherical harmonic is a scalar multiple of the Gegenbauer polynomial S n 1 x 1, x n C k x 1, which shows the claim.. The L -model We recall the L -model Schrödinger model of the minimal representation of G which is unitarily equivalent to ϖ see [6, 8]. Consider the isotropic cone C = {x, x R p 1 R q 1 : x = x } R p+q. Then the group G acts unitarily in a non-trivial way on the Hilbert space L C, dµ and defines a minimal representation of G. Here dµ is the Op 1, q 1-invariant measure on C which is in bipolar coordinates R + S p S q C, r, ω, η rω, rη normalized by dµ = 1 rp+q 5 dr dω dη. dω and dη denote the Euclidean measures on S p and S q, respectively. The representation of the whole group G on L C does not come from the geometry C, but the action of the subgroup K is given by rotation in the argument. Hence the K -invariant functions only depend on the radial parameter r R + and the space of K -invariants in L C is identified as L C K = L R +, 1 rp+q 5 dr. Let W be the V -isotypic component in L C. In this model it is more difficult to find explicit K-finite vectors. By highlighting K -fixed vectors, the following result was proved in [4, Section 8]: Lemma.. In each K-type W N the subspace W L C K one-dimensional and given by the radial functions is ur,. where u is an L -solution of 1. with µ = p 3, ν = q 3. 6

7 .3 The G-intertwiner Let T : L C H be the intertwining operator as given in [6, Section.]. It is the composition of the Fourier transform S R p+q S R p+q and an operator coming from the conformal transformation from the flat indefinite Euclidean space R p 1,q 1 to M. For radial functions f L R +, 1 rp+q 5 dr = L C K this operator can be written by means of the Hankel transform cf. [6, Lemma 3.3.1]: T fv, v, v, v p+q 1 = J p 3 1 v + v p+q 1 p+q 4 v r v + v p+q 1 J q 3 fr v r v + v p+q 1 r p+q 5 dr.3 for v, v, v, v p+q 1 M with v + v p+q 1 >. Since T intertwines the actions of G on both models, it clearly maps K -invariant functions to K - invariant functions and also preserves K-types, i.e. T W = V. Hence we obtain the following diagram: W V T : L C H K L C K H K. Now W L C K and V H K are one-dimensional and we have formulas. and.1 for their generators. Hence the intertwiner T has to map the functions. to multiples of the functions.1. Thus, for any L -solution u of 1. and v + v p+q 1 > we obtain ur J p 3 v r v + v p+q 1 J q 3 v r = const v + v p+q 1 p+q 4 v + v p+q 1 p C v r p+q 5 dr q C + p q Substituting x = r, µ = p 3 and ν = q 3 and putting cos ϑ = v, cos ϕ = v p+q 1, sin ϑ = v, sin ϕ = v. v p+q 1.4 we get 1.3 with a certain constant A µ,ν. This finishes the proof of Theorem A. 7

8 3 A closed formula for the constants In this section we find an explicit constant for the integral formula in Theorem A, namely we give a proof of Theorem B. Our method uses the generating function of L -eigenfunctions of D µ,ν. Remember that we assume the integrality condition 1.1. Let G µ,ν t, x = 1 1 t µ+ν+ Ĩ µ tx 1 t K ν x 1 t, 3.1 where Ĩαz := z α I α z and K α z := z α K α z denote the normalized I- and K-Bessel functions. Further, let Λ µ,ν x =,1,,... be the family of functions on R + which has G µ,ν t, x as its generating function: G µ,ν t, x := Λ µ,ν xt. 3. = Fact [4, Theorem A]. Λ µ,ν x is real analytic on R + and an L -solution of 1.. We will now compute the constants A µ,ν u and B µ,ν u for u = Λ µ,ν. Here we recall that A µ,ν and B µ,ν were defined in Theorem A and 1.4. From [4, Theorem 4.] we immediately obtain ν µ ν + B µ,ν Λ µ,ν Γ + ν 1 Γ for ν >, =!Γ µ+ µ ν + Γ 1 for ν =, 1 Γ ν 3.3 for ν = 1. For A µ,ν we have the following lemma: Lemma 3.1. For any N we have A µ,ν Λ µ,ν = 1 µ+ν Γ + µ ν+ πγ + µ Putting 3.3 and 3.4 together proves Theorem B. In the remaining part of this section, we give a proof of Lemma 3.1. Proof of Lemma 3.1. We put ϑ = ϕ = in 1.3. values we obtain J α = A µ,ν Λ µ,ν = 1 Γα + 1, Cλ n 1 = µ+ν!γ + µ ν+ πγ + µ + 1Γ + µ+ν+ Γn + λγλ Γn + 1Γλ, Together with the next lemma this finishes the proof. 8 Using the special Λ µ,ν xx µ+ν+1 dx.

