maximal symmetries for pseudo-differential operators · 2012-10-15 · maximal symmetries for...

MAXIMAL SYMMETRIES FORPSEUDO-DIFFERENTIAL OPERATORS

Maurice A. de Gosson

University of Vienna- NuHAG

15.10.2012

M. A. de Gosson (Institute) ESI 2012 15.10.2012 1 / 25

References

M. de Gosson. Symplectic Covariance Properties for Shubin andBorn-Jordan Pseudo-Differential Operators. (Dedicated to HansFeichtinger for his 60th birthday). Trans. Amer. Math. Soc. 2012[available online]

M. de Gosson, F. Luef. Preferred Quantization Rules: Born—Jordan vs.Weyl; Applications to Phase Space Quantization. Journal ofPseudodifferential Operators and Applications, Volume 2, Number 1,115—139, 2011

M. de Gosson, N. Dias, J. Prata. Maximal Covariance Group ofWigner Transforms and Pseudo-Differential Operators. Preprint 2012


Weyl Operators

Formally the Weyl correspondence AWeyl←→ a is defined by

Aψ(x) =1

(2π)n

∫Rn×Rn

e i (x−y )·ξa( 12 (x + y), ξ)ψ(y) dydξ. (1)

It makes sense for a ∈ S(R2n) and ψ ∈ S(Rn). A general definition, validfor a ∈ S ′(R2n) is

〈Aψ|φ〉 = 〈〈a,W (ψ, φ)〉〉where

W (ψ, φ) =1

(2π)n

∫Rne−iy ·ξψ(x + 1

2y)φ(x +12y)dy

is the cross-Wigner transform. The cross-Wigner transform extends to amapping

S ′(Rn)× S ′(Rn) −→ S ′(R2n).


Symplectic covariance

It is (more or less...) well-known that Weyl operators enjoy a symmetryproperty known as “symplectic covariance with respect to the metaplecticgroup”:

If AWeyl←→ a and S ∈ Sp(2n,R) (the symplectic group) then

S−1ASWeyl←→ a ◦ S where S ∈ Mp(2n,R) (the metaplectic

group) covers S .

This property is characteristic of the Weyl correspondence (Stein, Wong):

Let a 7−→ Op(a) be a linear mapping from S ′(R2n) to thespace of linear operators that is continuous in the topology ofS ′(R2n). Assume that: (i) if a = a(x), a ∈ L∞(Rn), thenOp(a) is multiplication by a(x); (ii) if S ∈ Sp(2n,R) thenOp(a ◦ S) = S−1 Op(a)S . Then a 7−→ Op(a) is the Weylcorrespondence.


Symplectic covariance

The property of symplectic covariance really singles out Weyl operatorsamong all possible “quantization schemes”; it can be proved using therelation

W (ψ, φ) ◦ S = W (S−1ψ, S−1φ) (2)

for the cross-Wigner transform.A remark: the symplectic covariance property, together with the fact thatto real symbols correspond (formally) self-adjoint operators, are at theorigin of the success of Weyl calculus as a privileged "quantizationscheme" in mathematical physics. There is however something whichseems to be a serious drawback for "real" physicists: the Weylquantization is perhaps "too good", because it allows to "dequantize"every "observable". But in quantum mechanics there are observableswhich have no classical counterpart... So, Born—Jordan quantization (laterin this talk) might be a better choice...


A question...

So far, so good. But can one expect to find a larger symmetry group forWeyl operators? That is, can we find, for a general invertible matrix M an

operator M such that if AWeyl←→ a then M−1AM

Weyl←→ a ◦M ? The answeris given by:

TheoremLet M ∈ GL(2n,R). Assume that there exists a unitary operatorM : L2(Rn) −→ L2(Rn) such that MAM−1

Weyl←→ a ◦M−1 for allA

Weyl←→ a ∈ S(R2n). Then M is symplectic or antisymplectic.

A matrix M is "symplectic" if MTMJ = −J where J =(0 I−I 0

)and

"antisymplectic" if MTMJ = −J.The proof of this result is based on a geometric Lemma, which is veryinteresting in its own right, because closely related to symplectic topology,more precisely, on the notion of symplectic capacity of an ellipsoid.


