EXPLICIT FORMULAE FOR THE WAVE KERNELS ...

IC/96/58

United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

EXPLICIT FORMULAE FOR THE WAVE KERNELS FOR THE LAPLACIANS AaP IN THE BERGMAN BALL Bn, n > 1

A. Intissar Department of Mathematics, Faculty of Science of Rabat, P.O. Box 1014 , Rabat, Marroco and M.V. Ould Moustapha1 International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT For the perturbed Bergman Laplacians A a ^ given by A a/3 = 4(1 - \z\2){(Si:i - ZiZj)-Q-f. +

aZi

~fo, + ^

in the unit ball Bn of Cn, we establish explicit formulae for the corresponding wave equations in Bn. The formulae obtained generalise, for arbitrary a, f3 G R, the formulae given in [2] and [5] for the wave equation associated to the shifted Bergman Laplacian AB = AOo in Bn. Moreover, using an analytic continuation argument, we are able to give explicit formulae for the solutions of the wave equation associated to a two parameter family of Laplacians Aa!/g on Cn which are natural deformations of the Fubini-Study Laplacian on the Projective space Pn(C)7 n > 1, viewed as the dual space of the Bergman ball Bn. MIRAMARE - TRIESTE April 1996

Permanent address: University of Nouakchott, P.O. Box 798, Nouakchott, Mauritania.

1-Introduction Let Bn = {z 6 Cn\ \z\ < 1} be the unit complex ball, endowed with the Bergman Kahler structure. For a, j3 real numbers, consider the following Cauchy problem of wave type on the ball Bn: t,z) VVn

'

\

(t,z) eRxBn dtu(t,z) = f(z)eC%°(Bn)

u(O,z) = O

[

'

where Aa/3 is the two real parameter family of "Laplacians" given by

Aa/3 = 4(1 - \z\2){ J2^V-EE ij=1

+ aE + PE-a[3} + a^

(1.2)

OZiZj

in which E = YJi=\z%~§^. is the complex Euler operator with E its complex conjugate and a^p = ( a + P + n ) 2 is the shift constant. For a = /3 = 0 the above problem (W®°) reduces to the wave equation in the n-dimensional complex hyperbolic space modeled by the Bergman ball Bn with A oo being the shifted Bergman invariant Laplacian. In this case, the unique solution u(t, z) of {W®°) is given by the explicit integral formula (see [6] for n = 1 , [8] for n = 2 and [2], [5] for n > 1) UI 6, Z I — I ^7T )

I ;

~ )

/

smh tot

ICOSnT-

COSn

p[Z^ IDJ J i

J [XJ(liJjyijU I

1 X.o )

J Bn

where p(z, w) is the Bergman distance for z , w in Bn and d/j,(w) is the Bergman volume form on Bn. Also, for a = /?, we can intertwine the operator Aaa with the shifted Bergman Laplacian AOo SO that the corresponding solution of the wave problem (W"a) can be given by q.(f

7

\ — (r>

\-r.

However, when a ^ /5, it becomes less obvious, even for the case n = 1, to deduce from above an explicit formula for the wave solution of the Cauchy problem iW^) . Thus, our main objective in this paper is to provide an explicit formula for the unique solution of the problem (W"P) for arbitrary a, /3 G R , encompressing the above formula (1.3) aa as a particular case, that is, we will establish explicit formulae for the distributional kernels K^ (t; z, w) of the integral operator:

y

which solves the Cauchy problem (W°P) (see theorem 2 section 2). Remark 1: The wave equation associated to the Laplacians Aap might be considered as the wave equation in the Bergman ball in the presence of a "constant" electromagnetic field B, corresponding to the differential form wap given by: 4

n

n dz

Wap = -. _ , 2{® J2 Zj 3 + P J2 Z3dzj} 1

\

Z

3=1

j=l

(L4)

for which we have dwa@ = B = 4(« — /3)ddLog(l — \z\2). i.e, a constant times the Kahler 2-form on Bn. Now we give an outline of the organization of our paper.In section 2, we fix notation and we state the explicit formulae for the wave kernels K"^(t; z, w) as well as some fundamental recursion formulae satisfied by these kernels, see theorem 1 and theorem 2. The proof of the stated results in section 2 will be postponed to section 4. Section 3 deals with some invariance properties of the Laplacians Aa/g and the wave kernels K^{t\ z, w) , with respect to a family of projective representations Ta^ of the group AutBn = PU(n, 1) on C°°(Bn). The established properties will allow us to use, in an appropriate way, the geodesic polar coordinates of the Bergman ball to obtain our explicit formula. Finally in section 5 , we will give some applications of the explicit formulae obtained and the method used in this paper to the Cauchy problem for the wave equation of the Laplacians Aap given on Cn (viewed as the affine part of the projective space Pn(C)) by AQi/g = 4(1 + \z\2){ Y^ ~—=- + EE - aE - fiE + a/3} - a2af} =

^ >j 1

(1.5)

^ -^

where a, /3 are non negative integers.