9 Lemma 3.. For every N we have Λ µ,ν L 1 R +, x µ+ν+1 dx and Λ µ,ν xx µ+ν+1 dx = 1 µ+ν Γ + µ+ν+ Proof. The fact that Λ µ,ν L 1 R +, x µ+ν+1 dx is derived from the asymptotic behaviour of Λ µ,ν x see [4, Theorem 4.]. To calculate the integral we use the following integral formula which is valid for Rλ + α ± β + 1 > and b > a > see e.g. [3, formula ]: I α axk β bxx λ dx = aα Γ λ+α+β+1 Γ λ+α β+1 1 λ b λ+α+1 Γα + 1 F 1 λ + α + β + 1!, λ + α β + 1 ; α + 1; a b where F 1 α, β; γ; z denotes the hypergeometric function. With 3.1 we obtain µ + ν + G µ,ν t, xx µ+ν+1 dx = µ+ν Γ 1 t µ+ν+ µ + ν + F 1, µ + ; µ + ; t µ + ν + = µ+ν Γ 1 + t µ+ν+ µ+ν Γ + µ+ν+ = t.! = Then, in view of 3., the claim follows by comparing coefficients of t. Hence, the proof of Theorem B is completed. 4 Applications and special values We conclude this article with some applications of Theorem A and discuss on special values of the integral formula. 4.1 The L -norm of Λ µ,ν As a first application of Theorem A we can give a closed formula for the L -norms of the orthogonal basis Λ µ,ν x N in L R +, x µ+ν+1 dx. The same result was obtained in [4, Theorem B] by different methods. Corollary 4.1. The L -norm of the eigenfunction Λ µ,ν Λ µ,ν. is given by, L R +,x µ+ν+1 dx = µ+ν 1 Γ + µ+ν+ Γ + µ ν+! + µ + 1Γ + µ + 1., 9

10 Proof. Let p = µ+3, q = ν +3. We define functions ϕ L C in bipolar coordinates by Then it is immediate that ϕ r, ω, η := Λ µ,ν r. Λ µ,ν L R +,x µ+ν+1 dx = µ+ν+3 vols p vols q ϕ L C. Now, the intertwining operator T : L C H is unitary up to a constant, namely see [6, Remark..]: T u H = 1 u L C. Using formula.3 for the intertwiner T and Theorem A one also obtains easily that T ϕ = 3µ+ν+ A µ,ν Λ µ,ν ψ with ψ as in.1. By [6, Fact.1.1 4] the G-invariant norm on H is given by u H = + p u L M, for u V. By using the formula see [3, ] we get π ψ L M = [ ] Cλ n cos ϑ sin λ ϑ dϑ = π1 λ Γn + λ, n!n + λ S p 1 S q 1 C p v = vols p vols q π q C + p q [ C p v p+q 1 dv cos ϑ] sin p ϑ dϑ [ C q cos ϕ] sin q ϕ dϕ π = π Γ + µ + 1Γ + µ+ν+! µ+ν + µ+1 Γ + µ ν+ + p q volsp vols q. Finally, putting all the steps together shows the claim. 1

11 4. The Poisson kernel of the Gegenbauer Polynomials As another application of Theorem A we get an integral formula for the generating function G µ,ν t, x of L -eigenfunctions, which is closely related to the Poisson kernel of the Gegenbauer polynomials. For this we put I µ,ν t, ϑ, ϕ := where a := sin ϑ cos ϑ+cos ϕ Ĩ µ and b := sin tx 1 t K ν ϕ cos ϑ+cos ϕ. x 1 t J µ ax J ν bxxµ+ν+1 dx, Corollary 4.. For cos ϑ + cos ϕ > and 1 < t < 1 we have the following formula: 1 t µ+ν+ cos ϑ + cos ϕ = µ+ν π = µ+ν+ µ ν+ Γ + 1 Γ + µ + 1 I µ,ν t, ϑ, ϕ µ+1 C cos ϑ ν+1 C + µ ν cos ϕt. For µ = ν = λ 1 this yields a formula for the Poisson kernel of the Gegenbauer polynomials. Recall that the Poisson kernel P λ t, ϑ, ϕ of the Gegenbauer polynomials is defined by see [1, formula 6.4.5] P λ t, ϑ, ϕ := n= n!n + λ Γn + λ C λ ncos ϑ C λ ncos ϕt n. For explicit formulas for the Poisson kernel of the Gegenbauer polynomials see e.g. [1, formula 7.5.6]. Now, Corollary 4. yields a new expression for P λ t, ϑ, ϕ λ Z, λ > : P λ t, ϑ, ϕ = where θ t = t t. π 8λ The bottom layer cos ϑ + cos ϕ λ θ t + λ [ ] 1 + t λ I λ 1,λ 1 t, ϑ, ϕ, As remarked in the introduction, for = the K-Bessel function ux = K ν x is a solution of the differential equation 1.. In this case the integral formula in Theorem A can be written as K ν xj x sin ϑ x sin ϕ µ J ν x µ+ dx cos ϑ + cos ϕ cos ϑ + cos ϕ = ν Γ µ ν+ π cos ϑ + cos ϕ sin µ ν ν+1 ϑ sin ϕ C cos ϕ. µ ν 11

12 This special case was already proved in [6, Lemma 7.8.1]. Another expression for this integral can be found in [, formula ]. References [1] G. E. Andrews, R. Askey, and R. Roy, Special functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, [] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Tables of integral transforms. Vol. II, McGraw-Hill Book Company, Inc., New York, [3] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press, New York, [4] J. Hilgert, T. Kobayashi, G. Mano, and J. Möllers, Special functions associated to a certain fourth order differential equation, preprint, 31 pp., arxiv: [5], Orthogonal polynomials associated to a certain fourth order differential equation, preprint, 14 pp., arxiv: [6] T. Kobayashi and G. Mano, Integral formula of the unitary inversion operator for the minimal representation of Op, q, Proc. Japan Acad. Ser. A 7, 7 31, the full paper to appear in the Mem. Amer. Math. Soc. is available at arxiv: [7] T. Kobayashi and B. Ørsted, Analysis on the minimal representation of Op, q. I. Realization via conformal geometry, Adv. Math. 18 3, [8], Analysis on the minimal representation of Op, q. III. Ultrahyperbolic equations on R p 1,q 1, Adv. Math. 18 3,