A Formula representing magnetic Berezin transforms on the Bergman ball of $\C^n$ as functions of the Laplace-Beltrami operator

0.17 MB·English

by  Allal Ghanmi

A FORMULA REPRESENTING MAGNETIC BEREZIN TRANSFORMS ON THE
BERGMAN BALL OF CN AS FUNCTIONS OF THE LAPLACE-BELTRAMI
OPERATOR
ALLALGHANMIANDZOUHAIRMOUAYN
ABSTRACT. WegiveaformulathatrepresentsmagneticBerezintransformsassociatedwith
generalized Bergmanspaces as functions of the Laplace-Beltrami operator on the Bergman
1 ballofCn.Inparticular,werecovertheresultobtainedbyJ.Peeter[J.Oper.Theory,24,1990].
1
0
Key-Words: Bergman ball; Magnetic Schrödinger operator; Generalized Bergman spaces;
2
GeneralizedBerezinTransforms;Laplace-Beltramioperator.

n
a Mathematical Subject Classification 2010:47B35;46B38;47G10
J

9
1

]
P
1 Introduction
S
.
h The Berezin transform was introduced and studied by Berezin [2] for certain classical
t symmetricdomainsinCnandnextextensivelyconsideredinthecontextofBergman,Hardy
a
m and Bargmann-Fock spaces [4, 7, 11]. It is closely related to Topelitz operators [19] and is
⋆
[ useful in the quantization theory and corresponding -product on a suitable algebras of

functionsaswellasinthecorrespondenceprinciple[2,4].
2
This transform is defined as follows. Let Ω be a domain of Cn with a Borel measure µ
v
8 and a closed subspace of L2(Ω,dµ) consisting of continuous functions and possessinga
4 H
reproducingkernelK( , ). Then,theBerezinsymbolσ(A)ofaboundedlinearoperatorAon
2 · ·
1 isthefunctiononΩgivenbyσ(A)(x) := Ae ,e ,wheree ( ) := K( ,x)K(x,x) 1/2 .
x x x −
1. THhus the Toeplitz operator Tf with symbohl f Li∞(Ω) is the· operat·or defined on ∈ Hby
∈ H
0
T (ϕ) = P(fϕ); ϕ , where P is the orthogonal projection from L2(Ω,dµ) into . The
1 f ∈ H H
1 Berezin transform associated to is then defined to be the positive self-adjoint operator
v: B := σT,whichturnsouttobeaHboundedoperatoron L2(Ω,K(x,x)dµ(x)).
Xi In this paper, we deal with the Bergman unit ball Ω = Bn in (Cn, , ) as domain en-
h· ·i
dowedwithitsLebesguemeasuredµ. SincetheBerezintransformcanbedefinedprovided
r
a that thereis agiven closedsubspaceof L2(Bn,dµ) possessingareproducingkernel,weare
concernedherewiththe L2-eigenspaces
A2m,ν(Bn) = ϕ ∈ L2(Bn,(1−|z|2)−n−1dµ); Hνϕ = ǫmν,nϕ , (1.1)
n o
associatedtothediscretespectrum
n
ǫν,n = 4ν(2m+n) 4m(m+n); m = 0,1,2, ,[ν ], (1.2)
m − ··· − 2
oftheSchrödingeroperatorwithuniformmagneticfieldonBn givenby
n ∂2 n ∂ ∂
H = 4(1 z 2) ∑(δ z z ) +ν∑ z z +ν2 +4ν2, (1.3)
ν ij i j j j
− −| | − ∂z ∂z ∂z − ∂z
(j=1 i j j=1(cid:18) j j(cid:19) )
1
2 ALLALGHANMIANDZOUHAIRMOUAYN
provided that ν > n/2. Above [x] denotesthe greatest integer not exceeding x. Note that
for m = 0, the space 2,ν(Bn) reduces further to be isomorphic to the weighted Bergman
spaceofholomorphicAfu0nctions gonBn satisfyingthegrowthcondition
g(z) 2(1 z 2)2ν−n−1dµ(z) < +∞. (1.4)
Bn| | −| |
Z
TheassociatedBerezintransformcanbeexpressedas
B[ϕ](z) = (2ν−n)Γ(2ν) (1−|z|2)(1−|ξ|2) 2ν ϕ(ξ) 1 ξ 2 −(n+1)dµ(ξ). (1.5)
πnΓ(2ν n+1) Bn 1 < z,ξ > 2 −| |
− Z (cid:18) | − | (cid:19)
(cid:0) (cid:1)
Moreover, it can be written as a function of the Laplace-Beltrami operator ∆Bn ([15, p.182])
as
2
B = Γ α+1+ n2 + 2i −∆Bn −n2 , (1.6)
(cid:12) (cid:16) Γ(α+1)Γ(αp+n+1) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where Γ(x) being the usual Gamma function and α = 2ν n 1 occurring in the measure
− −
weight ofthe Bergman space underconsideration in [15] by Peeterwho derived(1.6) from
a formula representing the Berezin transform on the Bergman ball as function of the ∆Bn,
whichwasfirstestablishedbyBerezinin[3].
