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*∂**t**u*=*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 L*^{2}*(R*^{n}*). Actually much more is true: if q**t* stands
for the heat kernel,

*q**t**(x)*=*(4πt)*^{−}^{n}^{2}*e*^{−}^{4t}^{1}^{|}^{x}^{|}^{2}

*then for f* ∈ *L*^{2}*(R*^{n}*), the solution u(x,t)* = *f* ∗*q**t**(x)*extends toC* ^{n}*as an entire

*function u(x*+

*i y,t)*which satisfies

R^{2n}

|*u(x*+*i y,t)|*^{2}*q**t**/*2*(y)d x d y*=*c**n*

R^{n}

|*f(x)|*^{2}*d x.*

*Conversely, any entire function F(x*+*i y)*onC* ^{n}*for which

R^{2n}

|F*(x*+*i y)|*^{2}*q**t**/*2*(y)d x d y<*∞

*can be factored as F(x)* = *f* ∗*q**t**(x)for some f* ∈ *L*^{2}*(R*^{n}*).*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 solution*v(x,t)*of the equation

*i∂**t**v*=*v, v(x,*0)= *f(x)*

*is given by the unitary group e*^{−}^{i t}^{}*, which is an isometry onto L*^{2}, 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∂**t**u*+1

2*∂*_{x}^{2}*u*=0, *u(x,*0)= *f(x),* *t,x*∈R

extends toC*as 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*^{x}^{2}*d x, then u(x,t)*extends toCas an entire function and

√1
*πt*^{2}

∞

−∞

∞

−∞

*e*^{−}

*y2*

*t 2*|*e*^{−}^{2t}^{i}^{z}^{2}*u(z,t)|*^{2}*d x d y*=
∞

−∞

|*f(x)|*^{2}*e*^{x}^{2} *d x*

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

√1*πα*
∞

−∞

∞

−∞

*e*^{−}^{y2}* ^{α}*|

*f(x*+

*i y)|*

^{2}

*d x d y*= ∞

*j*=0

*α*^{j}*j*!

∞

−∞

|∂*x*^{j}*f(x)|*^{2}*d 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∂**t**u*=*u,* *u(x,*0*)*= *f(x)*
is given by

*u(x,t)*=*(*4*πi t)*^{−}^{n}^{2}

R^{n}

*f(y)e*^{−}^{4i t}^{1} ^{|}^{x}^{−}^{y}^{|}^{2}*d y.*

*Therefore, u(x,t)e*^{4i t}^{1}^{|}^{x}^{|}^{2} is up to a constant multiple, the Fourier transform (at _{2t}^{1}*x)*
*of the function f(y)e*^{−}^{4i t}^{1}^{|}^{y}^{|}^{2}*.If we assume that f has a Gaussian decay, then it is*
well known that its Fourier transform extends toC* ^{n}*as 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}^{|}^{2}*d y, then a simple*
calculation shows that

*u(x,t)e*^{−}^{4t}^{i}^{}^{x}^{2}* ^{j}* =

*(4πi t)*

^{−}

^{n}^{2}

R^{n}

*f(y)e*^{4t}^{i}^{}^{y}^{2}^{j}

*e*^{−}^{2t}^{i}^{x}^{·}^{y}*d y,*

which can be written as

*u(x,t)e*^{−}^{4t}^{i}^{}^{x}^{2}* ^{j}* =

*(2i t)*

^{−}

^{n}

^{/}^{2}

*g*ˆ∗

*q*

*s*

1
*2tx*

*where g(y)*= *f(y)e*^{4t}^{i}^{|}^{y}^{|}^{2}*e*^{s}^{|}^{y}^{|}^{2}*. As g*∈ *L*^{2}*(R*^{n}*), it follows from the theorem of Segal*
*and Bargmann that F(x,t)*=*u(x,t)e*^{−}^{4t}^{i}^{}^{x}^{2}* ^{j}* extends toC

