Analyticity of the Schrodinger propagator on the Heisenberg group

DOI 10.1007/s00605-012-0424-7

Analyticity of the Schrödinger propagator on the Heisenberg group

S. Parui · P. K. Ratnakumar · S. Thangavelu

Received: 17 May 2011 / Accepted: 22 June 2012

© Springer-Verlag 2012

Abstract We discuss the analytic extension property of the Schrödinger propaga- tor for the Heisenberg sublaplacian and some related operators. The result for the sublaplacian is proved by interpreting the sublaplacian as a direct integral of an one parameter family of dilated special Hermite operators.

Keywords Schrödinger equation·Oscillatory group·Special Hermite expansion· Heisenberg group·Sublaplacian

Mathematics Subject Classification (1991) Primary 22E30;

Secondary 35G10·47A63

1 Introduction

It is well known that the heat equation is infinitely smoothing in the sense that the solution u(x,t)of the heat equation∂tu=u with initial condition u(x,0)= f(x)

Communicated by K. Gröchenig.

S. Parui (

B

School of Mathematics, National Institute of Science Education and Research, IOP Campus, Bhubaneswar 751005, India

e-mail: parui@niser.ac.in P. K. Ratnakumar

Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211019, India e-mail: ratnapk@hri.res.in

S. Thangavelu

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India e-mail: veluma@math.iisc.ernet.in

is real analytic even if f is only in L²(Rⁿ). Actually much more is true: if qt stands for the heat kernel,

qt(x)=(4πt)⁻ⁿ²e^−4t1|x|2

then for f ∈ L²(Rⁿ), the solution u(x,t) = f ∗qt(x)extends toCⁿas an entire function u(x+i y,t)which satisfies

R²ⁿ

|u(x+i y,t)|²qt/2(y)d x d y=cn

Rⁿ

|f(x)|²d x.

Conversely, any entire function F(x+i y)onCⁿfor which

R²ⁿ

|F(x+i y)|²qt/2(y)d x d y<∞

can be factored as F(x) = f ∗qt(x)for some f ∈ L²(Rⁿ).This is a celebrated theorem attributed to Bargmann [1].

In contrast to this, the Schrödinger equation cannot be expected to have such a smoothness property. In fact, the solutionv(x,t)of the equation

i∂tv=v, v(x,0)= f(x)

is given by the unitary group e^{−i t}, which is an isometry onto L², so without any improvement in the smoothing. In view of this, the following result of Hayashi and Saitoh [9] looks surprising at the first sight. In 1990, they have shown that the solution of the one dimensional Schrödinger equation

i∂tu+1

2∂_x²u=0, u(x,0)= f(x), t,x∈R

extends toCas an entire function, provided the initial condition f has enough decay at infinity. More precisely, they have shown that when f is square integrable with respect to e^x2d x, then u(x,t)extends toCas an entire function and

√1 πt²

∞

−∞

∞

−∞

e⁻

y2

t 2|e^−2tiz2u(z,t)|²d x d y= ∞

−∞

|f(x)|²e^x2 d x

for all t =0. Their proof is based on the identity

√1πα ∞

−∞

∞

−∞

e^−y2α|f(x+i y)|²d x d y = ∞

j=0

α^j j!

∞

−∞

|∂x^j f(x)|²d x.

However, the result can be directly proved in all dimensions without any recourse to the above identity. Indeed, the solution of the Schrödinger equation

i∂tu=u, u(x,0)= f(x) is given by

u(x,t)=(4πi t)⁻ⁿ²

Rⁿ

f(y)e^{−4i t1 |x−y|2}d y.

Therefore, u(x,t)e^{4i t1|x|2} is up to a constant multiple, the Fourier transform (at _2t¹x) of the function f(y)e^{−4i t1|y|2}.If we assume that f has a Gaussian decay, then it is well known that its Fourier transform extends toCⁿas a holomorphic function. There- fore, it is not surprising that the solution of the Schrödinger equation also has some smoothing properties, provided we start with initial conditions having enough decay.