2-Statement of the explicit formulae for the wave kernels

K^(t;z,w)

We start by fixing some necessary notation. Let p(z, w) be the Bergman distance given by cosh2 p(z, w) = (i_~^)/™ w 2\, where < z, w > = 5Z"=1 ZjUJ] is the Hermitian scalar product of Cn and let dp,(z) = (1 — \z\2)^n^1dm(z) be the Bergman volume form on Bn where dm(z) is the Lebesgue volume form on C n .Note that p(z,w) is AwtBn-bi-invariant and dp,(z) is AutBn invariant. Here AutBn is the Lie group of biholomorphic mappings of the complex ball. Also, for a,/3 real numbers and n G Z + , we will denote by I™@(t,p) the function defined on the interior of the light cone of i? 2 , that is, \p\ < \t\ by the formula Ta0(T n\ -

rri^Y,-(n+a+0)

„/ ^

o l i

t

,^-n+l/2

, ,

p (

,

-, x / O COSh t - ^uan y

cosh p

2 cosh p

where we have set a = l — n + a — (3, b = l — n + /3 — a while F(a, b; c, z) denotes the usual Gauss hypergeometric function. Now, we state the main recursion relations satisfied by the above explicit functions I"^(t,p) in (2.1). Theorem 1 The functions I^(t,p) (n> 0)

satisfy the following differential-difference equations:

ii) coth p§-plf (t, p) + {a + /3 + n)lf (t, p) = (2n - 1) cosh tl^+1 (t, p) Ma |

L_!

TOcp f-j- Q\ _I_ PQCVJ ~t T '

(~l~ fl\ -^z. ( C O S l l

f

COSll

O)I

o I ~t O)

Next, we state the explicit formulae for the wave kernels K"^(t;z,w). convenient to denote by K" (t; z, w) the function Kf(t;z,w)

= (l-)a(l-

< z,w>flf{t,p{z,w))

It will be (2.2)

( p) is the function given in (2.1), i.e for z, w in Bn such that p(z,w) < \t\ where I"" (£,

If(t,

p) = c o s h - ( ^ ) p(cosh21 - cosh2 p)-+1/2F(a -0,0-a;

1/2,

cosht

22 cosh p

Also, we set djj,a^(w) = (1 — \w\2)~(a+^ d/j,(w) where dfi{w) is the Bergman volume form on Bn. Theorem 2 i) Let u(t, z) be the solution of the wave Cauchy problem (W^) u(t, z) is given by the integral formula:

. Then

u{t,z) = (27r)~r where Ki(t;z,w) is as in (2.2). ii) Let K*P(t; z,w) be the distributional wave kernel of the Cauchy problem (W"^). K^ (t; z, w) is given by the explicit formula: KaP(+- 7 i/A — r (~\ — >\a(~\ — \P TaP (i n( 7 inX\ n \ i i ) — (-nvi ) yl ) ln yi, p\Z, W)j

Then (9 d\ ^.^±)a^

where I^f (t, p) is the function given in (2.1) and

3-Invariance properties of the Laplacians Aa^ In this section we will show that the Laplacians Aap and the wave kernel K^f{t\ z, w) are invariant with respect to a projective representation T a/3 of the group G = Aut(Bn) on the Hilbert space L2aji(Bn) = L2(Bn,djiafj) where dfiaf}(z) = (1 - \z\2)-(a+^dn(z) with dfi(z) being the G-invariant Bergman volume form of Bn Cf.[10]. The group G = Aut(Bn) consisting of all biholomorphic maps of the complex ball , acts naturally on Bn which we will denote by g.w for g £ Aut(Bn) and w £ Bn . This action is transitive. The isotropy group of the origin 0 £ Bn is the unitary group U(n) of Cn and for every g £ Aut(Bn) and every z, w in Bn, the following identity holds (see [10])

The above identity can be rewritten, up to a mild ambiguity for the powers depending only on g £ Aut(Bn), as follows: 1 - < g.z,g.w

>= {det{g'){z)}^i{det{g'){w)}^i(l-

< z,w > )