*as an entire function and*

^{n}

C^{n}

|F*(z,t)|*^{2}*q**s**/*2

*y*
*2t*

*d x d y* =*(2t)*^{n}

C^{n}

| ˆ*g*∗*q**s**(z)|*^{2}*q**s**/*2*(y)d x d y*

=*(2t)*^{n}

R^{n}

| ˆ*g(y)|*^{2}*d y,*

*which leads to the identity, with s* =^{1}_{2},

C^{n}

|u(z,*t)e*^{−}^{4t}^{i}^{}^{z}^{2}* ^{j}*|

^{2}

*e*

^{−}

1
*4t2*|*y*|^{2}

*d x d y* =*(4t*^{2}*π)*^{n}^{/}^{2}

R^{n}

|*f(y)|*^{2}*e*^{|}^{y}^{|}^{2}*d 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 L*^{2}*(H*^{n}*)*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|^{2}onR^{n}*. 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)P**k*

*where P**k**f* =

|α|=*k**f, *_{α}_{α}*, ** _{α}*being the normalised Hermite functions onR

^{n}*.*Using Mehler’s formula (see [25]) we obtain

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

R^{n}

*K**i t**(x,y)f(y)d y,*

where

*K**i t**(x,y)*=*(2πi sin 2t)*^{−}^{n}^{2}*e*^{i}^{4}^{(}^{cot 2t}^{)(|}^{x}^{|}^{2}^{+|}^{y}^{|}^{2}^{)−}^{i}^{(}^{csc 2t}^{)}^{x}^{·}^{y}*.*

*Note that K**i 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*^{−}^{4}^{i}^{(}^{cot 2t}^{)|}^{x}^{|}^{2}*e*^{i t H}*f(x)*=*(2πi sin 2t)*^{−}^{n}^{2}

R^{n}

*f(y)e*^{i}^{4}^{(}^{cot 2t}^{)|}^{y}^{|}^{2}*e*^{−}^{i}^{(}^{csc 2t}^{)}^{x}^{·}^{y}*d y.*

*If we assume that f(y)e*^{s}^{|}^{y}^{|}^{2} ∈ *L*^{2}*(R*^{n}*),then defining g(y)*= *f(y)e*^{s}^{|}^{y}^{|}^{2}*e*^{i}^{4}^{(}^{cot 2t}^{)|}^{y}^{|}^{2},
we obtain

*e*^{−}^{4}^{i}^{(}^{cot 2t}^{)|}^{x}^{|}^{2}*e*^{i t H}*f(x)*=*(*2*πi sin 2t)*^{−}^{n}^{2}*g*ˆ∗*q**s**((csc 2t)x)*

*where q**s* is the Euclidean heat kernel onR^{n}*.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 onR

^{n}*.*

We briefly recall the main theorem on Segal–Bargmann transform for the conve-
*nience of the reader. The operator which takes f* ∈ *L*^{2}*(R*^{n}*)*into the entire function
*f* ∗*q**t**(z)* onC* ^{n}* 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 L**^{2}*(R*^{n}*)onto the space of entire*
*functions F on* C^{n}*which are square integrable with respect to the measure*
*q**t**/*2*(y)d xd y.Moreover, for any f* ∈Ł^{2}*(R*^{n}*)*