If we assume that f is square integrable with respect to e^|y|2d y, then a simple calculation shows that

u(x,t)e^{−4ti x2j} =(4πi t)⁻ⁿ²

Rⁿ

f(y)e^{4ti y2j}

e^−2tix·yd y,

which can be written as

u(x,t)e^{−4ti x2j} =(2i t)^−n/2gˆ∗qs

1 2tx

where g(y)= f(y)e^4ti|y|2e^s|y|2. As g∈ L²(Rⁿ), it follows from the theorem of Segal and Bargmann that F(x,t)=u(x,t)e^{−4ti x2j} extends toCⁿas an entire function and

Cⁿ

|F(z,t)|²qs/2

y 2t

d x d y =(2t)ⁿ

Cⁿ

| ˆg∗qs(z)|²qs/2(y)d x d y

=(2t)ⁿ

Rⁿ

| ˆg(y)|²d y,

which leads to the identity, with s =¹₂,

Cⁿ

|u(z,t)e^{−4ti z2j}|²e⁻

1 4t2|y|²

d x d y =(4t²π)^n/2

Rⁿ

|f(y)|²e^|y|2d y.

Actually, the converse is also true.

In this article we consider the Schrödinger equation associated to the Hermite and special Hermite operators and also to the sublaplacian on the Heisenberg group.

The Hermite and Special Hermite cases are similar to the Euclidean Laplacian and straightforward, whereas the sublaplacian case is very different. We obtain an iso- metric isomorphism between certain subspaces of L²(Hⁿ)and a direct integral of weighted Bergman spaces related to special Hermite operator.

The paper is organised as follows. In Sect. 2, we treat the Hermite and special Hermite cases. In Sect.3, we consider the sublaplacian on the Heisenberg group. In Sect.4, the generalized sublaplacian has been treated.

2 Schrödinger equation for the Hermite and the special Hermite operators First we consider the regularising properties of the unitary group e^{i t H}, t ∈ Rgen- erated by the Hermite operator H = −+ |x|²onRⁿ. As is well known, e^{i t H} is a fractional Fourier transform and hence the holomorphic extendability of e^{i t H}f under suitable decay assumption on f is only but expected. The spectral decomposition of H is given by

H = ∞ k=0

(2k+n)Pk

where Pkf =

|α|=kf, _αα, _αbeing the normalised Hermite functions onRⁿ. Using Mehler’s formula (see [25]) we obtain

e^{i t H}f(x)=

Rⁿ

Ki t(x,y)f(y)d y,

where

Ki t(x,y)=(2πi sin 2t)⁻ⁿ²e^{i4(cot 2t)(|x|2+|y|2)−i(csc 2t)x·y}.

Note that Ki t(x,y)is well defined as long as 2t is not an integral multiple ofπ.

The one parameter group e^{i t H} is periodic in t and for 2t = kπ, k ∈ Z,e^{i t H} is given by the above integral. For such values of t,

e^{−4i(cot 2t)|x|2}e^{i t H}f(x)=(2πi sin 2t)⁻ⁿ²

Rⁿ

f(y)e^{i4(cot 2t)|y|2}e^{−i(csc 2t)x·y}d y.

If we assume that f(y)e^s|y|2 ∈ L²(Rⁿ),then defining g(y)= f(y)e^s|y|2e^{i4(cot 2t)|y|2}, we obtain

e^{−4i(cot 2t)|x|2}e^{i t H}f(x)=(2πi sin 2t)⁻ⁿ²gˆ∗qs((csc 2t)x)

where qs is the Euclidean heat kernel onRⁿ.Thus we see that e^{i t H} is related to the Segal–Bargmann transform or the heat kernel transform associated to the standard Laplacian onRⁿ.

We briefly recall the main theorem on Segal–Bargmann transform for the conve- nience of the reader. The operator which takes f ∈ L²(Rⁿ)into the entire function f ∗qt(z) onCⁿ is called the Segal–Bargmann transform. This is a variant of the Bargmann transform studied by Bargmann [1] and independently by Segal. The main theorem proved in [1] can be restated as follows.

Theorem 2.1 The Segal–Bargmann transform takes L²(Rⁿ)onto the space of entire functions F on Cⁿ which are square integrable with respect to the measure qt/2(y)d xd y.Moreover, for any f ∈Ł²(Rⁿ)

Cⁿ

|f ∗qt(x+i y)|²qt/2(y)d xd y=

Rⁿ

|f(x)|²d x.

