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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
Page 2
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Hermite polynomials
CMS quantum systems
Dunkl transformClassical Fourier transformDunkl transform
Translation operator for the Dunkl transformClassical Fourier transformDunkl translation
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 5
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 6
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 7
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Outline
Hermite polynomials
CMS quantum systems
Dunkl transformClassical Fourier transformDunkl transform
Translation operator for the Dunkl transformClassical Fourier transformDunkl translation
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 8
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 9
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 10
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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)
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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)
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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)
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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)
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 22
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Outline
Hermite polynomials
CMS quantum systems
Dunkl transformClassical Fourier transformDunkl transform
Translation operator for the Dunkl transformClassical Fourier transformDunkl translation
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 23
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 24
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 25
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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).
Hendrik De Bie Dunkl operators and Clifford algebras II
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Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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}
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 27
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 28
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
Outline
Hermite polynomials
CMS quantum systems
Dunkl transformClassical Fourier transformDunkl transform
Translation operator for the Dunkl transformClassical Fourier transformDunkl translation
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 29
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 30
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 31
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 32
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 33
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 34
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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〉
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 35
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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〉
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 36
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 37
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 38
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 39
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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 ′〉))
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 40
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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 ′〉))
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 41
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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 ′〉))
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 42
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 43
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 44
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 45
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 46
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl transform
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 47
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
Outline
Hermite polynomials
CMS quantum systems
Dunkl transformClassical Fourier transformDunkl transform
Translation operator for the Dunkl transformClassical Fourier transformDunkl translation
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 48
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 49
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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.
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 50
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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)
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 51
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 52
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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
Hendrik De Bie Dunkl operators and Clifford algebras II
Page 53
Hermite polynomialsCMS quantum systems
Dunkl transformTranslation operator for the Dunkl transform
Classical Fourier transformDunkl translation
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.
Hendrik De Bie Dunkl operators and Clifford algebras II