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Squeezed Coherent States and a Semiclassical Propagator for the Schr¨o dinger Equation in Phase Space Maurice A de Gosson Universit¨a t Potsdam,Inst.f.Mathematik Am Neuen Palais 10,D-14415Potsdam E-mail address:maurice.degosson@d3ddcad628ea81c758f57859 Serge M de Gosson V¨a xj¨o Universitet,MSI SE-35195V¨a xj¨o ,Sweden E-mail address:sergedegosson@d3ddcad628ea81c758f57859 February 1,2008Abstract We construct semiclassical solutions of the symplectically covariant Schr¨o dinger phase-space equation rigorously studied in a previous paper;we use for this purpose an adaptation of Littlejohn’s nearby-orbit method.We take the opportunity to discuss in some detail the so fruitful notion of squeezed coherent state and the action of the metaplectic group on these states.1Introduction

Let ψbe a square-integrable solution of Schr¨o dinger’s equation

i ??ψ

2π n/2e i

p ·x ′ψ(x ′)

?t = H ph Ψ(2)1

where the operator H ph is de?ned by the intertwining formula

Uφ H ph=H(x,?i ?x)Uφ;

A straightforward calculation shows that we have

Uφ(xψ)=(1

2

p?i ?x)Uφψ;(3) these relations motivate the notation

H ph=H(12p j?i ?x)

and we may thus rewrite(2)as the phase-space Schr¨o dinger equation

i ?

2

x+i ?p,1

?x1

,...,?

?p1

,...,?

(p0·x?1

The Wigner–Moyal transform of a pair(ψ,φ)of functions in the Schwartz space S(R n x)is de?ned by

W(ψ,φ)(z)= 1 p·yψ(x+1φ(x?1

π n/4e?1

(p0·x?1

2π n

φ z0(x)

2π n (ψ,φ z0)L2φ z0(x)d2n z0(10)

and

||ψ||2L2= 1

2π n

Aφ z0(x)

π n/4(det X)1/4e?1

where X and Y are real symmetric n ×n matrices with X >0;we have ||φ

(X,Y )||L 2=1.We will ?nd it convenient to set

M =i (X +iY ),X =X T >0,Y =Y T .

and to write φ (X,Y )=φ

M ;thus:

φ

M (x )=

12 Mx 2,X =Im M.The Wigner transform of φ

M is given by the formula

W φ

M (z )= 1

Gz 2(14)

where G is the symmetric matrix G = X +Y X ?1Y Y X ?1

X ?1Y X ?1 ;(15)

an essential remark is that G is in addition symplectic (this fact was apparently ?rst observed by Bastiaans [1]).More precisely,we have G =S T S with

S = X 1/20

X ?1/2Y X ?1/2 ∈Sp(n )(16)

as results from a direct calculation;notice that S belongs to the isotropy sub-group of the Lagrangian plane ?p =0×R n in Sp(n ).For z 0∈R 2n we set

φ z 0,M = T (z 0)φ

M ;

the Wigner transform of φ

z 0,M is given by

W φ z 0,M (z )=W φ

M (z ?z 0);(17)

it is thus a Gaussian centered at z 0.

Let us generalize formula (14)by calculating the Wigner-Moyal transform W (φ z 0,M ,φ

z ′0,M ′)of a pair of squeezed coherent states;recall for this purpose

the “Fresnel formula” 1 ξ·x e ?1

2 K ?1ξ2

(18)

valid for ξ∈C n ,K =K T ,Im K >0;here (det K )?1/2=λ?1/21···λ?1/2n where

λ?1/2j is the square root with positive real part of the eigenvalue λ?1j

of K ?

1

(see [4,16]).Proposition 2We have

W (φ M ,φ

M ′)(z )=

1 F z 2(19)

where F is the matrix F = 2i M ′)?1M ?i (M +M ′)?1

?i (M ?M ′)2i (M ?