Our aim is then to generalize (1.5) and (1.6) to the case of the spaces in (1.1). We pre-
cisely attach to each of these eigenspaces a Berezin transform B following the formalism
m
describedaboveas
Γ(n)m!(2(ν m) n)Γ(2ν m) (1 z 2)(1 ξ 2) 2(ν−m)
B [ϕ](z) : = − − − −| | −| | (1.7)
m πnΓ(n+m)Γ(2ν m n+1) Bn 1 z,ξ 2
− − Z (cid:18) | −h i| (cid:19)
2
Pm(n−1,2[ν−m])(1 2 ξ 2) ϕ(ξ)(1 ξ 2)−(n+1)dµ(ξ),
× − | | −| |
(cid:16) (cid:17)
(α,β)
where P ( ) denotes the Jacobi polynomial [10]. Moreover, we have to prove that such
m
·
transformcanalsobeexpressedasafunctionoftheLaplace-Beltramioperator∆Bn interms
ofthe ̥ -sumas
3 2
B = Cν,n ∑2m( 2)jA Γ(n+j)Γ 2(ν−m)− 12 n−i −∆Bn −n2 (1.8)
m m j=0 − j Γ 2(ν−(cid:16)m)+j+ 21 n(cid:16)+i p−∆Bn −n2 (cid:17)(cid:17)
(cid:16) (cid:16) p (cid:17)(cid:17)
̥ 12 n+i −∆Bn −n2 , n+j, 12 n+i −∆Bn −n2 1 ,
3 2
×  (cid:16) (νp−m)+j+ 21(cid:17) n+i −(cid:16)∆Bn −pn2 , n (cid:17) (cid:12)(cid:12) 
 (cid:16) p (cid:17) (cid:12) 
where (cid:12)
(2(ν m) n)Γ(n+m)
Cν,n = − − (1.9)
m m!Γ(2ν n m+1)Γ(2ν n)
− − −
and
min(m,j) (m!)2Γ(2ν m)Γ(2ν m+j p)
A = 2 j ∑ − − − . (1.10)
j − (j p)!(m+p j)!p!(m p)!Γ(n+j p)Γ(n+ p)
p=max(0,j m) − − − −
−
Inparticular,whenm = 0,thetransform(1.7)reducestothewellknownBerezintransform
B in(1.5)ontheunitball[2]andtheexpression(1.8)becomestheformula(1.6).
The paper is organized as follows. In Section 2, we fix some notations and we review
brieflysomeneededtoolsonthebi-holomorphicmapsoftheball,thespectralfunctionofthe
MAGNETICBEREZINTRANSFORMS 3
Laplace-BeltramioperatorandtheFourier-Jacobitransformofthisoperator. InSection3,we
are dealing with the L2-spectraltheoryoftheSchrödingeroperatorwith uniformmagnetic
fieldontheunitballandweareconcernedinparticularwiththereproducingkernelsofthe
eigenspaces associated with the discrete spectrum. In Section 4, we construct for each of
theseeigenspacesaBerezintransformthatweexpressintermsofthebi-holomorphicmaps
of the ball. In Section 5, we give a formula representingthe constructedBerezin transform
asfunctionoftheLaplace-Beltramioperator.
2 SOME NOTATIONS AND TOOLS
Herewefixsomenotationsandwerecallsomeneededtoolswewillbeusingsuchasthe
bi-holomorphicmapsoftheball,thespectralfunctionoftheLaplace-Beltramioperatorand
theFourier-Jacobitransform.
