Dunkl operators and Clifford algebras II

Hermite polynomialsCMS quantum systems

Dunkl transformTranslation operator for the Dunkl transform

Dunkl operators and Clifford algebras II

Hendrik De Bie

Clifford Research GroupDepartment of Mathematical Analysis

Ghent University

Hong Kong, March, 2011

Hendrik De Bie Dunkl operators and Clifford algebras II



Hermite polynomials

CMS quantum systems

Dunkl transformClassical Fourier transformDunkl transform

Translation operator for the Dunkl transformClassical Fourier transformDunkl translation




Summary previous lecture:Data:

I root system R in Rm, encoding finite reflection group GI multiplicity function k : R → C

Dunkl operators Ti , i = 1, . . . ,m

Ti f (x) = ∂xi f (x) +∑α∈R+

kααif (x)− f (σα(x))

〈α, x〉

Dunkl Laplacian

∆k =m∑

i=1

T 2i

Euler operator

E =m∑

i=1

xi∂xi




Summary previous lecture:Data:

I root system R in Rm, encoding finite reflection group GI multiplicity function k : R → C

Dunkl operators Ti , i = 1, . . . ,m

Ti f (x) = ∂xi f (x) +∑α∈R+

kααif (x)− f (σα(x))

〈α, x〉

Dunkl Laplacian

∆k =m∑

i=1

T 2i

Euler operator

E =m∑

i=1

xi∂xi




Define the following parameter:

µ =1

2∆k |x |2 = m + 2

∑α∈R+

kα ∈ C

Then we have the following operator identities:

Theorem

The operators ∆k , |x |2 and E + µ/2 generate the Lie algebra sl2[∆k , |x |2

]= 4(E +

µ

2)[

∆k ,E +µ

2

]= 2∆k[

|x |2,E +µ

2

]= −2|x |2




This lecture:

I Can we define Hermite polynomials related to the Dunkloperators?

I Is there a related quantum system?

I Same question for Fourier transform




Outline

Hermite polynomials

CMS quantum systems






Dunkl harmonics

I H` space of `-homogeneous null-solutions of ∆k

I weight wk(x) =∏α∈R+

|〈α, x〉|2kα

Then orthogonality:

Theorem

Let H` and Hn be Dunkl harmonics of different degree. Then onehas ∫

Sm−1

H`(x)Hn(x)wk(x)dσ(x) = 0.

We use Dunkl harmonics as building blocks for our Hermitepolynomials




Dunkl harmonics

I H` space of `-homogeneous null-solutions of ∆k

I weight wk(x) =∏α∈R+

|〈α, x〉|2kα

Then orthogonality:

Theorem

Let H` and Hn be Dunkl harmonics of different degree. Then onehas ∫

Sm−1

H`(x)Hn(x)wk(x)dσ(x) = 0.

We use Dunkl harmonics as building blocks for our Hermitepolynomials




Let {H(n)` }, n = 0, . . . , dimH` be ONB of H`.

Then

Definition

The Hermite polynomials related to this basis are given by

ψj ,`,n := D jH(n)` , j = 0, 1, . . .

withD := ∆k + 4|x |2 − 2(2E + µ)

(Natural generalization of rank one case)




Let {H(n)` }, n = 0, . . . , dimH` be ONB of H`.

Then

Definition

The Hermite polynomials related to this basis are given by

ψj ,`,n := D jH(n)` , j = 0, 1, . . .

withD := ∆k + 4|x |2 − 2(2E + µ)

(Natural generalization of rank one case)




Theorem (Rodrigues formula)

The Hermite polynomials take the form

ψj ,`,n = exp(|x |2/2)(−∆k − |x |2 + 2E + µ)j exp(−|x |2/2)H`

= exp(|x |2)(−∆k)j exp(−|x |2)H`.

Theorem (Differential equation)

ψj ,`,n is a solution of the following PDE:

[∆k − 2E]ψj ,`,n = −2(2j + `)ψj ,`,n.




Note thatψj ,`,n = fj ,`(|x |2)H

(n)` (x)

More precisely

Theorem

The Hermite polynomials can be written in terms of thegeneralized Laguerre polynomials as

ψj ,`,n = cjLµ2

+`−1

j (|x |2)H(n)` (x),

with

Lαt (x) =t∑

i=0

Γ(t + α + 1)

i !(t − i)!Γ(i + α + 1)(−x)i .

and cj a normalization constant.




Note thatψj ,`,n = fj ,`(|x |2)H

(n)` (x)

More precisely

Theorem

The Hermite polynomials can be written in terms of thegeneralized Laguerre polynomials as

ψj ,`,n = cjLµ2

+`−1

j (|x |2)H(n)` (x),

with

Lαt (x) =t∑

i=0

Γ(t + α + 1)

i !(t − i)!Γ(i + α + 1)(−x)i .

and cj a normalization constant.