C^{n}

|*f* ∗*q**t**(x*+*i y)|*^{2}*q**t**/*2*(y)d xd y*=

R^{n}

|*f(x)|*^{2}*d 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* =

^{k}_{2}

*π,k*∈Z,

*e*

^{i t H}*defines an isomorphism between L*

^{2}

*(R*

^{n}*,*

*e*

^{|}

^{x}^{|}

^{2}

*d x)and the Hilbert space of entire functions on*C

^{n}*that are square integrable*

*with respect to the weight function(2π|sin 2t|)*

^{−}

^{n}*e*

^{−(}

^{csc}

^{2}

^{2t}

^{)|}

^{y}^{|}

^{2}

*e*

^{(}

^{cot 2t}

^{)}

^{x}^{·}

^{y}*.*

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

C^{n}

|F*(x*+*i y,t)|*^{2}*q**s**/*2*((csc 2t)y)d x d y*

=*(2π*^{−}^{n}*)|sin 2t|*^{n}

C^{n}

| ˆ*g*∗*q**s**(x*+*i y)|*^{2}*q**s**/*2*(y)d x d y*

=*(*2*π)*^{−}* ^{n}*|

*sin 2t*|

^{n}R^{n}

| ˆ*g(y)|*^{2}*d y*

=*(2π)*^{−}* ^{n}*|

*sin 2t|*

^{n}R^{n}

|*f(y)|*^{2}*e*^{2s}^{|}^{y}^{|}^{2}*d y.*

*Taking s*=^{1}_{2} we observe that

C^{n}

|F*(x*+*i y)|*^{2}*e*^{−(}^{csc}^{2}^{2t}^{)|}^{y}^{|}^{2} *d x d y*

=*c**n**(2π)*^{−}* ^{n}*|

*sin 2t|*

^{n}R^{n}

|*f(y)|*^{2}*e*^{|}^{y}^{|}^{2}*d y.*

This can be rewritten as

C^{n}

|e^{i t H}*f(x*+*i y)|*^{2}*e*^{(}^{cot 2t}^{)}^{x}^{·}* ^{y}*|2π

*sin 2t*|

^{−}

^{n}*e*

^{−((}

^{csc}

^{2}

^{2t}

^{)|}

^{y}^{|}

^{2}

^{)}*d x d y*

=*c**n*

R^{n}

|*f(y)|*^{2}*e*^{|}^{y}^{|}^{2}*d 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* = ^{π}_{4}*,* *e*^{i t H}*f is a constant multiple of* *f and the*ˆ
above result reduces to

R^{n}

R^{n}

| ˆ*f(x*+*i y)|*^{2}*e*^{−|}^{y}^{|}^{2} *d x d y*=*c**n*

R^{n}

|*f(y)|*^{2}*e*^{|}^{y}^{|}^{2}*d 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** _{λ}*defined by

*L** _{λ}*= −+

*λ*

^{2}

4 *(|x|*^{2}+ |u|^{2}*)*−*iλ*
*n*

*j*=1

*(x**j**∂**u**j* −*u**j**∂**x**j**)*

whereis the Laplacian onR* ^{2n}*and

*λ*∈R\{0}. The spectral decomposition of L

*is given by*

_{λ}*L*_{λ}*f* =*(2π)*^{−}^{n}^{∞}

*k*=0

*(2k*+*n)|λ|f* ∗_{λ}*ϕ**k*^{λ}

where the*λ*twisted convolution between two functions is defined as
*f* ∗_{λ}*g(x,u)*=

C^{n}

*f(x*−*x,u*−*u)g(x,u* *)e*^{i}^{λ}^{2}^{(}^{u}^{·}^{x}^{−}^{u}^{·}^{x}^{)}*d x* *du,*

so that the Schrödinger equation is solved by the function

*e*^{i t L}^{λ}*f(x,u)*=*c** _{λ}*
∞

*k*=0

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

*Again the kernel of e*^{i t L}^{λ}*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 L}

^{λ}*f*=

*f*∗

*λ*

*p*

^{λ}

_{i t}*,*where

*p*^{λ}_{i t}*(x,u)*=*c**n*

*λ*
sin*λt*

*n*

*e*^{−}^{i}^{λ}^{4}^{(}^{cot t}^{λ)(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*.*

*As in the case of H , e*^{i t L}^{λ}*is also periodic in t and the above kernel exists as long as tλ*
is not an integral multiple of*π. Also note that for such t, p*_{i t}^{λ}*(x,u)*extends toC* ^{n}*×C

*as an entire function.*

^{n}*For t* ∈R, λ∈R\{0}, and*ξ, η*∈R* ^{2n}*, set

*ω*

*t*

^{λ}*(ξ, η)*=

*e*

^{(λ}^{cot}

^{λ}

^{t}

^{)ξ·η}*q*

_{sin2 tλ}*λ*2 *(η)*where
*q**s**(η)*is the Euclidean heat kernel onR^{2n}*. Consider the Hilbert space S*_{t}^{λ}*(C*^{2n}*)*of all
entire functions for which

F^{2}_{S}*λ*

*t* =

C^{2n}

|F(ξ+*iη)|*^{2}*ω*^{λ}*t**(ξ, η)dξ* *dη <*∞.