2.1 TheMoebiusgroupofBn. LetCnbeendowedwithitsstandardinnerproduct z,w =
z w + z w + + z w for every z = (z , ,z ), w = (w , ,w ) in Cn,hso thiat
1 1 1 2 1 n 1 n 1 n
··· ··· ···
z = z,z = z 2+ + z 2. We denote by Bn = z Cn : z < 1 the unit ball
1 n
o|f|Cn. Thhe Mioebius|gr|oup·o·f·Bn|, de|noted Aut(Bn), i.e. the{set∈of the|b|iholo}morphic map-
p p
pings ϕ : Bn Bn (called also the autormorphisms of Bn) acts transitively on Bn. For
fixed a Bn, l−e→t P be the orthogonalprojection of Cn onto the subspace < a > generated
a
by a and∈let Q = I P betheprojectionontotheorthogonalcomplementof< a >. Tobe
a a
quite explicit, we ha−ve P (z) = z,a a/ a,a if a = 0 and P (z) = 0. For every a Bn, set
a 0
h i h i 6 ∈
s = (1 a 2)1/2 anddefine
−| |
a P (z) (1 a 2)1/2Q (z)
ϕ (z) = − a − −| | a . (2.1)
a
1 z,a
−h i
Then, the maps ϕ have the following properties [16]: (i) ϕ (0) = a and ϕ (a) = 0, (ii)
a a a
ϕ (0) = s2P sQ and ϕ (a) = 1P 1Q , (iii) ϕ is an involution: ϕ (ϕ (z)) = z,
′a − a− a ′a −s2 a− s a a a a
(iv) ϕa : Bn → Bn is an homeomorphismand belongsto Aut(Bn), (v) The Jacobian |DDwξ| of
| |
thetransformationξ = ϕ (w)isequalto
a
n+1
2
Dw 1 a
| | = −| | (2.2)
Dξ 1 a,ξ 2!
| | | −h i|
and(vi)Theidentity
(1 a,a )(1 w,w )
′
1 ϕa(w),ϕa(w′) = −h i −h i (2.3)
− (1 w,a )(1 a,w )
′
−h i −h i
(cid:10) (cid:11)
holdsforeveryw,w Bn.
′
The elements of A∈ut(Bn) are the maps ϕ ; a Bn, and their unitary transformations.
a
∈
Indeed,ifψ Aut(Bn)and a = ψ 1(0),thenthereisauniqueU intheunitarygroupU(n)
−
∈
suchthatψ = U.ϕ . Moreover,inviewof(2.3),ψsatisfiestheidentity
a
(1 a,a )(1 w,w )
1 ψ(w),ψ(w′) = −h i −h ′i ; w,w′ Bn. (2.4)
− (1 w,a )(1 a,w ) ∈
−h i −h ′i
(cid:10) (cid:11)
Forourpurpose,wewillbeconcernedwiththeautomorphism
A w+z
z
ϕ (w) := , (2.5)
z
1+ w,z
h i
4 ALLALGHANMIANDZOUHAIRMOUAYN
where A isthen nmatrixgivenby
z
×
zz
Az := (1 z 2)12In+ ∗ , (2.6)
−| | 1+ 1 z 2
−| |
I beingtheidentitymatrix. p
n
2.2 TheLaplace-Belramioperator. RecallthattheBergmankerneloftheunitballisgiven
by
n!
K(z,w) = , (2.7)
πn(1 z,w )n+1
−h i
sothatonedefines
n+1
g = ∂ ∂ (logK(z,z)) = 1 z 2 δ z z , (2.8)
ij i j (1 z 2)2 −| | ij− i j
−| |
(cid:0)(cid:0) (cid:1) (cid:1)
wherewehaveusedthenotation∂ and∂ tomean
k k
∂ 1 ∂ ∂ ∂ 1 ∂ ∂
∂ = = i and ∂ = = +i (2.9)
k k
∂z 2 ∂x − ∂y ∂z 2 ∂x ∂y
k (cid:18) k k(cid:19) k (cid:18) k k(cid:19)
withz = x +iy C. Therefore,theBergmanmetriconBn definedthrough
k k k
∈
n
ds2 := ∑∂ ∂ logK(K(z,z))ξ ξ = (1 z 2) 2 ∑ (1 z 2)δ +z z dz dz , (2.10)
z i j i j −| | − −| | i,j i j i⊗ j
i,j i,j=1
(cid:0) (cid:1)
sothattheBergmanball(Bn,ds2)carriesaKählerianstructure. Thecorrespondinginvariant
volumemeasureis
n! dV
dg(z) = K(z,z)dV(z) = πn (1 z 2)n+1 = (1−|z|2)−(n+1)dµ(z) (2.11)
−| |
andthegeodesicdistanced(z,w) reads
1 z,w 2
cosh2(d(z,w)) = | −h i| . (2.12)
(1 z 2)(1 w 2)
−| | −| |
TheLaplace-Beltramioperatorassociatedto(Bn,ds2)isdefinedtobe
∆Bn = 4∑gij∂j∂i = 4(1 z 2)∑ δij zizj ∂i∂j; (2.13)
− −| | −
i,j i,j
(cid:0) (cid:1)