Proof.Setting W M,M′=W(φ M,φ M′)we have

W M,M′(z)= 1 pyφ M(x+1φ M′(x?1

2 (M+

py e i

π 2n(det XX′)1/4

Φ(x,y)=M(x+1M′(x?1

π n(det XX′)?1/4e?1

M′?(M?M′)?1(M?M′(M+

Noting the following formula,which is an easy consequence of the de?nitions

of the Wigner–Moyal transform and the Heisenberg–Weyl operators:

Lemma3For all z0,z′0in R2n z and f,g in L2(R2n z)we have

W( T(z0)ψ, T(z′0)ψ′)(z)=e i2σ(z0,z′0))×W(ψ,ψ′)(z?1

π n e i2σ(z0,z′0))(det XX′)?1/4e?12(z0+z′0))2 where F is given by(20);hence in particular

W(φ z

0,M ,φ z

0,M′

)(z)= 1 F(z?z0)2.

2.2Phase-space coherent states

For eachφ∈S(R n x)the operator

Uφψ(z)= π 2z) is an isometry of L2(R n x)onto its range Hφ;it follows that:

5

Proposition5LetΦ z

0=Uφ(φ z

).For eachΨ∈Hφwe have

Ψ(z)= 1

2π n |(Ψ,Φ z0)L2|2d2n z0.(22)

Proof.Letψbe de?ned byΨ=Uφψ;In view of part(ii)of Proposition1 (formula(10))we have

ψ= 1

2π n (ψ,φ z0)L2Uφ(φ z0)d2n z0

= 1

3Metaplectic Group and Coherent States

The metaplectic group Mp(n)is a faithful unitary representation of Sp2(n),the double cover of the symplectic group Sp(n).There are several di?erent ways to describe the elements of Mp(n)(see for instance[9,16,23]);for our purposes the most adequate de?nition makes use the notion of generating function for a free symplectic matrix because it is the simplest way to arrive at the Weyl symbol of metaplectic operators(and hence to their extension to phase space).

The interest of the metaplectic representation comes from the fact that it links in a crucial way classical(Hamiltonian)mechanics to quantum mechanics (see for instance[18]or[9]and the references therein).Assume in fact that H is a Hamiltonian function which is a quadratic polynomial in the position and momentum variables(with possibly time-dependent coe?cients):thus

H(z,t)=1

2

H′′(t)z2

where H′′(t)is a symmetric matrix(it is the Hessian matrix of H).The associ-ated Hamilton equations˙z=?z H(z,t)determine a(generally time-dependent)?ow consisting of symplectic matrices S t.We thus have a continuous path

6

t?→S t in the symplectic group Sp(n)passing through the identity at time t=0:S0=I.Following general principles,this path can be lifted(in a unique way)to a path t?→ S t in Mp(n)such that S0= I(the identity in Mp(n)). Choose now an initial wavefunctionψ0in,say,S(R n x)and setψ(x,t)= S tψ0(x). The functionψsatis?es Schr¨o dinger’s equation

?ψ

i

(x,i ?x)T H′′(t)(x,i ?x).

2

3.1Description of Mp(n)and IMp(n)

Let W be quadratic form of the type

Qx2

W(x,x′)=1

2

with P=P T,Q=Q”,det L=0;(we have set P x2=P x·x,etc.).To each such quadratic form we associate the generalized Fourier transform

S W,mψ(x)= 1|det L| e i

DB?1x2?(B?1)T x·x′+1

2

?We have

S T(z0)= T(Sz0) S(27) for all S∈Mp(n)and z0∈R2n;

?If S∈Mp(n)has projection S=πMp( S)then

W( Sψ, Sφ)=W(ψ,φ)?S?1.

The group generated by the Weyl–Heisenberg operators T(z0)and the op-erators S∈Mp(n)is a group of unitary operators in L2(R n);it is denoted by IMp(n)and called the inhomogeneous metaplectic group.

3.2The action of Mp(n)on coherent states

To describe the action of S∈Mp(n)on the squeezed coherent statesφ z0,M we will need the following Lemma.Let us denote byΣ(n)the Siegel half-space, that is

Σ(n)={M:M=M T,Im M>0}

(M is a complex n×n matrix).