We now prove

**Theorem 2.4 The operator e**^{i t L}^{λ}*is an isometric isomorphism between L*^{2}*(R*^{2n}*,*
*e*^{|ξ|}^{2}*dξ)and the Hilbert space S*_{t}^{λ}*(C*^{2n}*)*:

C^{2n}

|e^{i t L}^{λ}*f(ξ*+*iη)|*^{2}*ω*^{λ}_{t}*(ξ, η)dξdη*=

R^{2n}

|*f(ξ)|*^{2}*e*^{|ξ|}^{2}*dξ.* (2.2)

*Moreover, the weight functionω**t*^{λ}*is unique.*

*Proof For f* ∈*L*^{2}*(R*^{2n}*,e*^{|ξ|}^{2}*dξ)*, consider

*e*^{i t L}^{λ}*f(ξ)*= *f* ∗_{λ}*p*_{i t}^{λ}*(ξ), ξ*=*(x,u)*
which is given by the integral

*c**n*

*λ*
sin*λt*

*n*

R^{2n}

*f(a,b)e*^{i}^{λ}^{4}^{(}^{cot t}^{λ)[|}^{x}^{−}^{a}^{|}^{2}^{+|}^{u}^{−}^{b}^{|}^{2}^{]}*e*^{i}^{λ}^{2}^{(}^{a}^{·}^{u}^{−}^{b}^{·}^{x}^{)}*da db.*

Defining*v**t**(ξ)*=*e*^{i t L}^{λ}*f(ξ)p*_{−}^{λ}_{i t}*(ξ)*, we can write the above as
*v**t**(ξ)*=

*λ*
sin*λt*

*2n*

R^{2n}

*g*_{λ}*(η)e*^{−}^{i}^{λ}^{2}^{(}^{A}^{η·ξ)}*dη*

*where g*_{λ}*(η)*= *f(η)e*^{i}^{λ}^{4}^{(}^{cot t}^{λ)|η|}^{2} *and A is the 2n*×*2n matrix*
*A*=*A(λ)*=