OrthogonalityDefine Hermite functions by

φj ,`,n := ψj ,`,n exp(−|x |2/2)

Theorem

One has ∫Rm

φj1,`1,n1 φj2,`2,n2wk(x)dx ∼ δj1j2δ`1`2δn1n2

Proof: split integral in spherical and radial part; use orthogonalityof Laguerre polynomials.




OrthogonalityDefine Hermite functions by

φj ,`,n := ψj ,`,n exp(−|x |2/2)

Theorem

One has ∫Rm

φj1,`1,n1 φj2,`2,n2wk(x)dx ∼ δj1j2δ`1`2δn1n2

Proof: split integral in spherical and radial part; use orthogonalityof Laguerre polynomials.




Why so important?

Proposition

The Hermite polynomials {ψj ,`,n} form a basis for the space of allpolynomials P.

It can be proven that{φj ,`,n}

is dense in both

I L2(Rm,wk(x)dx)

I S(Rm)




Why so important?

Proposition

The Hermite polynomials {ψj ,`,n} form a basis for the space of allpolynomials P.

It can be proven that{φj ,`,n}

is dense in both

I L2(Rm,wk(x)dx)

I S(Rm)




Alternative definition:Let {φν , ν ∈ Zm

+} be a basis of P such that φν ∈ P|ν|.

Definition

The generalized Hermite polynomials {Hν , ν ∈ Zm+} associated

with {φν} on Rm are given by

Hν(x) := e−∆k/4φν(x) =

b|ν|/2c∑n=0

(−1)n

4nn!∆n

kφν(x).

Rosler M.,

Generalized Hermite polynomials and the heat equation for Dunkl operators.Comm. Math. Phys. 192, 3 (1998), 519–542.




Alternative definition:Let {φν , ν ∈ Zm

+} be a basis of P such that φν ∈ P|ν|.

Definition

The generalized Hermite polynomials {Hν , ν ∈ Zm+} associated

with {φν} on Rm are given by

Hν(x) := e−∆k/4φν(x) =

b|ν|/2c∑n=0

(−1)n

4nn!∆n

kφν(x).

Rosler M.,

Generalized Hermite polynomials and the heat equation for Dunkl operators.Comm. Math. Phys. 192, 3 (1998), 519–542.




Reduces to previous definition by choosing as basis for Ppolynomials of the form

|x |2jH(l)n

i.e.e−∆k/4|x |2jH

(l)n ∼ ψj ,`,n




Outline

Hermite polynomials

CMS quantum systems






Recall quantum harmonic oscillator in Rm:

−∆

2ψ +

|x |2

2ψ = Eψ

(PDE, E is eigenvalue called energy)

This invites us to consider:

−∆k

2ψ +

|x |2

2ψ = Eψ

(Replace Laplacian by Dunkl Laplacian!)This new equation contains differential AND difference terms




Recall quantum harmonic oscillator in Rm:

−∆

2ψ +

|x |2

2ψ = Eψ

(PDE, E is eigenvalue called energy)

This invites us to consider:

−∆k

2ψ +

|x |2

2ψ = Eψ

(Replace Laplacian by Dunkl Laplacian!)This new equation contains differential AND difference terms




These QM systems derive from actual physical problems

Class of systems = Calogero-Moser-Sutherland (CMS) models

I m identical particles on line or circle

I external potential

I pairwise interaction

T. H. Baker and P. J. Forrester,

The Calogero-Sutherland model and generalized classical polynomials,Comm. Math. Phys. 188 (1997) 175–216.

J.F. van Diejen and L. Vinet,

Calogero-Sutherland-Moser Models(CRM Series in Mathematical Physics, Springer-Verlag, 2000).




Theorem

The Hermite functions {φj ,`,n} form a complete set ofeigenfunctions of

H := −1

2(∆k − |x |2)

satisfying

Hφj ,`,n =(µ

2+ 2j + `

)φj ,`,n.

Proof: use the sl2 relations.

Consequences:

I complete decomposition of Hilbert space L2(Rm,wk(x)dx)into H-eigenspaces

I alternative proof of orthogonality of {φj ,`,n}




Some more references:

Explicit examples of physical systems and reduction to Dunkl case:

C.F. Dunkl,

Reflection groups in analysis and applications.Japan. J. Math. 3 (2008), 215–246.

M. Rosler,

Dunkl operators: theory and applications.Lecture Notes in Math., 1817,Orthogonal polynomials and special functions, Leuven, 2002, (Springer, Berlin, 2003) 93–135. Online:arXiv:math/0210366.