(3.1)'

where g'(z) is the complex Jacobian of the mapping g £ Aut(Bn). Finally we recall that Aut(Bn) is generated by the group U(n) and the mapping gz £ Aut(Bn) given by the explicit formulae: Qz-w =

A-w - z 1— < w,z >

where Az is the n x n matrix given by:

f

3.2

a

/ is the identity of Cn, when z ranges over Bn. Here z* = (z\, • • •, zn) =tz and we should note that for the mapping gz in (3.2) we have g^1 = g-z with g~1.0 = z and (I'

y

'

J

(3.3)

{l+

Now, for a, (3 real numbers, we consider the projective representation T a/3 of the group G = Aut(Bn) on C 0 0 ^ ) defined by: \z)

(3.4)

where }^

(3.5)

which satisfies, up to a factor depending only on G and of modulus one, the following chain rule: Ja/5(gig2,z) = JaP(gi,g2z)JaP(g2,z) (3.6) Thus using the above notation we have Proposition 3.1 i) The action Taf5 in (3.4) is a unitary projective representation of the group G = Aut(Bn) on the Hilbert space L^(Bn) = L2(Bn, d/j,ap). ii) The Laplacians Aap is Ta@ -invariant. That is we have: rpa

for every g £ Aut(Bn). Proof i) Let g e G and nd / e L2a/3(Bn). Then we have to show that ||T a/3 (#)/|| L 2 = For this we make the change of variable g~x.z = w. Using the G-invariance of the Bergman measure d/j, as well as the fact that Ja^{g^1,g.w) = {Jal3(g, w)}^1 , we can use identity (3.1)' for z = w to get the unitarity of Ta^{g). ii)The Ta/3-invariance of Aap, can be obtained easily using the chain rule (3.6) for J a/3 and the following observation made in [1] which defines Aap through the action T a/3 as (3.7)

for some g G G such that 5r~10 = z and where Aw = YJj=i dJ^dw- ^s ^ e u s u a ^ Euclidean Laplacian of Cn. In fact formula (3.7) holds by a direct computation with the choice of gz £ G given by (3.2). This is independent of such a choice since the Euclidean Laplacian is invariant under the group of isotropy U(n) hence we have proved proposition (3.1) . As a consequence of the Ta/3-invariance of the Laplacians Aap and the uniqueness of the solution of the Cauchy problem {W^) with initial conditions in C^{Bn) we have the following property for the wave kernels K^f[t\ z, w) Corollary: For t £ -R, let K^{t;z,w) tor Wf(t) =

Sin(t

\/~ Z ^ ) ;

be the Schwartz kernel of the integral opera-

w h i c h s o i v e s t he

Cauchy Problem (Wf).

Then for every

g £ G we have: Kf(t;gz,gw) = {r\g,z)}-1{J^{g,w)}-1K^{t-z,w)

(3.8)

where JaP(g,z)

is defined by (3.5).

Remark 2 Combining (3.8) and the identity (3.1)', we see that the wave kernel K^{t\ z, w) is of the form: Kf{t;z,w)

= {l-)a(l-

flf(t;z,w)

(3.9)

where /"^(i; z, w) is bi-invariant under G. Below, we will state without proof some spectral properties of the Laplacians Aap in the Hilbert space L2a^(Bn) which enables us to have a clear meaning of the integral operator W^ (t) =

sin(ty/-Aaafi p)

/

\/-AaP

for t G R

Proposition 3.2 i) For every 7 G R we have the intertwining relation

ii)The Laplacian Aap is self-adjoint in the Hilbert space L ^ ( B n ) iii)The spectrum of the Laplacian — Aap in L2a^(Bn) consists of a continuous part given by [0,+oo[ and at most a finite discrete part of eigenvalues. More precisely, the point spectrum of — Aap is given by Ea? = {iik = -(2k + n - \a - f3\f; 0 < k
n. Otherwise it is empty. Thus, by proposition (3.2), we may apply the mapping spectral theorem to define 1

a?

for every fixed t G R as the operator associated to the even function

X

a

for /i ranging over sp(Aa^) = [0, +oo[UE P.

Remark on the proof of the proposition (3.2): By a direct computation we obtain i), from which we see that the Laplacian Aap is unitarily equivalent to the Laplacian Dk = A_feiyt with k = ^ ^ ( t a k e 7 = ^ ^ ) - Thus to prove ii) and iii) we may work with Dk instead of A a ^. Indeed for n = 1, the statements ii) and iii) are contained in the work of Elstrodt [3], see also Patterson [9], and for n > 2, we may refer the reader to a recent work of Zhang [11] where a weighted Plancherel formula is established, see also Boussejra and Intissar [1] for an L2-concrete spectral theory of the invariant Laplacian Aap in the ball Bn.