cot(tλ)*I**n* −I*n*

*I**n* cot(t*λ)I**n*

*.*

We would like to express*v**t* *as the convolution of a function h** _{λ}*∈

*L*

^{2}

*(R*

^{2n}*)*with the Euclidean heat kernel onR

^{2n}*, still denoted by q*

*s*

*, for suitable s. Let B*=

*(A*

^{t}*)*

^{−}

^{1}and

*C*=

^{2}

_{λ}*B. We then have*

*v**t**(ξ)*=
*λ*

sin*λt*
*2n*

R^{2n}

*g*_{λ}*(Cη)|det C*|e^{−}^{i}^{ξ·η}*dη*

=

R^{2n}

*h*ˆ_{λ}*(η)e*^{−}^{s}^{|η|}^{2}*e*^{−}^{i}^{ξ·η}*dη*

where*h*ˆ_{λ}*(η)*= _{sin}^{λ}_{λ}_{t}*2n*

|*det C*|*g*_{λ}*(Cη)e*^{s}^{|η|}^{2} *for a suitable s to be chosen later.*

Therefore,*v**t**(ξ)*=*h** _{λ}*∗

*q*

*s*

*(ξ),*the convolution with the Euclidean heat kernel on R

^{2n}*. If h*

*∈*

_{λ}*L*

^{2}

*(R*

^{2n}*)*, then it follows from the Bargmann–Segal theorem that

*v*

*t*

*(ξ)*extends toC

*as an entire function and*

^{2n}

C^{2n}

|v*t**(ξ*+*iη)|*^{2}*q**s**/*2*(η)dξdη*=

R^{2n}

|h_{λ}*(η)|*^{2}*dη.*

Now,

R^{2n}

|*h*_{λ}*(η)|*^{2}*dη*=

R^{2n}

| ˆ*h*_{λ}*(η)|*^{2}*dη*

=
*λ*

sin*λt*
*4n*

|*det C*|^{2}

R^{2n}

|g_{λ}*(Cη)|*^{2}*e*^{2s}^{|η|}^{2}*dη*

=
*λ*

sin*λt*
*4n*

|*det C*|

R^{2n}

|g_{λ}*(η)|*^{2}*e*^{2s}^{|}^{C}^{−1}^{η|}^{2}*dη.*

*After computing the determinant of C and*|C^{−}^{1}*η|*we end up with

R^{2n}

|h_{λ}*(η)|*^{2}*dη*= *λ*^{4n}*(sin tλ)*^{4n}

2^{2n}

*λ*^{2n}*(sin tλ)*^{2n}

R^{2n}

|g_{λ}*(η)|*^{2}*e*^{λ}

2

2 |η|^{2}*s csc*^{2}*t**λ**dη*

=

2^{n}*λ*^{n}*(sin tλ)*^{n}

*2n*

R^{2n}

|*f(η)|*^{2}*e*^{|η}^{2}^{|}*dη*

*by choosing s*=2^{sin}_{λ}^{2}2^{t}* ^{λ}*. Thus we have proved the isometry

C^{2n}

|e^{i t L}^{λ}*f(ξ*+*iη)|*^{2}|*p*^{λ}_{−}_{i t}*(ξ*+*iη)|*^{2}*q**sin2 t**λ*
*λ*2

*(η)dξdη*

=*c*_{λ}

R^{2n}

|*f(η)|*^{2}*e*^{|η|}^{2} *dη,*

*where c** _{λ}*=

_{sin t}^{2}

^{λ}

_{λ}*2n*

. Note that

|*p*_{−}^{λ}_{i t}*(ξ*+*iη)|*^{2}=
*λ*

*sin tλ*
*2n*

|e^{−}^{i}^{λ}^{4}^{(}^{cot t}^{λ)(ξ+}^{i}^{η)}^{2}|^{2}=
*λ*

*sin tλ*
*2n*

*e*^{λ}^{2}^{(}^{cot t}* ^{λ)ξ·η}*
and hence

*ω*

^{λ}

_{t}*(ξ, η)*=

*e*

^{λ(}

^{cot t}

^{λ)ξ·η}*q*

*sin2 t*

*λ*

*λ*2

*(η).*

Observe that the equation

*v**t**(ξ)*=*e*^{i t L}^{λ}*f(ξ)p*_{−}^{λ}_{i t}*(ξ)*=*h** _{λ}*∗

*q*

*s*

*(ξ)*

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

*F(ξ*+*iη)*=*h** _{λ}*∗

*q*

*s*

*(ξ*+

*iη),*

*h*

*∈*

_{λ}*L*

^{2}

*(R*

^{2n}*)*

*define f in terms of h*

*by the equations*

_{λ}*h*ˆ_{λ}*(η)*= |*det C*|g_{λ}*(Cη)e*^{s}^{|η|}^{2}*,* *f(η)*=*g*_{λ}*(η)e*^{−}^{i}^{λ}^{4}^{(}^{cot t}^{λ)|η|}^{2}*.*
In other words,

*f(η)*= ˆ*h*_{λ}*(C*^{−}^{1}*η)e*^{−}^{s}^{|}^{C}^{−}^{1}^{η|}^{2}|*det C*|^{−}^{1}*e*^{−}^{i}^{λ}^{4}^{(}^{cot t}^{λ)|η|}^{2}*.*
*Since 2s|C*^{−}^{1}*η|*^{2}= |η|^{2}*,*we see that

R^{2n}

|*f(η)|*^{2}*e*^{|η|}^{2}*dη*=*c*_{λ}

R^{2n}

| ˆ*h*_{λ}*(C*^{−}^{1}*η)|*^{2}*dη <*∞.

Clearly,

*e*^{i t L}^{λ}*f(ξ)p*^{λ}_{−}_{i t}*(ξ)*=*h** _{λ}*∗

*q*

*s*

*(ξ)*=

*F(ξ)*

*as the above calculation shows, and hence e*

^{i t L}*is onto.*

^{λ}We now prove the uniqueness for the weight function. Let*F**s**(C*^{2n}*)*be the image of
*L*^{2}*(R*^{2n}*)under the transform h*→*h*∗*q**s**(ζ )where s*=2^{sin}_{λ}^{2}2^{λ}^{t}*.*In view of the above
remarks it is enough to prove the weight function characterising the space*F**s**(C*^{2n}*)*

is unique. We know from [1] that this space is a weighted Bergman space with the
*weight function q**a**/*2*(η), η*∈R^{2n}*.*The space*F**s*is invariant under the action ofR^{2n}*.*
*Consequently, if there exists another nonnegative weight function W**s**(ξ, η)*such that

