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Journal of Mathematical Analysis and Applications

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Metaplectic group, symplectic Cayley transform, and fractional Fourier transforms v-/

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Maurice A. de Gosson a'*, Franz Luefb

a University of Vienna, Faculty of Mathematics (NuHAG), 1090 Vienna, Austria

b Department of Mathematics, Norwegian University of Science and Technology, 7491 Trondheim, Norway

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abstract

Article history: Received 15 February 2013 Available online 14 March 2014 Submitted by P.G. Lemarie-Rieusset

Keywords:

Metaplectic and symplectic group Fractional Fourier transform Weyl operator Feichtinger algebra

We begin with a survey of the standard theory of the metaplectic group with some emphasis on the associated notion of Maslov index. We thereafter introduce the Cayley transform for symplectic matrices, which allows us to study in detail the spreading functions of metaplectic operators, and to prove that they are basically quadratic chirps. As a non-trivial application we give new formulae for the fractional Fourier transform in arbitrary dimension. We also study the regularity of the solutions to the Schrödinger equation in the Feichtinger algebra.

© 2014 The Authors. Published by Elsevier Inc. This is an open access article

under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

Contents

1. Introduction......................................................................948

2. Metaplectic operators and their Maslov indices ..............................................950

2.1. Free symplectic matrices and a factorization result..............................................................................950

2.2. Quadratic Fourier transforms and the metaplectic group......................................................................951

2.3. The Maslov index..............................................................953

3. The Weyl representation of metaplectic operators..........................................................................................954

3.1. The symplectic Cayley transform...................................................954

3.2. The operators A(v).............................................................956

3.3. The spreading function of a metaplectic operator................................................................................959

4. Applications ......................................................................961

4.1. The Schrödinger equation........................................................961

4.2. Fractional Fourier transform......................................................963

4.3. Multiple-angle FRFT...........................................................965

5. Conclusions, discussion, and conjectures...................................................967

Acknowledgments.......................................................................968

References............................................................................968

* Corresponding author. http://dx.doi.org/10.1016/j.jmaa.2014.03.013

0022-247X/© 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

The metaplectic group Mp(d) has a long and rich history; its definition goes back to I. Segal [20] and D. Shale [21], and its study has been taken up from an abstract viewpoint by A. Weil [23] in connection with C. Siegel's work in number theory. Its theory has subsequently been developed and used by many authors working in various areas of mathematics and physics (see [13,18] for applications to quantum mechanics); a non-exhaustive list of contributions is (in alphabetical order) [4,6,11,17,22], and the references therein. The metaplectic group also intervenes in time-frequency analysis, especially in connection with the theory of Gabor frames (see in particular K.H. Grochenig's treatise [14] for applications to Gabor analysis).

The aim of this article is to study the metaplectic representation from the point of view of the Weyl calculus of pseudodifferential operators, and to apply our results to a systematic treatment of the fractional Fourier transform. Some of our results have been exposed in [12,13] with quantum-mechanical applications in mind (the semiclassical Gutzwiller formula), but in a different spirit and with different notation. The exposition we give in this paper is conceptually simpler, thanks to a more systematic use of the symplectic Cayley transform, which is an object of intrinsic genuine mathematical interest. We have carefully avoided sloppy statements about the phase factors intervening in the expression of the involved metaplectic operators; while it does not matter what these factors are as long as these operators appear in conjugation formulae (e.g. various symplectic covariance formulae in Weyl calculus), this lack of rigor is completely fatal in many other circumstances, for instance in quantum mechanics where one considers sums of terms involving each a different operator.