Study of Hermite polynomials in superspace; extensive comparisonwith Dunkl case

K. Coulembier, H. De Bie and F. Sommen

Orthogonality of Hermite polynomials in superspace and Mehler type formulae,Accepted in Proc. LMS, arXiv:1002.1118.




Classical Fourier transformDunkl transform

Outline

Hermite polynomials

CMS quantum systems







The classical Fourier transformDefinition:

F(f )(y) = (2π)−m2

∫Rm

e i〈x ,y〉f (x)dx

Here, K (x , y) = e i〈x ,y〉 is the unique solution of the system

∂xj K (x , y) = iyjK (x , y), j = 1, . . . ,m

K (0, y) = 1





The Dunkl transform (k > 0)Consider the system

Txj K (x , y) = iyjK (x , y), j = 1, . . . ,m

K (0, y) = 1

One proves that this system has a unique solution

K (x , y) = Vk

(e i〈x ,y〉

)

Then

Definition

The Dunkl transform is defined by

Fk(f )(y) = ck

∫Rm

K (x , y)f (x)wk(x)dx

with wk(x)dx the G-invariant measure and ck a constant.





The Dunkl transform (k > 0)Consider the system

Txj K (x , y) = iyjK (x , y), j = 1, . . . ,m

K (0, y) = 1

One proves that this system has a unique solution

K (x , y) = Vk

(e i〈x ,y〉

)Then

Definition

The Dunkl transform is defined by

Fk(f )(y) = ck

∫Rm

K (x , y)f (x)wk(x)dx

with wk(x)dx the G-invariant measure and ck a constant.





No explicit expression of K (x , y) known, except special cases

Properties:

I |K (x , y)| ≤ 1, for all x , y ∈ Rm

I Fk well-defined on L1(Rm,wk(x))

I K (x , y) = K (y , x)

I K (g · x , g · y) = K (x , y) for all g ∈ G

de Jeu, M.F.E.

The Dunkl transform.Invent. Math. 113 (1993), 147–162.





No explicit expression of K (x , y) known, except special cases

Properties:

I |K (x , y)| ≤ 1, for all x , y ∈ Rm

I Fk well-defined on L1(Rm,wk(x))

I K (x , y) = K (y , x)

I K (g · x , g · y) = K (x , y) for all g ∈ G

de Jeu, M.F.E.

The Dunkl transform.Invent. Math. 113 (1993), 147–162.





Proposition

Let f ∈ S(Rm). Then

Fk(Txj f ) = −iyjFk(f )

Fk(xj f ) = −iTyjFk(f ).

Moreover, Fk leaves S(Rm) invariant.

Proof: use Txj K (x , y) = iyjK (x , y) and

〈Tj f , g〉 = −〈f ,Tjg〉





Proposition

Let f ∈ S(Rm). Then

Fk(Txj f ) = −iyjFk(f )

Fk(xj f ) = −iTyjFk(f ).

Moreover, Fk leaves S(Rm) invariant.

Proof: use Txj K (x , y) = iyjK (x , y) and

〈Tj f , g〉 = −〈f ,Tjg〉





Spectrum of Dunkl transform:Dunkl transform acts nicely on Hermite functions {φj ,`,n}

Theorem

One hasFkφj ,`,n = i2j+`φj ,`,n

Proof:

I Recall φj ,`,n = (−∆k − |x |2 + 2E + µ)j exp(−|x |2/2)H`

I Compute Fk(H`e−|x |2/2)

Corollary

One hasF4

k = id.






Theorem


Proof:



Corollary

One hasF4

k = id.






Theorem


Proof:



Corollary

One hasF4

k = id.





Bochner formula? Dunkl transform of f (|x |)H` with H` ∈ H`

Theorem

Let H` ∈ H` and f (|x |) of suitable decay. Then one has

Fk(f (|x |)H`)(y) = c`H`(y)F`+µ/2−1(f )(|y |)

with

Fα(f )(s) :=

∫ +∞

0f (r)(rs)−αJα(rs)r 2α+1dr

the Hankel transform and c` a constant only depending on `.

Proof based on decomposition

K (x , y) =∞∑`=0

d`J`+µ/2−1(|x ||y |)Vk(Cµ/2−1` (〈x ′, y ′〉))





Bochner formula? Dunkl transform of f (|x |)H` with H` ∈ H`Theorem


Fk(f (|x |)H`)(y) = c`H`(y)F`+µ/2−1(f )(|y |)

with

Fα(f )(s) :=

∫ +∞




K (x , y) =∞∑`=0

d`J`+µ/2−1(|x ||y |)Vk(Cµ/2−1` (〈x ′, y ′〉))





Bochner formula? Dunkl transform of f (|x |)H` with H` ∈ H`Theorem


Fk(f (|x |)H`)(y) = c`H`(y)F`+µ/2−1(f )(|y |)

with

Fα(f )(s) :=

∫ +∞




K (x , y) =∞∑`=0

d`J`+µ/2−1(|x ||y |)Vk(Cµ/2−1` (〈x ′, y ′〉))





Heisenberg uncertainty principle:S(Rm) is dense in L2(Rm,wk), so Fk extends to L2

Theorem

Let f ∈ L2(Rm,wk(x)). Then

|| |x |f ||2 || |x |Fk(f ) ||2 ≥µ

2||f ||22 .