C^{2n}

|G(ξ+*iη)|*^{2}*W**s**(ξ, η)dξdη*=

C^{2n}

|G(ξ+*iη)|*^{2}*q**s**/*2*(η)dξdη*

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

R^{2n}

|*f(ξ)|*^{2}*dξ* =

C^{2n}

|*f* ∗*q**s**(ξ*+*iη)|*^{2}*W**s**(η)dξdη*

=

R^{2n}

R^{2n}

|*f(ξ)|*^{2}*e*^{−}^{2}^{ξ·η}*e*^{−}^{2s}^{|ξ|}^{2} *W**s**(η)dξdη.*

*Taking f to be the Gaussian e*^{−}^{1}^{2}^{|}^{x}^{|}^{2} we see that

R^{n}

*W**s**(η)e*^{(}^{1}^{+}^{2s}^{)}^{−}^{1}^{|η|}^{2}*dη <*∞.

*Moreover, W**s**(η)*is given by the equation
*e*^{2s}^{|ξ|}^{2} =

R^{2n}

*e*^{−}^{2}^{ξ·η}*W**s**(η)dη.*

*In view of the above integrability of W**s**,*both sides of this equation extend toC* ^{n}*as
entire functions. Consequently we see that

*e*^{−}^{2s}^{|ξ|}^{2} =

R^{2n}

*e*^{−}^{2i}^{ξ}^{·η}*W**s**(η)dη.*

*and hence W**s**is equal to q**s**/*2*(η). Thus the weightω*^{λ}*t* *is unique for S*_{t}^{λ}*(C*^{2n}*).*

**3 Schrödinger equation for the sublaplacian**
*Let X**j* = _{∂}^{∂}_{x}* _{j}* −

^{1}

_{2}

*u*

*j*

*∂*

*∂ξ**,* *U**j* =_{∂}^{∂}_{u}* _{j}* +

^{1}

_{2}

*x*

*j*

*∂*

*∂ξ* *and Z*= _{∂ξ}* ^{∂}* be the left invariant vector

*fields on H*

*forming a basis for the Heisenberg Lie algebra. Then the sublaplacian*

^{n}*on H*

*is defined by*

^{n}*L*= −
*n*

*j*=1

*(X*^{2}*j*+*U*^{2}_{j}*)*

and it plays the role of Laplacian onR* ^{n}*. It is well known that

*L*is a nonnegative

*essentially self adjoint operator which generates a semigroup e*

^{−}

^{t}*of convolution*

^{L}*with the smooth kernel p*

*t*:

*p**t**(x,u, ξ)*=
∞

−∞

*e*^{−}^{i}^{λξ}*p*^{λ}_{t}*(x,u)dλ*

*where p*^{λ}_{t}*is the kernel associated to the special Hermite operator L** _{λ}*studied in Sect.2.

*Note that e*^{−}^{t}^{L}*and e*^{−}^{t L}* ^{λ}*are related via

*(e*^{−}^{t}^{L}*f)*^{λ}*(x,u)*=*e*^{−}^{t L}^{λ}*f*^{λ}*(x,u).*

*Here f*^{λ}*(x,u)*=

R *f(x,u, ξ)e*^{i}^{λξ}*dξ*. At the convolution level*(f* ∗*p**t**)*^{λ}*(x,u)*=
*f** ^{λ}*∗

*λ*

*p*

_{t}

^{λ}*(x,u).*In [14], the authors have characterised the image of L

^{2}

*(H*

^{n}*)*under

*the semigroup e*

^{−}

^{t}

^{(L+}

^{Z}^{2}

*. 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 L*

^{2}

*(H*

^{n}*)under e*

^{i t}*.*

^{L}Formally*v**t**(x,u, ξ)*=*e*^{i t}^{L}*f(x,u, ξ)*solving the Schrödinger equation
*i∂**t**v**t**(x,u, ξ)*=*Lv**t**(x,u, ξ), v*0*(x,u, ξ)*= *f(x,u, ξ)*
*is given by f* ∗*p**i t**(x,u, ξ)*which can also be written as

*f* ∗*p**i t**(x,u, ξ)*=
∞

−∞

*e*^{−}^{i}^{λξ}*f** ^{λ}*∗

*λ*

*p*

^{λ}

_{i t}*(x,u)dλ.*

*Note that the kernel p**i t* has singularities at the points where*λt is an integral multiple*
of*π. Therefore, the above formula is not valid even for fixed t, unless the function*
*λ*→ *f** ^{λ}*is compactly supported so that the singularities are avoided. Observe that