At the most elementary level the metaplectic group explains the connection between two very classical notions, the Fourier transform and the symplectic group: to the Fourier transform

Ff M= /" e-2™^ f (x) dx

(or rather a slight variant thereof, see formula (5)) the metaplectic representation associates the symplectic rotation

J = ( 0 1

J v -I 0

At a mathematically more sophisticated level, the metaplectic group Mp(d) can be defined in at least two different ways, both making (implicitly or explicitly) use of the following basic fact: the symplectic group Sp(d) is connected and ^[Sp(d)] is isomorphic to the integer group (Z, +); from this follows that Sp(d) has (connected) covering groups Spg(d) of all orders q = 2,3,..., +<. Having this in mind one can proceed as follows:

• Either one uses representation theory: let p be the Schrödinger representation of the Heisenberg group H(d) [8,13,14]; to every (z,t) = (x,u,r) G H(d) it associates the unitary operator

p(z,t ) = e2niT enix'u TXMU

where Tx and Mu are, respectively, the time-shift and modulation operators; a theorem basically due to Shale and Weil [21,23] then asserts the existence of a unitary representation on L2(Rd) of Sp2(d) such that

[M]([A])p(z,r )[M]([A]) 1 = p(Az,r)

if [A] G Sp2(d) covers A G Sp(d). The mapping is the metaplectic (or Shale-Weil) representation, the group Mp(d) = [^](Sp2(d)) is called the metaplectic group, and its elements metaplectic operators1 or metaplectomorphisms;

• Or one constructs directly, using both geometric and analytical methods, a family of unitary operators on L2(Rd) by using the machinery of generating functions for free symplectic matrices; the group generated by these generalized Fourier transforms is then the metaplectic group Mp(d): this is the approach that will be used in the present paper. This way of introducing the metaplectic representation has the advantage of producing relatively simple integral formulae; a crucial result is that each metaplectic operator can be written as the product of two Fourier integral operators with quadratic phases.

The first approach is perhaps the most well-known (it is exposed with great clarity in Gröchenig [14]), and it is certainly also the most elegant. There are however hidden difficulties, especially in the interpretation of the cocycles which appear. The second approach is, in a sense, more constructive; it has the advantage of being the shortest bridge between classical and quantum dynamics because it allows an immediate calculation of the solutions of Schrödinger equations associated with quadratic Hamiltonian operators (see [13]). For us, it has a major advantage in the context of time-frequency analysis, because it will allow us to arrive at a simple and useful description of metaplectic operators in terms of only time-shifts, modulation operators, and symplectic matrices. This will allow us to write explicit formulae for the spreading function of metaplectic operators, and to apply these to the fractional Fourier transform.

This article is structured as follows: in Section 2 we review (mostly without proofs) the theory of the metaplectic group from the point of view of quadratic Fourier transforms as exposed in Leray [17], de Gos-son [11]; the key result of this section is Theorem 3 which says that every metaplectic operator can be factored as the product of exactly two quadratic Fourier transforms. This gives us the opportunity to discuss (briefly) the related topic of Maslov index. We thereafter proceed to study the elements of Mp(d) from the viewpoint of pseudodifferential calculus. It will follow from the Weyl representation of metaplectic operators that the two metaplectic operators corresponding to a symplectic matrix A with no eigenvalue equal to one, can be represented, for a suitable choice of -\/det(A — I), as A(v) and A(v+2) = —A(v) with

where we have set p(z) = eniuxTxMu if z = (x, w). This formula - which is after all surprisingly simple - is as far as we know new in time-frequency analysis. It turns out that it can, after some simple manipulations, be rewritten in the form

is a real symmetric matrix, called the symplectic Cayley transform of A. It shows that the spreading function of a metaplectic operator A(v) with det(A- I) = 0 is just the "chirp"

Ma = 2 J (A +1 )(A — I)

1 In Gröchenig [14] they are called symplectic operators.

The quantum physicist's Schrôdinger equation is seldom considered in TFA. This equation is however closely related to the notion of fractional Fourier transform, and its generalizations. It therefore deserves to be discussed in some detail. In passing we state and prove regularity results for the solutions of that equation in Feichtinger's algebra.

Notation. Our notation is standard (it is basically that of [8,14]); the elements of M2d = Rd x Rd are denoted by z = (x,w) and the symplectic product of z,z' is [z,z'] = w • xX — w' • x, where the dot • stands for the usual Euclidean scalar product. When M is a symmetric matrix and u a vector we write Mu2 for Mu • u.