The Lemma

The key is:

Lemma

Let M ∈ GL(2n,R) and assume that MTGM ∈ Sp(n) for every

G =(X 00 X−1

)∈ Sp+(n). (3)

Then M is either symplectic, or anti-symplectic.

Its proof uses elementary symplectic geometry, but is not totally trivial. Ithas the following consequence, independent of this talk:

Corollary

Let k : R2n −→ R2n be a linear automorphism taking every symplectic ballto another symplectic ball. Then k is either symplectic or antisymplectic.


Proof of the Theorem, using another theorem

Recall that 〈Aψ|ψ〉 = 〈〈a,W (ψ,ψ)〉〉. It therefore suffi ces to prove thefollowing result for the Wigner transform Wψ = W (ψ,ψ):

TheoremLet M ∈ GL(2n,R). (i) Assume that M is antisymplectic:

S = CM ∈ Sp(n) where C =(I 00 −I

); then for every ψ ∈ S ′(Rn)

Wψ(Mz) = W (S−1ψ)(z) (4)

where S is any of the two elements of Mp(n) covering S. (ii) Conversely,assume that for any ψ ∈ S(Rn) there exists ψ′ ∈ S ′(Rn) such that

Wψ(Mz) = Wψ′(z). (5)

Then M is either symplectic or antisymplectic.


Proof of the "other Theorem"

(i) Wψ(Cz) = Wψ(z) is trivial. (ii) Choosing for ψ a Gaussian of theform

ψX (x) =( 1

π h

)n/4(detX )1/4e−

12 h Xx ·x (6)

(X is real symmetric and positive definite) we have

WψX (z) =( 1

π h

)ne−

1h Gz ·z (7)

where G =(X 00 X−1

)i∈ Sp(n). ConditionWψ(Mz) = Wψ′(z) implies

that we must have

Wψ′(z) =( 1

π h

)ne−

1h M

TGMz ·z .


The Wigner transform of a function being a Gaussian if and only if thefunction itself is a Gaussian: the matrix MTGM being symmetric andpositive definite we can use a Williamson diagonalization: there existsS ∈ Sp(n) such that

ST (MTGM)S = ∆ =(

Σ 00 Σ

)(8)

and henceWψ′(Sz) =

( 1π h

)ne−

1h ∆z ·z .

In view of the symplectic covariance of the Wigner transform, we have

Wψ′(Sz) = Wψ′′(z) , ψ′′ = S−1ψ′

where S ∈ Mp(n) is one of the two elements of the metaplectic groupcovering S .


We now show that the equality

Wψ′′(z) = π−ne−∆z ·z (9)

implies that ψ′′ must be a Gaussian of the form (6) and hence Wψ′′ mustbe of the type (7). That ψ′′ must be a Gaussian follows from Wψ′′ ≥ 0and Hudson’s theorem. If ψ′′ were of the more general type

ψX ,Y (x) = π−n/4(detX )1/4e−12 (X+iY )x ·x (10)

(X ,Y are real and symmetric and X is positive definite) the matrix G in(7) would be

G =(X + YX−1Y YX−1

X−1Y X−1

)(11)

which is only compatible with (9) if Y = 0.


In addition, due to the parity of Wψ′′, ψ′′ must be even hence Gaussiansmore general than ψX ,Y are excluded. It follows from these considerationsthat we have

∆ =(

Σ 00 Σ

)=

(X 00 X−1

)so that Σ = Σ−1. Since Σ > 0 this implies that we must have Σ = I , andhence, using formula (8), ST (MTGM)S = I . It follows that we must have

MTGM ∈ Sp(n) for every G =(X 00 X−1

)∈ Sp+(n). In view of the

Lemma the matrix M must then be either symplectic or antisymplectic.