*p*^{λ}_{t}*(x,u)*=*c**n*

*λ*
*sin tλ*

*n*

*e*^{−}^{λ}^{4}^{(}^{cot t}^{λ)(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}

so that there is no problem at*λ* =0. Therefore, if we assume that*λ* → *f** ^{λ}* is sup-
ported in, say|λ| ≤

*R, the above integral representation for f*∗

*p*

*i 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*∗

*p*

*i t*

*(x,u, ξ)*can be holomorphically extended to the whole ofC

^{2n}^{+}

^{1}

*which is the complexification of H*

*.*

^{n}*For R>*0 let*H**R**stand for the Hilbert subspace of L*^{2}*(H*^{n}*)*consisting of functions
*f having the following two properties:*

(i) *f** ^{λ}*is supported in|λ| ≤

*R*(ii)

R* ^{2n+1}*|

*f(x,u, ξ)|*

^{2}

*e*

^{(|}

^{x}^{|}

^{2}

^{+|}

^{u}^{|}

^{2}

^{)}*d x du dξ <*∞.

Define*f*_{H}*R*to be the square root of the quantity in (ii). Then*H**R*becomes a Hilbert
space with the inner product

*(f,g)*=

R^{2n}^{+}^{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 of*H**R**under e*^{i t}* ^{L}*.

*Let T*

*t*

*stand for the map which takes f*∈

*L*

^{2}

*(H*

^{n}*)*into

*(e*

^{i t L}

^{λ}*f*

^{λ}*)*

_{λ∈[−}*R*

*,*

*R*]

*.*

**Theorem 3.1 Let R***>* *0 and* |t| *<* ^{π}_{R}*. Then T**t* : *H**R* → _{R}

−*R*⊕S_{t}^{λ}*dλ* *is, up to a*
*constant multiple, an isometric isomorphism.*

*Proof The theorem follows from Theorem 2.4. Indeed, consider the map f*→*(e*^{i t}^{L}*f)** ^{λ}*
from

*H*

*R*→

_{R}−*R*⊕S*t*^{λ}*dλ.*As*(e*^{i t}^{L}*f)** ^{λ}*=

*e*

^{i t L}

^{λ}*f*

*, we have*

^{λ}*e*

^{i t L}

^{λ}*f*

^{λ}^{2}

_{S}*λ*

*t* =*c**n*

R^{2n}

|*f*^{λ}*(x,u)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du.*

Integrating with respect to*λ*and using Plancherel in the*ξ*variable, we obtain
*R*

−*R*

(e^{i t}^{L}*f)*^{λ}^{2}_{S}*λ*

*t**dλ*=*c**n*

R^{2n}^{+}^{1}

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

To prove the converse, suppose *(F*_{λ}*)**λ∈[−**R**,**R*] belong to *R*

−*R*⊕*S*_{t}^{λ}*dλ.* Then using
Theorem 2.4 for each*λ*∈ [−*R,R*]*there exists unique f** _{λ}*∈

*L*

^{2}

*(R*

^{2n}*,e*

^{|}

^{x}^{|}

^{2}

^{+|}

^{u}^{2}

^{|}

*d x du)*

*such that e*

^{i t L}

^{λ}*f*

*=*

_{λ}*F*

*and the equality*

_{λ}

R^{2n}

|*f*_{λ}*(x,u)|*^{2}*e*^{|}^{x}^{|}^{2}^{+|}^{u}^{2}^{|}*d x du*= F_{λ}^{2}_{S}*λ*
*t*

*holds. We define f on H** ^{n}*by

*f(x,u,t)*=
*R*

−*R*

*f*_{λ}*(x,u)e*^{−}^{i}^{λ}^{t}*dλ.*

*We see that f* ∈*H**R**and T**t**f is(F*_{λ}*)*_{λ∈[−}*R**,**R*]*.Now the function f is unique as f** ^{λ}*is
so. This proves the surjectivity.