Lemma6Let S∈Sp(n)be given by(25)and M∈Σ(n).Then det(A+BM)= 0,det(C+DM)=0and

α(S)M=(C+DM)(A+BM)?1∈Σ(n)(28) (in particularα(S)M is symmetric),and

α(SS′)M=α(S)α(S′)M.(29) The action Sp(n)×Σ(n)?→Σ(n)de?ned by(28)is transitive:if M,M′∈Σ(n) then there exists S∈Sp(n)such that M′=α(S)M.

For a proof see[4,12];also see the preprint[2]by Combescure and Robert. Notice that(29)implies that S?→α(S)is a true representation of the sym-plectic group in the Siegel half-space.

Let S∈Mp(n)have projection

S=πMp( S)= A B C D

on Sp(n);then

Sφ (x)= 1 2 α(S)x2

where the branch cut of the square root of det(A+iB)is taken to lie just under the positive real axis;m( S)is the Maslov index(24)of d3ddcad628ea81c758f57859ing this formula

8

Sφ

z 0(x )is easily calculated:since by de?nition φ z 0= T (z 0)φ the metaplectic covariance formula (27)immediately

yields Sφ

z 0(x )= T (Sz 0) Sφ (x )

(see Littlejohn [17]for an explicit formula).

The results above can be generalized to arbitrary squeezed coherent states:Proposition 7Let φ

z 0,M ,M ∈Σ(n ),be a squeezed coherent state and S

∈

Mp(n ),S =πMp ( S )

.We have Sφ M =φ

α(S )M , Sφ

z 0,M = T (Sz 0)φ α(S )M .(30)

(see for instance [2,4,17]).The Gaussian character of a wavepacket is thus preserved by metaplectic op-erators;as a consequence the solution of a Schr¨o dinger equation with Gaussian initial value remains Gaussian when the Hamiltonian operator is associated to a quadratic Hamiltonian function.

3.3The Weyl symbol of a metaplectic operator

For S ∈Sp(n )such that det(S ?I )=0we set

M S =

12π n i m ?Inert W xx |det(S ?I )| e

i

2π n i ν( S )

|det(S ?I )| e i

Formulae (31)and (32)yield the

Weyl representations of the metaplectic operators;for a de?nition and a precise study of the Conley–Zehnder index

ν( S

)appearing in (32)see [9].In [6]it was also proven that formula (32)can be rewritten alternatively as

S = 1|det(S ?I )| e ?i

2π n i ν( S

)

?t = Hψ,ψ(t =0)=ψ0

(35)10

is given by

U(t,z0)ψ0=e i

2

σ(z(t′),˙z(t′))?H(z(t′),t′)dt′.(37)

One shows([3,17])that the functionψ=U(t,z0)ψ0is the solution of the Schr¨o dinger equation associated to the Hamiltonian function

H z

(z,t)=H(f t(z0),t)

+H′(f t(z0),t)(z?f t(z0))+

1

γ(z0,t).

Let us work out explicitly formula(36)when the initial wavepacketψ0is a squeezed coherent state:

Proposition9Letφ z

0,M

be an arbitrary squeezed coherent state.We have:

U(t,z0)φ z

0,M

=e i

γ(z0,t) T(f t(z0)) S t(z0)φ M

that is,taking the?rst formula(30)into account,

U(t,z0)φ z

0,M

=e i

γ(z0,t)φ

f t(z0),α(S t(z0))M 11

which completes the proof.

2π n (ψ0,φ z0)L2φ z0(x)d2n z0

and one then takes as semiclassical solution

U(t)ψ0(x)= 1

2π n ψ0,φ z0 L2 T(f t(z0))e i

(39) provided that the Hamiltonian function H satis?es uniform estimates of the type

|?αz H(z,t)|≤C′α,T(1+|z|)m(40) for|α|≥m,|t|

(It is in fact possible to obtain precise bounds for the constant C T;see[3]).