4- Proof of the explicit formulae for the wave kernels K^f (t; z, w) In this section, we give the proof of theorem 1 and theorem 2 stated in section 2: Proof of theorem 1 The proof of the recursion formulae for the functions I^(t,p) is essentially based on two recursion identities for the Gauss hypergeometric function F(a,b;c,z) (see [7]) dm dz ['-^l

- z)a+b-cF(a,

b; c, z)} = (c - m ) m ^ m - 1 F ( a - m,b - m;c- m,z)

F{a, b- c, z) = ( a + 1 ) ( f r + 1 ) z ( 1 _ z ) F ( a + 2 ,ft+ 2; c + 2, z)

c(c+ I)

(A)

+ -[c - (a + b + l)z]F{a + 1, b + 1; c + 1, z) c Now we recall that the functions I"^(t, p) are given by

{B)

which we will rewrite in a convenient way in order to use identities (A) and (B) . Noting that a = 1 — n + a — /3, b = 1 — n + /3 — a, c = | — n and z = | ( 1 — ™^ *) we see that I%P(t,p) can be written as I f (i, p) = 7 (n) cosh-( J I + a + « p ^ - ^ l - z) a + "- c F(a, b; c, z)

(4.1)

with a, 6, c as above and where j(n) = e^1/2_" . Note t h a t a + b-c= c-1 = 1/2 - n and 1 - z = ±(1 + ^ J ) . Thus, to see that the recursion formulae in i) and ii) of theorem 1 hold, we apply the identity (A) with m = 1 to the functions /"^(t, p) in the form (4.1) and the fact that the derivatives in t and p are given, respectively,by Id sinh tdt

lid 2 cosh p dz

d 1 cosht d cosh/)—- = — op 2 cosh p dz For iii) in theorem 1, we may set a = 1 + a — f3 — n — 2, b = 1 + /3 — a — n — 2, c = | — n — 2 in the identity (B) with z as above to see that the recursion formula iii) is only a rewording of (B). With this we have the proof of theorem 1. Proof of t h e o r e m 2: We begin by recalling that from the Ta^-invariance properties established in section 3, the wave kernel is of the form Kf{t-z,w)

= cn(l - < z,w>)a(l-

flf(t;z,w)

(4.2) aj8

where I^(t; z,w)) is now bi-invariant under the natural action of G on Bn. Thus our main task is to explicite the above function I"^(t;z,w) . For this we set I"^(t;z,w)) = I%P(t,p(z,w)) where I"&(t,p) is the function given in (2.1) ,see also (4.1), and p(z,w) is the geodesic distance in the Bergman ball, and we will show below that the corresponding function K^{t;z,w) in (4.2) satisfies the wave equation %K?(t;z,w) = AaPKf(t;z,w)

(4.3)

for every t 6 R and z, w in Bn such that p(z, w) < \t\. To prove (4.3), we use the Ta/3-invariance property of the Laplacians Aap as well as those of K^f defined by (4.2) to see that we need only to prove the following wave equation | ^ ( t , P) = Lflf(t,

P)

where L^f(p, -f-) is the radial part of Aap given by

0 0 we consider the generalised integrals „(£, p) given by f+o°

-i\

, n + iX+\f\

n + iX — \v\

2

, , n , t < m h p )

Z

\sm^i/^^\M-2

Z

A

where CU(X) is given by the formula

Here Cn(A) is the analogue of the C-Harish Chandra function,related to the operator Dv = A J/ /2,-J//2 in the Bergman ball Bn. Proposition 5.1 For \v\ < n the generalised function cf)u(t, p) is given by the following explicit formula 4>v{t, p) = 7t/,w c o s h " p ( .

^^^)n-1(cosh21 — coshp)^ ' F(\v,—\u\;l/2,z)

(5.2)

sinn i

where 7^^ is a constant and z = (coshi — cosh p)/2 cosh p Proof: We will only sketch the proof of proposition 5.1. To begin, one has to perform a Fourier-Helgason transform on Bn by setting for (A, a;) £ i ? x dBn:

f(X,u) = f f(z)Pv(X;z,u)dfx(z) , / e C™{Bn)

(5.3)

where P"(X; z,co) is the generalised Poisson kernel on Bn x dBn given by the formula i-\2/

y

l-'

and d/j,(z) is the Bergman volume form on _Bn.For v = 0 see [4] . Then, there is an inversion formula of the Fourier-Helgason transform (5.3) which is given for |^| < n by the following formula: f(z)=