Using the above theorem of Segal and Bargmann we can easily prove the following result for the operators e^{i t H}.

Theorem 2.2 For any t = ^k₂π,k∈Z, e^{i t H}defines an isomorphism between L²(Rⁿ, e^|x|2d x)and the Hilbert space of entire functions onCⁿ that are square integrable with respect to the weight function(2π|sin 2t|)⁻ⁿe^{−(csc22t)|y|2}e^{(cot 2t)x·y}.

Proof As g ∈ L²(Rⁿ),the function F(x,t)= e^{−4i(cot 2t)|x|2}e^{i t H}f(x)has an entire extension toCⁿand according to the theorem of Bargmann, we have

Cⁿ

|F(x+i y,t)|²qs/2((csc 2t)y)d x d y

=(2π⁻ⁿ)|sin 2t|ⁿ

Cⁿ

| ˆg∗qs(x+i y)|²qs/2(y)d x d y

=(2π)⁻ⁿ|sin 2t|ⁿ

Rⁿ

| ˆg(y)|²d y

=(2π)⁻ⁿ|sin 2t|ⁿ

Rⁿ

|f(y)|²e^2s|y|2d y.

Taking s=¹₂ we observe that

Cⁿ

|F(x+i y)|²e^{−(csc22t)|y|2} d x d y

=cn(2π)⁻ⁿ|sin 2t|ⁿ

Rⁿ

|f(y)|²e^|y|2d y.

This can be rewritten as

Cⁿ

|e^{i t H}f(x+i y)|²e^{(cot 2t)x·y}|2πsin 2t|⁻ⁿe^{−((csc22t)|y|2)}d x d y

=cn

Rⁿ

|f(y)|²e^|y|2d y.

The converse follows from the fact that the Segal–Bargmann transform is onto. The proof of this fact is involved which uses the action of Bargmann transform on Hermite functions. See also Section 6, Chapter 1 of Folland [4] for details about Bargmann transform.

Remark 2.3 We remark that when t = ^π₄, e^{i t H}f is a constant multiple of f and theˆ above result reduces to

Rⁿ

Rⁿ

| ˆf(x+i y)|²e^−|y|2 d x d y=cn

Rⁿ

|f(y)|²e^|y|2d y.

Therefore, we can view Theorem 2.1 as a result for the fractional Fourier transform.

We now turn our attention to the special Hermite operator L_λdefined by

L_λ= −+λ²

4 (|x|²+ |u|²)−iλ n

j=1

(xj∂uj −uj∂xj)

whereis the Laplacian onR²ⁿandλ∈R\{0}. The spectral decomposition of L_λis given by

L_λf =(2π)^−n∞

k=0

(2k+n)|λ|f ∗_λϕk^λ

where theλtwisted convolution between two functions is defined as f ∗_λg(x,u)=

Cⁿ

f(x−x,u−u)g(x,u )e^{iλ2(u·x−u·x)}d x du,

so that the Schrödinger equation is solved by the function

e^{i t Lλ}f(x,u)=c_λ ∞ k=0

e^{(2k+n)|λ|i t} f ∗_λϕ^λk(x,u). (2.1)

Again the kernel of e^{i t Lλ} is explicitly known in terms of the heat kernel p_t^λ(x,u) associated to the special Hermite operator L_λ(see [18]): e^{i t Lλ}f = f ∗λ p^λ_{i t},where

p^λ_{i t}(x,u)=cn

λ sinλt

n

e^{−iλ4(cot tλ)(|x|2+|u|2)}.

As in the case of H , e^{i t Lλ}is also periodic in t and the above kernel exists as long as tλ is not an integral multiple ofπ. Also note that for such t, p_{i t}^λ(x,u)extends toCⁿ×Cⁿ as an entire function.