(gij) being the inverse matrix of (gij). Note that ∆Bn is invariant under the action of the
group SU(n,1) and its spectrum σ(∆Bn) in the Hilbert space L2(Bn,(1 z 2)−n−1dµ) is
− | |
purelycontinuousandequalsto] ∞, n2]. Insteadof∆Bn,wehave todealwiththenon-
− −
negativeellipticself-adjoint operator Ln := ∆Bn n2 sothat σ( Ln) = [0,+∞[. Hence,
foragivensuitablerealvaluedfunctio−n f : R− R−,theoperator f−( L )isdefinedby
n
−→ −
f( L )[ϕ](z) := K (z,w)ϕ(w)(1 w 2) n 1dµ(w), (2.14)
n f − −
− Bn −| |
Z
whosethekernelfunctionhastheintegralrepresentation
+∞
K (z,w) = Ψ(z,w;λ)f(λ)dλ, (2.15)
f
Z0
MAGNETICBEREZINTRANSFORMS 5
wherethespectralkernelisgivenby([13,p.13]):
Γ(n+iλ) 4 n+iλ, n iλ
Ψ(z,w;λ) = | 2 | ̥ 2 −2 sinh2(d(z,w)) , (2.16)
4πn+1Γ(n) Γ(iλ) 2 2 1 n −
| | (cid:20) (cid:12) (cid:21)
(cid:12)
intermsofthe2̥1( )-Gausshypergeometricfunction[10]. (cid:12)(cid:12)
·
2.3 TheFourier-JacobiTransform. LetusrecallthattheJacobifunctionsaredefinedby
φλ(α,β)(t) := 2̥1 12 (α+β+1+iλα)+, 211(α+β+1−iλ) −sinh2(|t|) . (2.17)
(cid:20) (cid:12) (cid:21)
(cid:12)
Theparticularcaseoft = iθ andλ = i(2n+α+β+1)leadsto(cid:12)(cid:12)
φ(α,β) (iθ) := ̥ −n,n+α+β+1−iλ sin2( θ ) (2.18)
i(2n+α+β+1) 2 1 α+1 | |
(cid:20) (cid:12) (cid:21)
(cid:12)
n! (α,β) (cid:12)
= (α+1) Pn (cos(|2θ|)), (cid:12) (2.19)
n
( 1,1)
anormalizedJacobipolynomial. Wehavethefollowingspecialcasesφ −2 2 (t) = cos( λt )
λ | |
(0,0)
and φ (t) = P (cosh( 2t )), where P ( ) is the Legendre function. Now, let us as-
λ 21(iλ−1) | | ν ·
sumethatα > 1and β R iR,andset
− ∈ ∪
∆ (t) := (2sinh( t ))2α+1(2cosh( t ))2β+1. (2.20)
α,β
| | | |
Then the Jacobi function φ(α,β) in (2.17) is an even C∞-function on R satisfying the second
λ
orderdifferentialequation
d2 + ∆′α,β(t) d +λ2+(α+β+1)2 φ(α,β)(t) = 0 (2.21)
dt2 ∆ (t)dt λ
α,β !
with φ(α,β)(0) = 1, and is uniquely definedby theseproperties. Moreover, for α > 1 and
λ −
β [ α 1,α+1] iR,wehavetheintegraltransformpair
∈ − − ∪
+∞
g(λ) = f(t)φ(α,β)(t)∆ (t)dt (2.22)
λ α,β
Z
0
+∞
f(t) = (2π)−1 g(λ)φλ(α,β)(t) cα,β(λ) −2dλ, (2.23)
Z0
(cid:12) (cid:12)
where (cid:12) (cid:12)
2α+β+1 iλΓ(α+1)Γ(iλ)
−
c (λ) := . (2.24)
α,β Γ 1 (α+β+1+iλ) Γ 1 (α β+1+iλ)
2 2 −
Thisestablishesaone-to-oneco(cid:0)rrespondencebetw(cid:1)ee(cid:0)nthespaceofeven(cid:1)C∞-functionsonR
withcompactsupportanditsimageundertheFourier-cosinetransform,ascharacterizedby
theclassical Paley-Wienertheorem. Themapping f g, whichiscalled theFourier-Jacobi
7→
transform,can beextendedtoan isometryoftheHilbertspacesbetween L2 R ,∆ (t)dt
+ α,β
and L2 R+,(2π)−1 cα,β(λ) −2dλ . Notethatin thecase ofα > 1and β (cid:0)> α+1, (2.22(cid:1))
− | |
(cid:16) (cid:12) (cid:12) (cid:17)
(cid:12) (cid:12)
6 ALLALGHANMIANDZOUHAIRMOUAYN
remains valid except that we have to add to the right hand side of the second formula in
(2.23)theterm
∑ g(λ)d (λ), (2.25)
α,β
λ D
∈ α,β
where D is afinitesubsetofiR and theweightsd (λ) can be givenexplicitly. Formore
α,β α,β
information on the Fourier-Jacobi transform see Koornwinder’s paper [12] and the refer-
encestherein.