We now address the problem of characterising the image of*H**R* *under e*^{i t}* ^{L}*as a

*weighted Bergman space. Note that the operator e*

^{i t}*commutes with left translations by elements of the Heisenberg group (being a right convolution operator) but the space*

^{L}*H**R*is not invariant under left translations. However, it is invariant under translation in
the last variable. This means that*H**R*is invariant under translation by elements of the
centre of the Heisenberg group. This shows that the image of*H**R**under e*^{i t}* ^{L}*is also
invariant under the action of centre of Heisenberg group. Suppose now that there is a

*nonnegative weight function W*

*t*

*(z, w,s)*onC

*×C*

^{n}*×Csuch that*

^{n}

C^{2n}^{+}^{1}

|e^{i t}^{L}*f(z, w, ζ )|*^{2}*W**t**(z, w, ζ )d z dwdζ* (3.1)

=*c**n*

R^{2n+1}

|*f(x,u, ξ)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du dξ*

*for all f* ∈*H**R**.Replacing f byτ**a**f(x,u, ξ)*= *f(x,u, ξ*−*a),a*∈Rand using the
*fact that e*^{i t}* ^{L}*commutes with

*τ*

*a*we see that

C^{2n+1}

|e^{i t}^{L}*f(z, w, ζ )|*^{2}*W**t**(z, w, ζ*+*a)d z dwdζ* (3.2)

=*c**n*

R^{2n}^{+}^{1}

|*f(x,u, ξ)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du dξ*

*for all f* ∈*H**R**and a* ∈ *R.Since the right hand side is independent of a we see that*
*W**t**(z, w, ζ*+*a)*=*W**t**(z, w, ζ )for all a which means that W**t**(z, w, ζ )*depends only
on the imaginary part of*ζ.*Thus if we let*ζ* =*ξ* +*iη*then

C^{2n}^{+}^{1}

|e^{i t}^{L}*f(z, w, ζ )|*^{2}*W**t**(z, w,iη)d z dwdζ* (3.3)

=*c**n*

R^{2n+1}

|*f(x,u, ξ)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du dξ*

Using the Euclidean Plancherel theorem in the*ξ* variable, the above leads to

R

R

C^{2n}

|*e*^{i t L}^{λ}*f*^{λ}*(z, w)|*^{2}*e*^{2}^{λη}*W**t**(z, w,iη)d z dwdηdλ*

=*c**n*

R

R^{2n}

|*f*^{λ}*(x,u)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du dλ.*

Let*ϕ**k* be a sequence of Schwartz functions onRsuch that*(ϕ*ˆ*k**)*^{2}converges to the
Dirac mass at*λ*0*.Replacing f by f(x,u, ξ)ϕ**k**(ξ)in the above equation and letting k*

tend to infinity we obtain

R

C^{2n}

|e^{i t L}^{λ0}*f*^{λ}^{0}*(z, w)|*^{2}*e*^{2}^{λ}^{0}^{η}*W**t**(z, w,iη)d z dwdη* (3.4)

=*c**n*

R^{2n}

|*f*^{λ}^{0}*(x,u)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du.*

This is true for any*λ*0*,*|λ0| ≤*R.Since the weight function for the image of L*^{2}*(R*^{2n}*)*
*under e*^{i t L}* ^{λ}*is unique, the above leads to the condition

∞

−∞

*e*^{2}^{λη}*W**t**(z, w,iη)dη* (3.5)

=*c**n*

*λ*^{2}
sin^{2}*tλ*

*n*

*e*^{−}

1 4

*λ*2
*sin2 t**λ*

*(|**y*|^{2}+|v|^{2}*)*

*e*^{λ(}^{cot t}^{λ)(}^{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 W***t**(z, w,iη)such*
*that*

C^{2n}^{+}^{1}

|*e*^{i t}^{L}*f(z, w, ζ )|*^{2}*W**t**(z, w,iη)d z dwdζ*

=*c**n*

R^{2n+1}

|*f*^{λ}*(x,u)|*^{2}*e*^{(|}^{x}^{|}^{2}^{+|}^{u}^{|}^{2}^{)}*d x du dλ*

*for all f* ∈*H**R**.*

*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 W**t**(z, w,iη)*as a function of*η.*In view
of (3.5) it is integrable as a function of*ηas we are assuming that W**t* is nonnegative.

Moreover,

*(λ*+*i s)*→

R

*e*^{2}^{(λ+}^{i s}^{)η}*W**t**(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^{2}^{λη}*W**t**(z, w,iη)dη.*