For t ∈R, λ∈R\{0}, andξ, η∈R²ⁿ, setωt^λ(ξ, η)=e^{(λcotλt)ξ·η}q_{sin2 tλ}

λ2 (η)where qs(η)is the Euclidean heat kernel onR²ⁿ. Consider the Hilbert space S_t^λ(C²ⁿ)of all entire functions for which

F²_Sλ

t =

C²ⁿ

|F(ξ+iη)|²ω^λt(ξ, η)dξ dη <∞.

We now prove

Theorem 2.4 The operator e^{i t Lλ} is an isometric isomorphism between L²(R²ⁿ, e^|ξ|2dξ)and the Hilbert space S_t^λ(C²ⁿ):

C²ⁿ

|e^{i t Lλ}f(ξ+iη)|²ω^λ_t(ξ, η)dξdη=

R²ⁿ

|f(ξ)|²e^|ξ|2dξ. (2.2)

Moreover, the weight functionωt^λis unique.

Proof For f ∈L²(R²ⁿ,e^|ξ|2dξ), consider

e^{i t Lλ}f(ξ)= f ∗_λ p_{i t}^λ(ξ), ξ=(x,u) which is given by the integral

cn

λ sinλt

n

R²ⁿ

f(a,b)e^{iλ4(cot tλ)[|x−a|2+|u−b|2]}e^{iλ2(a·u−b·x)}da db.

Definingvt(ξ)=e^{i t Lλ}f(ξ)p₋^λ_{i t}(ξ), we can write the above as vt(ξ)=

λ sinλt

2n

R²ⁿ

g_λ(η)e^{−iλ2(Aη·ξ)}dη

where g_λ(η)= f(η)e^{iλ4(cot tλ)|η|2} and A is the 2n×2n matrix A=A(λ)=

cot(tλ)In −In

In cot(tλ)In

.

We would like to expressvt as the convolution of a function h_λ∈ L²(R²ⁿ)with the Euclidean heat kernel onR²ⁿ, still denoted by qs, for suitable s. Let B =(A^t)⁻¹and C= ²_λB. We then have

vt(ξ)= λ

sinλt 2n

R²ⁿ

g_λ(Cη)|det C|e^−iξ·ηdη

=

R²ⁿ

hˆ_λ(η)e^−s|η|2e^−iξ·ηdη

wherehˆ_λ(η)= _sin^λ_λt2n

|det C|g_λ(Cη)e^s|η|2 for a suitable s to be chosen later.

Therefore,vt(ξ)=h_λ∗qs(ξ),the convolution with the Euclidean heat kernel on R²ⁿ. If h_λ ∈ L²(R²ⁿ), then it follows from the Bargmann–Segal theorem thatvt(ξ) extends toC²ⁿ as an entire function and

C²ⁿ

|vt(ξ+iη)|²qs/2(η)dξdη=

R²ⁿ

|h_λ(η)|²dη.

Now,

R²ⁿ

|h_λ(η)|²dη=

R²ⁿ

| ˆh_λ(η)|²dη

= λ

sinλt 4n

|det C|²

R²ⁿ

|g_λ(Cη)|²e^2s|η|2dη

= λ

sinλt 4n

|det C|

R²ⁿ

|g_λ(η)|²e^2s|C−1η|2dη.

After computing the determinant of C and|C⁻¹η|we end up with

R²ⁿ

|h_λ(η)|²dη= λ⁴ⁿ (sin tλ)⁴ⁿ

2²ⁿ

λ²ⁿ(sin tλ)²ⁿ

R²ⁿ

|g_λ(η)|²e^λ

2

2 |η|²s csc²tλdη

=

2ⁿλⁿ (sin tλ)ⁿ

2n

R²ⁿ

|f(η)|²e^|η2|dη

by choosing s=2^sin_λ²2^tλ. Thus we have proved the isometry

C²ⁿ

|e^{i t Lλ}f(ξ+iη)|²|p^λ_{−i t}(ξ+iη)|²qsin2 tλ λ2

(η)dξdη

=c_λ

R²ⁿ

|f(η)|²e^|η|2 dη,

where c_λ= _{sin t}^2λ_λ2n

. Note that

|p₋^λ_{i t}(ξ+iη)|²= λ

sin tλ 2n

|e^{−iλ4(cot tλ)(ξ+iη)2}|²= λ

sin tλ 2n

e^{λ2(cot tλ)ξ·η} and henceω^λ_t(ξ, η)=e^{λ(cot tλ)ξ·η}qsin2 tλ

λ2

(η).