3 THE GENERALIZED BERGMAN SPACES ν (Bn)
Am
Associated to the Kählerian manifold (Bn,ds2), modeling the complex hyperbolic space
ofnegativeholomorphicsectionalcurvature,thereisacanonicalvectorpotential
θ(z) = √ 1(∂ ∂¯)Log(1 z 2). (3.1)
− − − −| |
Thus,themagneticSchrödingeroperator,describingachargedparticlemovinginBn under
theactionofthederivedmagneticfielddθ,canbedefinedas
Hν := (d+√ 1νθ)∗(d+√ 1νθ). (3.2)
− −
ItappearsthenastheLaplace-Beltramioperatorin(2.13)perturbedbyafirstorderdifferen-
tialoperator. Itsexpressioninthecomplexcoordinatesisgivenby([9]):
n ∂2 n ∂ ∂
H = 4 1 z 2 ∑ (δ z z¯ ) +ν∑ z z¯ +ν2 +4ν2. (3.3)
ν ij i j j j
− −| | − ∂z ∂z¯ ∂z − ∂z¯
(i,j=1 i j j=1(cid:18) j j(cid:19) )
(cid:16) (cid:17)
Different aspects of the spectral analysis of H have been studied by many authors under
ν
different names and notations (like Maass Laplacians, Landau Hamiltonians, ... ), see for
example [18, 1, 5] for n 2 and [6, 14] for n = 1. For instance, note that H is an elliptic
ν
≥
denselydefinedoperatorontheHilbertspaceL2(Bn,(1 z 2) n 1dµ)andadmitsaunique
− −
−| |
self-adjoint realization denoted also by H . Moreover, it is a known fact that the discrete
ν
spectrumof H on L2(Bn,(1 z 2) n 1dµ) isnottrivialwheneverν > n/2andconsistsof
ν − −
−| |
afinitenumberofinfinitelydegenerateeigenvaluesoftheform
ǫν,n := 4ν(2m+n) 4m(m+n) (3.4)
m −
for varying integer m such that 0 m < ν n/2. The corresponding L2-eigenfunctions,
≤ −
i.e.,theelementsofthespace 2,ν(Bn)in(1.1)canbewrittenexplicitlyintermsoftheJacobi
m
A
polynomials P(α,β)(x) and the complex spherical harmonics hp,q(z,z¯) [8, 16]. Namely, from
j
[18,9]wecheckthatthefunctions
ψνp,,mq(z) := (1−|z|2)ν−mPm(n+qp+q−1,2[ν−m]−n)(1−2|z|2)hp,q(z,z¯), (3.5)
−
constitute an orthogonal basis of the space in (1.1), for varying integers p and q such that
p = 0,1,2, , and q = 0,1,2, ,m, whose the square norms in L2(Bn,(1 z 2) n 1dµ)
− −
··· ··· −| |
aregivenby
2 πnΓ(n+m+p)Γ(2ν n m q+1)
ψν,m = − − − . (3.6)
p,q 2n!(2[ν m] n)(m q)!Γ(2ν m+p)
(cid:13) (cid:13) − − − −
(cid:13) (cid:13)
Furthermore, the(cid:13)space(cid:13) 2,ν(Bn) is a reproducing kernel Hilbert space, i.e., for every fixed
m
z Bn,thereexistsafunActionKν belongingtothisspacesuchthat
∈ m,z
f(z) = f,Kmν,z = Bn Kmν,n(z,w)f(w)(1−|z|2)−n−1dµ(w) (3.7)
Z
(cid:10) (cid:11)
MAGNETICBEREZINTRANSFORMS 7
for every f 2,ν(Bn), where we have set Kν,n(z,w) = Kν (w). Its expressionis given in
∈ Am m m,z
termsofthehyperbolicdistancein(2.12)as
ν
1 z,w
Kmν,n(z,w) = γmν,n 1−hz,wi (cosh(d(z,w)))−2(ν−m) (3.8)
!
−h i
Pm(n−1,2[ν−m]−n) 1 2tanh2(d(z,w)) ,
× −
(cid:16) (cid:17)
where
(2(ν m) n)Γ(2ν m)
γν,n = − − − . (3.9)
m πnΓ(2ν m n+1)
− −
For m = 0, the L2-eigenspaceassociatedto thebottomeigenvalue ǫν,n = 4νn is isomorphic
0
to the usual weighted Bergman Hilbert space of holomorphic functions g on Bn such that
(1.4)holds. Thecorrespondingreproducingkernelreads
(2ν n)Γ(2ν) 2ν
Kν,n(z,w) = − cosh(d(z,w)) − . (3.10)
0 πnΓ(2ν n+1)
− (cid:16) (cid:17)
Definition3.1. Assumethat ν > n/2andlet m bean integer suchthat0 m < ν n/2. Then
≤ −
thespace 2,ν(Bn)iscalledhereageneralized Bergmanspaceofindexm.