4.2Weyl operators on phase space

In[8](also see[9]for details)we noticed that the wave-packet transform(1)is related to the Wigner–Moyal transform(6)by the simple formula

Uφψ(z)= π 2z),ψ∈L2(R n x)(41) It follows that we have

(Uφψ,Uφψ′)L2(R2n

z )

=(ψ,ψ′)L2(R n

x

)

(42)

12

and hence each of the linear mappings U φis an isometry of L 2(R n x )onto a closed subspace H φof L 2(R 2n z )(the square integrable functions on phase space).It follows that U ?φU φis the identity operator on L 2(R n x )and that P φ=U φU ?φis

the orthogonal projection onto the Hilbert space H φ.De?ning T

ph (z 0)by T

ph (z 0)Ψ(z )=e ?i 2π

n a σ(z 0) T

(z 0)ψ(x )d 2n z 0the phase space operator

A ph Ψ(z )= 12π n/2i ν(S )|det(S ?I )| e i

2π

n i ν(S )

2σ(Sz,z )

T ph ((S ?I )z )d 2n z (48)and S ph = 1|det(S ?I )| T ph (Sz ) T ph (?z )d 2n z .(49)Notice that the well-known “metaplectic covariance”relation A ?S = S ?1 A S valid for any S ∈Mp(n )with projection S ∈Sp (n )extends to the phase-space Weyl operators A

ph :we have S ph

T ph (z 0) S ?1ph = T ph (Sz ), A ?S ph = S ?1ph A ph S ph .(50)

In Subsection 2.2we de?ned coherent states in phase space.The metaplectic action on coherent states described in Proposition 7carries over to this case without di?culty,yielding the formulae

S ph Φ M =Φ α(S )M , S ph Φ z 0,M = T ph (Sz 0) S ph Φ α(S )M .13

4.3Nearby-orbit method in phase space

Let us now state and prove the main result of this paper:

Proposition 12(i)Let U (t,z 0)be the semiclassical propagator for Schr¨o dinger’s equation and set ψ=U (t,z 0)ψ0.The wavepacket transform U φtakes ψto the function Ψde?ned by

Ψ=e i

γ(z 0,t )

T ph (f t (z 0))( S t (z 0))ph Φ 0.(52)

Proof.Set ψ=U (t,z 0)ψ0;by de?nition of U (t,z 0)we have

ψ=e i

γ(z 0,t )

U φ( T

(f t (z 0)) S t (z 0) T (z 0)?1ψ0=e i

The result above therefore suggests that the phase-space version of the semi-classical nearby-orbit propagator U (t,z 0)should be given by the formula

U ph (t,z 0)=e i

?t

= H ph Ψ,Ψ(t =0)=Φ z 0with Φ

z 0=U φ(φ z 0).Suppose that H satis?es the conditions (40)in Proposition 11.Then,for |t |

?t

= Hψ,ψ(t =0)=φ z 0;since U φis a linear isometry we have ||U ph (t,z 0)Φ z 0?Ψ(·,t )||L 2(R 2n z )=||U (t,z 0)φ z 0?ψ(·,t )||L 2(R n x )

≤C T √14

5Conclusion and Discussion

There are several problems and questions we have not discussed in this paper, and to which we will come back in forthcoming publications.Needless to say, there is one outstanding omission:we haven’t analyzed the domain of validity of the nearby-orbit method much in detail.There are many results in the literature for the standard nearby-orbit method.For instance,Hagedorn[10,11]obtains precise estimates using the Lie–Trotter formula;similar results were rediscovered and sharpened by Combescure-Robert[3]using the Duhamel principle.Also see [19](Ch.2,§2.1).It shouldn’t be too di?cult to obtain corresponding estimate for the phase-space Schr¨o dinger equation,using the properties of the wavepacket transform.It is on the other hand well-known that there are problems with long times when the associated classical systems exhibits a chaotic behavior;as Littlejohn points out in[17],the nearby orbit methods probably fails for long times near classically unstable points.

Acknowledgement14This work has been supported by the FAPESP grant 2005/51766–7during the author’s stay at the University of S?a o Paulo.I take the opportunity to thank Professor Paolo Piccione for his kind and generous invitation.

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