P"(X;z,9)fu{X,9)\Cu(X)\-2dXd9

/ J — oo

(5.4)

JdBn

where the function CV{X) is, up to a constant, the analogue of C-Harish Chandra function given above. Therefore, using the fact that the above Fourier-Helgason diagonalises the operator A ^ 2 _ ^ 9

with —A2 as eigenvalue, it becomes easy to see that the wave kernel K^2' v^2(t; z, w) can be given by the integral formula

f JdBn

But the integral over the sphere dBn = S2n 1 can be computed in terms of the hypergeometric functions. For instance by taking w = 0, we have /

Pv A; z, 9)da{9) = 1 - \z\2) — F(

JdBn

l

-; n, \z\2)

\ I

I

where do(9) is the normalised measure on dBn. Thus using the explicit formula for K^2'~vl2{t, z, 0) given in theorem 2 and comparing it to that above, we get the desired formula for 4>v(t,p) in (5.2). Remark 3 For the case \v\ > n, we still have an explicit formula for

where Az is the n x n matrix given by: zz* z2 For this transformation we have gz.O = z and it is biholomorphic from Cn\H+ onto Cn\H~ where Hf are the complex hyperplanes (affine): Hf = {w G C n ; < z,w > = ±1}- Finally, we will give below some remarks on the further uses of the formulae obtained for the wave equation of the Laplacians Aap.

5.3 Some further remarks To start with, let F C G be a cocompact discrete group acting on Bn without fixed points and set M = Bn/T the quotient manifold of Bn by F. Then we can restrict the Laplacians Aa/3 to give rise to a family of Laplacians A^jf on M acting on Ta/3—invariant functions i.e on functions satisfying Ta^{^)f = f for every 7 £ F. Therefore it becomes natural to study the distributional traces of the operators associated to the wave equation of A£f i.e TrC1^/1^-)

and Tr(cos(ti/-A^).

Indeed the case of a = 0 = 0 has been

carried out by the authors of the article [2] in which the explicit formula of for the wave kernel K^{t\ z.w) plays an essential role. Thus the case of arbitrary a, j3 with a — f3 ^ 0 might be of great interest for further applications. For instance, the family of operators Aj,/2,-i,/2 = Du had been discussed in great details by Patterson [9] on the hyperbolic disk B\ in the context of stationary spectral theory. Thus the study of the corresponding wave equation of Dv on Bn and M particularly for \v\ > n, might provide a time approch to the spectral scattering theory. We hope then that matters will be clarified and carried out in a near future.

References [1] Boussejra,A. and Intissar,A.: L2—concrete spectral analysis of the invariant Laplacians A a £ in the unit complex ball Bn. Preprint, Faculty of sciences of Rabat (1995) [2]-Bunke,U. Olbrich,M. and Juhl,A.:The wave kernel for the Laplacian on locally symmetric spaces of rank one, Theta functions, Trace formulas and the Selberg zeta function. Ann. Global Anal. Geom. 12 (1994) 357-405 [3]-Elstrodt,J.: Die Resolvente zum Eigenwertproblem da automorphen formen in der hyperbolishen Ebene I. Math.Ann. 203 (1973) 295-330 [4]-Helgason,S.: "Groups and Geometric analysis". Academic Press, New York 1984 11

[5]-Intissar,A. et Ould Moustapha,M.V.: Solution explicite de l'equation des ondes dans l'espace symetrique de type non compact de rang 1. C.R.Acad.Sci.Paris 321 (1995) 77-81 [6]-Lax,P.D. and Phillips,R.S.:The asymptotic distribution of lattice points in Euclidean and non Euclidean spaces. J.Funct.Anal.46 (1982) 280-350. [7]-Magnus , W. , Oberhettinger, F. and Soni R. P.:Formulas and Theorems for The special functions of Mathematical Physics. Third enlarged edition Springer-Verlag Berlin Heidelbeg New York 1966 [8]-Ould Moustapha,M.V.: Analyse spectrale des operateurs de Laplace-Beltrami dans la boule de Lobacheveski et la boule de Bergman dans R2n. These de troisieme cycle, Facultes des Sciences de Rabat (1992). [9]-Patterson, S.J.:The Laplacians operator on a Riemann surface, I. Composito Math. 31 (1975) 83-107. [10]-Rudin, W.:Function Theory in the unit ball of C", Springer-Verlag New York (1980). [ll]-Zhang, G.: A weighted Plancherel formula II. The case the ball. Studia Mathematica 102 (1992) 103-120.

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