m
A
4 GENERALIZED BEREZIN TRANSFORMS ON THE UNIT BALL
WeadopttotheformalismpresentedinSection1tothecaseoftheBergmanballΩ = Bn
as domain of Cn and 2,ν(Bn) as a closed subspace of L2(Bn,(1 z 2) n 1dµ). Then, the
m − −
A −| |
Berezinsymbolofaboundedoperator Aon 2,ν(Bn)isthefunction
m
A
σ (A)(z) := A K , K ; z Bn. (4.1)
m m m
z z L2(Bn,(1 z2) n 1dµ) ∈
D (cid:16) (cid:17) (cid:16) (cid:17) E −| | − −
f f
Here(K ) denotesthenormalizedreproducingkernelwithevaluationatz,thatis
m z
Kν,n(z,w)
f (K ) (w) = m , w Bn, (4.2)
m z Kν,n(z,z) ∈
m
whereKν,n(z,w) isgiveninf(3.8). Therefpore,forgivencomplex-valuedfunction φ suchthat
m
φK L2(Bn,(1 z 2) n 1dµ), the Berezin transform of φ is defined to be the Berezin
m − −
∈ −| |
symbol of the Toeplitz operator T (φ) with symbol φ defined on 2,ν(Bn) by T (φ)[f] :=
m m m
Pf[φf]. Thatis, A
m
B [φ](z) = σ (T (φ))(z) = T (φ) K , K = P φ. K , K . (4.3)
m m m m m m m m m
z z z z
D (cid:16) (cid:17) (cid:16) (cid:17) E D (cid:16) (cid:16) (cid:17) (cid:17) (cid:16) (cid:17) E
Byexpressing(4.3)asanintegral,wegetf f f f
Bm[φ](z) = Bm(z,w)φ(w)(1 w 2)−n−1dµ(w), (4.4)
Bn −| |
Z
where
Kν,n(z,w) 2
B (z,w) := | m | , z,w Bn. (4.5)
m Kν,n(z,z) ∈
m
8 ALLALGHANMIANDZOUHAIRMOUAYN
Explicitly,
Bm(z,w) = Γm(m!Γ+(nn))γmν,n(coshd(z,w))−4(ν−m) Pm(n−1,2[ν−m]) 1−2tanh2(d(z,w)) 2,
(cid:16) (cid:16) (cid:17)(cid:17)(4.6)
whereγν,n isasin(3.9). Wethenstatethefollowing
m
Definition 4.1. Let m = 0,1,2, ,[ν n] and let γν,n as in (3.9). Then, the integral operator
··· − 2 m
actingonφ L∞(Bn)as
∈
m!Γ(n)
Bm[φ](z) =Γ(m+n)γmν,n Bn(cosh(d(z,w)))−4(ν−m) (4.7)
Z
Pm(n−1,2[ν−m]) 1 2tanh2(d(z,w)) 2φ(w) 1 w 2 −n−1dµ(w).
× − −| |
(cid:16) (cid:16) (cid:17)(cid:17)
(cid:0) (cid:1)
iscalledthegeneralized Berezintransformofindexm.
Proposition4.2. ThetransformB definedin(4.7)canbewrittenas
m
m!Γ(n)
Bm[φ](z) = Γ(m+n)γmν,n Bn(1−|w|2)2(ν−m) (4.8)
Z
2
Pm(n−1,2[ν−m])(1 2 w 2) φ(ϕz(w))(1 w 2)−n−1dµ(w),
× − | | −| |
(cid:16) (cid:17)
where ϕ (w)isasin(2.5).
z
Proof. Fixz Bn andsetw = ϕ 1(ξ),where ϕ (w)isasin(2.5). Wehave ϕ Aut(Bn)and
∈ −z z z ∈
ϕ (0) = z. Thequantityintherighthandsideof(4.8)reads,intermsofξ,as
z
Γm(m!Γ+(nn))γmν,n 1− ϕ−z 1(ξ) 2 2(ν−m) (4.9)
(cid:18) (cid:12) (cid:12) (cid:19)
(cid:12) (cid:12) 2 n 1
× Pm(n−1,2[ν−m](cid:12)) 1−2(cid:12) ϕ−z 1(ξ) 2 φ(ξ) 1− ϕ−z 1(ξ) 2 − − dµ ϕ−z 1(ξ) .