Again consequence of Hermite functions being solutions of CMSsystem





Heisenberg uncertainty principle:S(Rm) is dense in L2(Rm,wk), so Fk extends to L2

Theorem

Let f ∈ L2(Rm,wk(x)). Then

|| |x |f ||2 || |x |Fk(f ) ||2 ≥µ

2||f ||22 .

Again consequence of Hermite functions being solutions of CMSsystem





There exists interesting other expression for Dunkl TF on S(Rm):

Fk = eiπ4

(−∆k+|x |2−µ)

Indeed, check that

eiπ4

(−∆k+|x |2−µ)φj ,`,n = i2j+kφj ,`,n

using the CMS system:

−1

2(∆k − |x |2)φj ,`,n =

(µ2

+ 2j + `)φj ,`,n.

Ben Saıd S.

On the integrability of a representation of sl(2,R).J. Funct. Anal. 250 (2007), 249–264.





There exists interesting other expression for Dunkl TF on S(Rm):

Fk = eiπ4

(−∆k+|x |2−µ)

Indeed, check that

eiπ4

(−∆k+|x |2−µ)φj ,`,n = i2j+kφj ,`,n

using the CMS system:

−1

2(∆k − |x |2)φj ,`,n =

(µ2

+ 2j + `)φj ,`,n.

Ben Saıd S.

On the integrability of a representation of sl(2,R).J. Funct. Anal. 250 (2007), 249–264.





The formulationFk = e

iπ4

(−∆k−|x |2−µ)

is ideal for further generalizations

One only needs operators generating sl2

See

S. Ben Saıd, T. Kobayashi and B. Ørsted,

Laguerre semigroup and Dunkl operators.Preprint: arXiv:0907.3749, 74 pages.

H. De Bie, B. Orsted, P. Somberg and V. Soucek,

The Clifford deformation of the Hermite semigroup.Preprint, 27 pages, arXiv:1101.5551.




Classical Fourier transformDunkl translation

Outline

Hermite polynomials

CMS quantum systems







Convolution for the classical Fourier transformDefinition of convolution:

(f ∗ g)(x) =

∫Rm

f (x − y)g(y)dy ,

Crucial to prove inversion of the FT for other function spaces thanS(Rm)(take g the heat kernel approximation of delta distribution)

? similar approach for Dunkl transform





Convolution depends on the translation operator

τy : f (x) 7→ f (x − y)

Under the Fourier transform, τy satisfies

F (τy f (x)) (z) = e i〈z,y〉F(f )(z)

I use as definition in case of Dunkl TFI very powerful technique!

M. Rosler,

A positive radial product formula for the Dunkl kernel.Trans. Amer. Math. Soc. 355 (2003), 2413–2438.

S. Thangavelu and Y. Xu,

Convolution operator and maximal function for the Dunkl transform.J. Anal. Math. 97 (2005), 25–55.





Formal definition:

Definition

The translation operator for the Dunkl transform is defined by

τy f (x) =

∫Rm

K (−x , z)K (y , z)Fk(f )(z)dz .

I If f ∈ S(Rm), then also τy f (x)

I Very complicated to compute τy f (x)





Main result:

Theorem

Let f (|x |) be a radial function. Then

τy (f )(x) = Vk (f (|x − y |)) .

Proof:Note that Dunkl TF of radial function is again radialThen use decomposition

K (x , y) =∞∑`=0

d`J`+µ/2−1(|x ||y |)Vk(Cµ/2−1` (〈x ′, y ′〉))

combined with orthogonality of Gegenbauers and addition formulafor Bessel function





Main result:

Theorem

Let f (|x |) be a radial function. Then

τy (f )(x) = Vk (f (|x − y |)) .

Proof:Note that Dunkl TF of radial function is again radialThen use decomposition

K (x , y) =∞∑`=0

d`J`+µ/2−1(|x ||y |)Vk(Cµ/2−1` (〈x ′, y ′〉))

combined with orthogonality of Gegenbauers and addition formulafor Bessel function





Many difficult problems and open questions related to translationoperator:

I explicit formula for specific G?

I boundedness of translation on certain function spaces

I translation of non-radial functions, explicit examples?

I positivity? (in general: NO)

I etc.


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Dunkl operators and Clifford algebras II