Observe that the equation

vt(ξ)=e^{i t Lλ}f(ξ)p₋^λ_{i t}(ξ)=h_λ∗qs(ξ)

with s chosen as above sets up a one to one correspondence between the image of L²(R²ⁿ,e^|ξ|2dξ)under e^{i t Lλ}and the image of L²(R²ⁿ)under the heat kernel transform viz. h→h∗qs(ξ+iη). Indeed, if

F(ξ+iη)=h_λ∗qs(ξ+iη), h_λ∈L²(R²ⁿ) define f in terms of h_λby the equations

hˆ_λ(η)= |det C|g_λ(Cη)e^s|η|2, f(η)=g_λ(η)e^{−iλ4(cot tλ)|η|2}. In other words,

f(η)= ˆh_λ(C⁻¹η)e^{−s|C−1η|2}|det C|⁻¹e^{−iλ4(cot tλ)|η|2}. Since 2s|C⁻¹η|²= |η|²,we see that

R²ⁿ

|f(η)|²e^|η|2dη=c_λ

R²ⁿ

| ˆh_λ(C⁻¹η)|²dη <∞.

Clearly,

e^{i t Lλ}f(ξ)p^λ_{−i t}(ξ)=h_λ∗qs(ξ)=F(ξ) as the above calculation shows, and hence e^{i t Lλ}is onto.

We now prove the uniqueness for the weight function. LetFs(C²ⁿ)be the image of L²(R²ⁿ)under the transform h→h∗qs(ζ )where s=2^sin_λ²2^λt.In view of the above remarks it is enough to prove the weight function characterising the spaceFs(C²ⁿ)

is unique. We know from [1] that this space is a weighted Bergman space with the weight function qa/2(η), η∈R²ⁿ.The spaceFsis invariant under the action ofR²ⁿ. Consequently, if there exists another nonnegative weight function Ws(ξ, η)such that

C²ⁿ

|G(ξ+iη)|²Ws(ξ, η)dξdη=

C²ⁿ

|G(ξ+iη)|²qs/2(η)dξdη

then we should have Ws(ξ, η)=Ws(η)([14, p. 308]). Using the Plancherel formula in theξ variable we have

R²ⁿ

|f(ξ)|²dξ =

C²ⁿ

|f ∗qs(ξ+iη)|²Ws(η)dξdη

=

R²ⁿ

R²ⁿ

|f(ξ)|²e^−2ξ·ηe^−2s|ξ|2 Ws(η)dξdη.

Taking f to be the Gaussian e^−12|x|2 we see that

Rⁿ

Ws(η)e^{(1+2s)−1|η|2}dη <∞.

Moreover, Ws(η)is given by the equation e^2s|ξ|2 =

R²ⁿ

e^−2ξ·ηWs(η)dη.

In view of the above integrability of Ws,both sides of this equation extend toCⁿas entire functions. Consequently we see that

e^−2s|ξ|2 =

R²ⁿ

e^−2iξ·ηWs(η)dη.

and hence Wsis equal to qs/2(η). Thus the weightω^λt is unique for S_t^λ(C²ⁿ).

3 Schrödinger equation for the sublaplacian Let Xj = _∂^∂_xj −¹₂uj ∂

∂ξ, Uj =_∂^∂_uj +¹₂xj ∂

∂ξ and Z= _∂ξ^∂ be the left invariant vector fields on Hⁿforming a basis for the Heisenberg Lie algebra. Then the sublaplacian on Hⁿis defined by

L= − n

j=1

(X²j+U²_j)

and it plays the role of Laplacian onRⁿ. It is well known thatL is a nonnegative essentially self adjoint operator which generates a semigroup e^−tL of convolution with the smooth kernel pt:

pt(x,u, ξ)= ∞

−∞

e^−iλξp^λ_t(x,u)dλ

where p^λ_t is the kernel associated to the special Hermite operator L_λstudied in Sect.2.

Note that e^−tLand e^{−t Lλ}are related via

(e^−tLf)^λ(x,u)=e^{−t Lλ}f^λ(x,u).