(cid:18) (cid:18) (cid:12) (cid:12) (cid:19)(cid:19) (cid:18) (cid:12) (cid:12) (cid:19) (cid:16) (cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
Bywritingdowntheidentity(2.4)(cid:12)forψ =(cid:12)ϕ 1andw = w ,(cid:12)weget(cid:12)
−z ′
2 (1 z 2) 1 ξ 2
1− ϕ−z 1(ξ) = −1| | ξ(cid:0),z−2| | (cid:1) = cosh−2(d(z,ξ)) = 1−tanh2(d(z,ξ)). (4.10)
(cid:12) (cid:12) | −h i|
(cid:12) (cid:12)
Also, by(cid:12)making(cid:12)use of (4.11) giving the explicit expression of the Jacobian of the transfor-
mationξ = ϕ (w)isgiventhrough(4.11),weseethat
z
n 1 n 1
1− ϕ−z 1(ξ) 2 − − dµ ϕ−z 1(ξ) = 1− ϕ−z 1(ξ) 2 − − |DDwξ|dµ(ξ)
(cid:18) (cid:12) (cid:12) (cid:19) (cid:16) (cid:17) (cid:18) (cid:12) (cid:12) (cid:19) | |
(cid:12)(cid:12) (cid:12)(cid:12) = (1 ξ(cid:12)(cid:12) 2)−n−1(cid:12)(cid:12)dµ(ξ). (4.11)
−| |
Therefore,theexpressionin(4.9)canalsobewrittenas
2(ν m)
m!Γ(n) (1 z 2) 1 ξ 2 −
γν,n −| | −| | (4.12)
Γ(m+n) m ZBn 1 ξ(cid:0),z 2 (cid:1)!
| −h i|
Pm(n−1,2[ν−m]) 1 2tanh2(d(z,ξ)) 2φ(ξ) 1 ξ 2 −n−1dµ(ξ)
× − −| |
(cid:16) (cid:16) (cid:17)(cid:17)
(cid:0) (cid:1)
MAGNETICBEREZINTRANSFORMS 9
orequivalentlyas
m!Γ(n)
Γ(m+n)γmν,n Bn (coshd(z,ξ))−4(ν−m) (4.13)
Z
Pm(n−1,2[ν−m]) 1 2tanh2(d(z,ξ)) 2φ(ξ) 1 ξ 2 −n−1dµ(ξ),
× − −| |
whichisexactlyt(cid:16)heexpression(cid:16)ofB asdefinedin(cid:17)(4(cid:17).7). (cid:0) (cid:1) (cid:3)
m
5 Bm AS FUNCTION OF THE LAPLACE-BELTRAMI OPERATOR
Our task here is to express the generalized Berezin transform B as a function of the
m
Laplace-Beltramioperator∆Bn. Precisely,weestablishthefollowingresult.
Theorem 5.1. Let m = 0,1,2, ,[ν n]. Then, the transform B defined in (4.7) can be ex-
··· − 2 m
pressed intermsoftheLaplace-Beltarmi operator ∆Bn as
B = Cν,n ∑2m( 2)jA Γ(n+j)Γ 2(ν−m)− 21 n−i −∆Bn −n2
m m j=0 − j Γ 2(ν−(cid:16)m)+j+ 12 n(cid:16)+i p−∆Bn −n2 (cid:17)(cid:17)
(cid:16) (cid:16) p (cid:17)(cid:17)
̥ 21 n+i −∆Bn −n2 , n+j, 12 n+i −∆Bn −n2 1 ,
3 2
×  (cid:16) (νp−m)+j+ 21(cid:17) n+i −(cid:16)∆Bn −pn2 , n (cid:17) (cid:12)(cid:12) 
 (cid:16) p (cid:17) (cid:12) 
where (cid:12)
(2(ν m) n)Γ(n+m)
Cν,n = − −
m m!Γ(2ν n m+1)Γ(2ν n)
− − −
min(m,j) (m!)2Γ(2ν m)Γ(2ν m+j p)
Aj = 2−j ∑ (j p)!(m+ p j)!−p!(m p)!−Γ(n+j− p)Γ(n+p).