Shubin operators

Given a symbol a the τ-pseudo-differential operator Aτ = Opτ(a) isformally defined by

Aτψ(x) =∫∫

e2πiξ(x−y )a(τx + (1− τ)y , ξ)ψ(y)dξdy ; (12)

for τ = 12 we recover the Weyl correspondence: A1/2 = AWeyl = A where

Aψ(x) =∫∫

e2πiξ(x−y )a( 12 (x + y), ξ)ψ(y)dξdy . (13)

When τ = 1 we get the Kohn—Nirenberg operators

A(x ,D)ψ(x) =∫∫

e2πiξ(x−y )a(x , ξ)ψ(y)dξdy

=∫e2πiξxa(x , ξ)ψ(ξ)dξ.


The operators Tτ(z)Let T (z0) be the Heisenberg operator: for ψ ∈ S ′(Rn)

T (z0)ψ(x) = e i (ξ0x−12 ξ0x0)ψ(x − x0) (14)

where z0 = (x0, ξ0). More generally:

Tτ(z0) = ei2 (2τ−1)ξ0x0 T (z0). (15)

and we have Tτ(z0)−1 = T1−τ(−z0). Commutation relations:

Tτ(z0)Tτ(z1) = e iσ(z0,z1)Tτ(z1)Tτ(z0) (16)

Tτ(z0 + z1) = e−i2 στ(z0,z1)Tτ(z0)Tτ(z1) (17)

whereστ(z0, z1) = 2(1− τ)ξ0x1 − 2τξ1x0.

Note that στ is a bilinear form, but not in general a symplectic form (itfails to be antisymmetric if τ 6= 1

2 ).


Shubin’s τ-operators can be written

Opτ(a)ψ =1

(2π)n

∫aσ(z)Tτ(z)ψdz (18)

where aσ is the symplectic Fourier transform of a, that is

aσ(z) =1

(2π)n

∫e−iσ(z ,z ′)a(z ′)dz ′.

Following the usage in the theory of Weyl operators, we will call aσ the“twisted symbol of Opτ(a)”. The distributional kernel of Aτ is given by

Kτ(x , y) = F−12 [aσ(x − y , ·)] (τx + (1− τ)y). (19)


Symplectic Cayley transform

Define

Sp(0)(2n,R) = {S ∈ Sp(2n,R) : det(S − I ) 6= 0}Sym(0)(2n,R) = {M ∈ Sym(2n,R) : det(M − 1

2J) 6= 0}.

The symplectic Cayley transform of S ∈ Sp(0)(2n,R) is the bijectionSp(0)(2n,R) −→ Sym(0)(2n,R) given by

M(S) = 12J(S + I )(S − I )

−1 (20)

The inverse of that bijection is given by

S = (M − 12J)−1(M + 1

2J). (21)

We have the properties M(S−1) = −M(S) and, when in additionS ′, SS ′ ∈ Sp(0)(2n,R):

M(SS ′) = M(S) + (ST − I )−1J(M(S) +M(S ′))−1J(S − I )−1. (22)


Intertwiners

Let S ∈ Sp(0)(2n,R) and define the operators

Rτ(S) =1

(2π)n1√

| det(S − I )|

∫ei2M (S )z

2Tτ(z)dz

Rτ(S) =√| det(S − I )|

∫Tτ(Sz)Tτ(−z)dz (23)

TheoremThe operator Rτ(S) is a continuous mapping S(Rn) −→ S(Rn) satisfying

Rτ(S)Tτ(z) = Tτ(Sz)Rτ(S) (24)

and we have, for a ∈ S ′(R2n),

Rτ(S)Opτ(a) = Opτ(a ◦ S)Rτ(S). (25)


Let S , S ′ ∈ Sp(0)(2n,R) and assume that SS′ ∈ Sp(0)(2n,R) as well. We

haveRτ(SS ′) = e i

π4 signM (SS ′)Rτ(S)Rτ(S ′) (26)

We also haveRτ(S−1) = Rτ(S)−1 = R1−τ(S)∗ (27)

so the operators Rτ(S) are not unitary if τ 6= 12 (if τ = 1

2 they are theusual metaplectic operators).