Here f^λ(x,u)=

R f(x,u, ξ)e^iλξ dξ. At the convolution level(f ∗pt)^λ(x,u)= f^λ∗λ p_t^λ(x,u).In [14], the authors have characterised the image of L²(Hⁿ)under the semigroup e^−t(L+Z2). There it has been proved that the image is not a weighted Bergman space as expected, but can be expressed as a direct integral of weighted (twisted) Bergman spaces. Here we consider a similar problem for the Schrödinger equation and try to characterise the image of L²(Hⁿ)under e^{i tL}.

Formallyvt(x,u, ξ)=e^{i tL}f(x,u, ξ)solving the Schrödinger equation i∂tvt(x,u, ξ)=Lvt(x,u, ξ), v0(x,u, ξ)= f(x,u, ξ) is given by f ∗pi t(x,u, ξ)which can also be written as

f ∗pi t(x,u, ξ)= ∞

−∞

e^−iλξf^λ∗λ p^λ_{i t}(x,u)dλ.

Note that the kernel pi t has singularities at the points whereλt is an integral multiple ofπ. Therefore, the above formula is not valid even for fixed t, unless the function λ→ f^λis compactly supported so that the singularities are avoided. Observe that

p^λ_t(x,u)=cn

λ sin tλ

n

e^{−λ4(cot tλ)(|x|2+|u|2)}

so that there is no problem atλ =0. Therefore, if we assume thatλ → f^λ is sup- ported in, say|λ| ≤ R, the above integral representation for f ∗pi t makes sense for all t,|t|< ^π_R.If we further assume that f^λ(x,u), has enough decay as a function of (x,u), then f ∗pi t(x,u, ξ)can be holomorphically extended to the whole ofC²ⁿ⁺¹ which is the complexification of Hⁿ.

For R>0 letHRstand for the Hilbert subspace of L²(Hⁿ)consisting of functions f having the following two properties:

(i) f^λis supported in|λ| ≤R (ii)

R²ⁿ⁺¹|f(x,u, ξ)|²e^(|x|2+|u|2)d x du dξ <∞.

Definef_HRto be the square root of the quantity in (ii). ThenHRbecomes a Hilbert space with the inner product

(f,g)=

R²ⁿ⁺¹

f(x,u, ξ)g(x,u, ξ)e^(|x|2+|u|2)d x du dξ.

We have the following direct integral characterisation of the image ofHRunder e^{i tL}. Let Tt stand for the map which takes f ∈L²(Hⁿ)into(e^{i t Lλ}f^λ)_λ∈[−R,R].

Theorem 3.1 Let R > 0 and |t| < ^π_R. Then Tt : HR → _R

−R⊕S_t^λdλ is, up to a constant multiple, an isometric isomorphism.

Proof The theorem follows from Theorem 2.4. Indeed, consider the map f→(e^{i tL}f)^λ fromHR →_R

−R⊕St^λdλ.As(e^{i tL}f)^λ=e^{i t Lλ}f^λ, we have e^{i t Lλ}f^λ2_Sλ

t =cn

R²ⁿ

|f^λ(x,u)|²e^(|x|2+|u|2)d x du.

Integrating with respect toλand using Plancherel in theξvariable, we obtain R

−R

(e^{i tL}f)^λ2_Sλ

tdλ=cn

R²ⁿ⁺¹

|f(x,u, ξ)|²e^(|x|2+|u|2)d x du dξ.

To prove the converse, suppose (F_λ)λ∈[−R,R] belong to R

−R⊕S_t^λdλ. Then using Theorem 2.4 for eachλ∈ [−R,R]there exists unique f_λ∈ L²(R²ⁿ,e^|x|2+|u2|d x du) such that e^{i t Lλ}f_λ=F_λand the equality

R²ⁿ

|f_λ(x,u)|²e^|x|2+|u2|d x du= F_λ²_Sλ t

holds. We define f on Hⁿby

f(x,u,t)= R

−R

f_λ(x,u)e^−iλtdλ.

We see that f ∈HRand Ttf is(F_λ)_λ∈[−R,R].Now the function f is unique as f^λis so. This proves the surjectivity.