p=max(0,j m) − − − −
−
Proof. Ononehand,thetransformB istheintegraloperator
m
B [ϕ](z) = B (z,w)ϕ(w)(1 w 2) n 1dµ(w) (5.1)
m m − −
Bn −| |
Z
withthekernelfunction
m!Γ(n) 2
Bm(z,w) = Γ(n+m)γmν,n(cosh(ρ))−4(ν−m) Pm(n−1,2[ν−m]) 1−2tanh2(ρ) , (5.2)
(cid:16) (cid:16) (cid:17)(cid:17)
where ρ = d(z,w) and γν,n is as in (3.9). On the other hand, from (2.14) combined with
m
(2.15),weobtain
+∞
f( ∆Bn n2)[ϕ](z) = Ψ(z,w;λ)f(λ)dλ ϕ(w)(1 w 2)−n−1dµ(w), (5.3)
− − ZBn(cid:18)Z0 (cid:19) −| |
Ψ(z,w;λ) being the spectral function given in (2.16). Now, by equating (5.1) and (5.3) we
get
+∞
Ψ(z,w;λ)f(λ)dλ = B (z,w) (5.4)
m
Z0
byuniquenessofthekernels. Explicitly,
+∞ Γ n+iλ 4 n+iλ, n iλ
2 ̥ 2 −2 sinh2(ρ) f(λ)dλ = h(ρ), (5.5)
Z0 (cid:12) Γ(cid:0)(iλ)(cid:1)2(cid:12) 2 1(cid:20) n (cid:12)− (cid:21)
(cid:12) | | (cid:12) (cid:12)
(cid:12)
(cid:12)
10 ALLALGHANMIANDZOUHAIRMOUAYN
wherehisusedtorepresentsthefunction4πn+1m!Γ(n)B (z,w),i.e.,
m
4πn+1m!Γ2(n) 2
h(ρ) := Γ(n+m) γmν,n(cosh(ρ))−4(ν−m) Pm(n−1,2(ν−m)−n)(1−2tanh2(ρ)) . (5.6)
(cid:16) (cid:17)
Todeterminethefunction f weneedtoinverttheintegralequation(5.5). Forthis,wemake
appealtotheFourier-Jacobitransform
h L2 R+,∆α,β(t)dt g L2 R+,(2π)−1 cα,β(λ) −2dλ (5.7)
∈ 7−→ ∈
(cid:0) (cid:1) (cid:16) (cid:12) (cid:12) (cid:17)
asdefinedin(2.22)by (cid:12) (cid:12)
+∞
g(λ) = h(t)φ(α,β)(t)∆ (t)dt. (5.8)
λ α.β
Z0
Itsinverseisgivenby
1 +∞
(α,β) 2
h(t) = 2π g(λ)φλ (t) cα,β(λ) − dλ. (5.9)
Z0
(cid:12) (cid:12)
For α = n 1, β = 0 and t = ρ, the involved quantit(cid:12)ies φ(α,β(cid:12))(t), ∆ (t) and c (λ) given
− λ α,β α,β
respectivelyby(2.17),(2.20)and(2.24)read
φλ(n−1,0)(ρ) = 2̥1 n+2iλ, 12(nn−iλ) −sinh2(ρ) (5.10)
(cid:20) (cid:12) (cid:21)
(cid:12)
∆n 1,0(ρ) = 2cosh(ρ)(sinh(ρ))2n−1 (cid:12)(cid:12) (5.11)
−
2n iλΓ(n)Γ(iλ)
−
c (λ) = . (5.12)
n−1,0 Γ2 n+iλ
2
(cid:0) (cid:1)(n 1,0)
Therefore,(5.5)canberewrittenintermsofφλ − (ρ)andcn 1,0(λ)as
−
1 +∞ 1
2π Z0 f(λ)φλ(n−1,0)(ρ)|cn−1,0(λ)|−2dλ = 2π22nΓ2(n)h(ρ). (5.13)
Inviewofthetransform(5.8),thefunction f takestheform
1 +∞
f(λ) = 2π22nΓ2(n) Z0 h(ρ)φλ(n−1,0)(ρ)∆n−1,0(ρ)dρ. (5.14)
Substitutionofh(ρ),φλ(n−1,0)(ρ)and∆n 1,0(ρ)bytheirexplicitexpressions,asgivenby(5.6),
−
(5.10)ad(5.11)respectively,yields
2πnm! +∞ n+iλ, n iλ
f(λ) = γν,n ̥ 2 −2 sinh2(ρ) (5.15)
Γ(n+m) m 2 1 n −
Z0 (cid:20) (cid:12) (cid:21)
(cid:12)
(sinh(ρ))2n−1(cosh(ρ))−4(ν−m)+1 (cid:12)(cid:12)Pm(n−1,2(ν−m)−n)(1 2tanh2(ρ)) 2dρ.
× −
(cid:16) (cid:17)
Bythechangeofvariable x = sinh2(ρ),Equation(5.15)becomes
πnm! +∞ n+iλ, n iλ
f(λ) = γν,n ̥ 2 −2 x (5.16)
Γ(n+m) m 2 1 n −
Z0 (cid:20) (cid:12) (cid:21)
(cid:12)
(cid:12) 1 x 2
×xn−1(1+x)−2(ν−m) (cid:12)Pm(n−1,2(ν−m)−n) 1−+x dx.
(cid:18) (cid:18) (cid:19)(cid:19)