Application: Wigner functions

The usual cross-Wigner function W (ψ, φ) has the following property: forall ψ, φ ∈ S(Rn) and S ∈ Sp(2n,R) we have

W (Sψ, Sφ)(z) = W (ψ, φ)(S−1z) (28)

where S ∈ Mp(2n,R) is any of the two metaplectic operators which coverS . In the τ-dependent case we define

Wτ(ψ, φ)(z) =1

(2π)n

∫e−iy ξψ(x + τy)φ(x − (1− τ)y)dy . (29)

and (28) must be modified as follows:

TheoremLet S ∈ Sp(0)(2n,R) and ψ, φ ∈ S(Rn). We have

Wτ(Rτ(S)ψ,R1−τ(S)φ)(z) = Wτ(ψ, φ)(S−1z). (30)

Notice that we recover the usual symplectic covariance formula for thecross-Wigner transform when τ = 1

2 .M. A. de Gosson (Institute) ESI 2012 15.10.2012 19 / 25

A boundedness result

Each operator Rτ(S) is bounded on L2(Rn) provided that the parameter τis suffi ciently close the “Weyl value” 1

2 . Intuitively speaking, it means thatsuch operators are small deformations of the corresponding metaplecticoperator S and therefore bounded on the square integrable functions:

TheoremLet S ∈ Sp(0)(2n,R). There exists ε > 0 such that

Rτ(S) : L2(Rn) −→ L2(Rn) for 12 − ε ≤ τ ≤ 12 + ε, and there exists a

constant C (ε, S) such that

||Rτ(S)ψ||L2 ≤ C (ε,S)||ψ||L2 , ψ ∈ L2(Rn). (31)

This is in accordance with the theory of the “ε-Wigner transform”

W (ε)(ψ, φ) =12ε

∫ 12+ε

12−ε

Wτ(ψ, φ)dτ (32)

developed by Boggiatto et al. who show that W (ε)(ψ, φ) has propertiessimilar to those of W (ψ, φ) as ε→ 0+.


Born and Jordan (1925) defined two years before Weyl:

xmξ`BJ−→ 1

`+ 1

`

∑k=0

Ξ`−k Xm Ξk . (33)

The BJ rule coincides with the Weyl rule when m+ ` ≤ 2: in both cases

xξ −→ 12

`

∑k=0

(X Ξ+ ΞX ).

But it is not a Shubin correspondence for any τ. However, the BJcorrespondence is obtained by averaging all the τ correspondences overthe interval [0, 1]. This leads to a general definition.


Born—Jordan operators

For a symbol a we set ABJ = OpBJ(a) with

ABJf =∫ 1

0Aτfdτ (34)

(averaging for τ ∈ [0, 1].) One verifies that one recovers the rule

xmξ`BJ−→ 1

`+ 1

`

∑k=0

Ξ`−k Xm Ξk (35)

as a particular case using the formula∫ 1

0(1− τ)kτ`−kdτ = B(k + 1, `− k + 1) = k !(`− k)!

(k + `+ 1)!.


Alternative definition

The operator ABJ = OpBJ(a) can also be defined in terms of Boggiatto etal.’s averaged Wigner transform: set

Wτ(f , g)(z) =∫e−2πiyωf (x + τy)g(x − (1− τ)y)dy (36)

and

WBJ(f , g) =∫ 1

0Wτ(f , g)dτ.

Then we can define Aτf and ABJf for a ∈ S ′(R2n), f , g ∈ S(Rn) by

(Aτf |g)L2 = 〈a,Wτ(f , g)〉 , (ABJf |g)L2 = 〈a,WBJ(f , g)〉. (37)


The metaplectic group Mp(2n,R) is generated by the modified Fouriertransform J = e i

nπ4 F , the multiplication operators V−P f = e

i2Px

2f

(P = PT ) and the unitary scaling operatorsML,m f (x) = im

√| det L|f (Lx) (det L 6= 0, mπ = arg det L). The

projections of these operators on Sp(2n,R) are, respectively, J,

V−P =(I 0P I

), and ML =

(L−1 00 L2

).

TheoremLet ABJ = OpBJ(a) with a ∈ S ′(R2n). We have

S OpBJ(a) = OpBJ(a ◦ S−1)S (38)

for every S ∈ Mp(2n,R) product of a (finite number) of operators J andML,m .


THANK YOU FOR YOUR KIND ATTENTION!


maximal symmetries for pseudo-differential operators · 2012-10-15 · maximal symmetries for...

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