We now address the problem of characterising the image ofHR under e^{i tL}as a weighted Bergman space. Note that the operator e^{i tL}commutes with left translations by elements of the Heisenberg group (being a right convolution operator) but the space

HRis not invariant under left translations. However, it is invariant under translation in the last variable. This means thatHRis invariant under translation by elements of the centre of the Heisenberg group. This shows that the image ofHRunder e^{i tL}is also invariant under the action of centre of Heisenberg group. Suppose now that there is a nonnegative weight function Wt(z, w,s)onCⁿ×Cⁿ×Csuch that

C²ⁿ⁺¹

|e^{i tL}f(z, w, ζ )|²Wt(z, w, ζ )d z dwdζ (3.1)

=cn

R²ⁿ⁺¹

|f(x,u, ξ)|²e^(|x|2+|u|2)d x du dξ

for all f ∈HR.Replacing f byτaf(x,u, ξ)= f(x,u, ξ−a),a∈Rand using the fact that e^{i tL}commutes withτawe see that

C²ⁿ⁺¹

|e^{i tL}f(z, w, ζ )|²Wt(z, w, ζ+a)d z dwdζ (3.2)

=cn

R²ⁿ⁺¹

|f(x,u, ξ)|²e^(|x|2+|u|2)d x du dξ

for all f ∈HRand a ∈ R.Since the right hand side is independent of a we see that Wt(z, w, ζ+a)=Wt(z, w, ζ )for all a which means that Wt(z, w, ζ )depends only on the imaginary part ofζ.Thus if we letζ =ξ +iηthen

C²ⁿ⁺¹

|e^{i tL}f(z, w, ζ )|²Wt(z, w,iη)d z dwdζ (3.3)

=cn

R²ⁿ⁺¹

|f(x,u, ξ)|²e^(|x|2+|u|2)d x du dξ

Using the Euclidean Plancherel theorem in theξ variable, the above leads to

R

R

C²ⁿ

|e^{i t Lλ}f^λ(z, w)|²e^2ληWt(z, w,iη)d z dwdηdλ

=cn

R

R²ⁿ

|f^λ(x,u)|²e^(|x|2+|u|2)d x du dλ.

Letϕk be a sequence of Schwartz functions onRsuch that(ϕˆk)²converges to the Dirac mass atλ0.Replacing f by f(x,u, ξ)ϕk(ξ)in the above equation and letting k

tend to infinity we obtain

R

C²ⁿ

|e^{i t Lλ0} f^λ0(z, w)|²e^2λ0ηWt(z, w,iη)d z dwdη (3.4)

=cn

R²ⁿ

|f^λ0(x,u)|²e^(|x|2+|u|2)d x du.

This is true for anyλ0,|λ0| ≤R.Since the weight function for the image of L²(R²ⁿ) under e^{i t Lλ}is unique, the above leads to the condition

∞

−∞

e^2ληWt(z, w,iη)dη (3.5)

=cn

λ² sin²tλ

n

e⁻

1 4

λ2 sin2 tλ

(|y|²+|v|²)

e^{λ(cot tλ)(x·y+u·v)} which should be valid for all|λ| ≤R.

We now show that the above condition cannot be satisfied by a non negative weight function leading to the following negative result.

Theorem 3.2 There does not exist a non negative weight function Wt(z, w,iη)such that

C²ⁿ⁺¹

|e^{i tL}f(z, w, ζ )|²Wt(z, w,iη)d z dwdζ

=cn

R²ⁿ⁺¹

|f^λ(x,u)|²e^(|x|2+|u|2)d x du dλ

for all f ∈HR.

Proof As we have observed, the weight function has to satisfy the condition (3.5) for all|λ| ≤ R.We fix z and wand consider Wt(z, w,iη)as a function ofη.In view of (3.5) it is integrable as a function ofηas we are assuming that Wt is nonnegative.

Moreover,

(λ+i s)→

R

e^{2(λ+i s)η}Wt(z, w,iη)dη

is holomorphic on the strip|λ|<R. To see this, we need only to verify the convergence of the integral

R

|η|e^2ληWt(z, w,iη)dη.