2. Metaplectic operators and their Maslov indices

We begin by recalling some elementary properties from the theory of free symplectic matrices (for a detailed exposition and a rather complete list of references see [13]).

2.1. Free symplectic matrices and a factorization result

Let A and J be the 2d-dimensional block matrices

A = (c d) , J = (-I 0

(all the blocks are of dimension d); the matrix A is symplectic if AT JA = J (or, equivalently, if AJAT = J). The symplectic matrices form a connected classical Lie group, Sp(d). One says that the symplectic matrix A is free if det B = 0; the free symplectic matrices form a dense subset of Sp(d). To a free symplectic matrix A one associates a quadratic form

W(x,xr) = 1 DB-1x2 - B-1x ■ x' + 1B-1 Cx'2 (1)

called the free generating function2 of A. The reason for this denomination is that this quadratic form entirely determines A: the relation (x,w) = A(x',u>') is equivalent to

w = dxW (x,x') and W = -dx, W (x,x') (2)

as is verified by a straightforward calculation. When A is a free symplectic matrix generated by W we will often write it AW. The basic example is A = J; the generating function is in this case W(x,x') = -x ■ x'. Since the inverse of the symplectic matrix A is given by the formula

A-1 = ( DT -BT A \-CT CT ,

it follows that the inverse of a free symplectic matrix also is free, and that AW1 = AW' with W'(x,x') = -W (x',x).

An important property of the free symplectic matrices in Sp(d) is that they generate Sp(d); even more important is that every symplectic matrix can be written as a product of exactly two free symplectic matrices; since this property will play an important role in many parts of this paper let us put it in italics:

Proposition 1. For every A G Sp(d) there exist two generating functions W and W' such that A = Aw Aw '.

2 The use of the letter W comes from the word Wirkung, which is German for action (which is what the generating function represents in mechanics).

In [13] we give a conceptual geometric proof of this result; we discourage the reader to try to work it out (except in the case d = 1) by using matrix multiplication!

Since we are in the business of generating functions, let us note that every free symplectic matrix AW can be factored as:

AW = V-DB-1 M B-1 JV-B-1A (3)

where we are using the notations

I 0\ . . fL 0 -PI), Ml =U (LT)

when P = PT and det L = 0 (that DB-1 and B-1A indeed are symmetric follows from the fact that A G Sp(d)). It follows from formula (3) and Proposition 1 that the set of all matrices Vp, ML and J generates Sp(d).

For further use here is a technical result that we will use in connection with the study of the spreading function of metaplectic operators:

Lemma 2. Let A = {AD) be a free symplectic matrix. We have

det(A-I) = (—1)d(det B)det(B-1A + DB-1 — B-1 — (BT )-1). (4)

In particular det(A — I) = 0 if and only if the Hessian matrix of the function x ——> W(x, x) is invertible. Proof. We have

i A — I B \ 0 B WC — (D — I )B-1(A — I) 0 \ V C D — i) VI D — IJ\ B-1(A — I) I)

and hence

det(A— I) = (det(—B)) det(C — (D — I)B-1(A — I)). Formula (4) follows noting that since A is symplectic we have C — DB-1A = —(BT)-1. 2.2. Quadratic Fourier transforms and the metaplectic group Let us introduce the following unitary operators

Vpf(x) = einPx f(x), Ml,mf(x) = imyj|detL\f(Lx) for P = PT, det L = 0 and argdet B = mn (mod 4), and J = i-d/2F, that is

JVf (x) = i-d/2 j e-2nixx f (x') dx'. (5)

These operators generate a group of unitary operators on L2(Rd), called the metaplectic group Mp(d). Note the obvious relations (Vp)-1 = V-P and (ML,m)-1 = ML-1,d-m.

To a free symplectic matrix A = AW we now associate following Leray [17] two operators iJ,m(AW) and Mm+2(Aw) defined, for f G S(Rd), by

Mm (Aw )f (x) = im-d/2^J |det B-1\ J e2niW(x'x>)f (xx') dx' (6)

where m corresponds, as above, to a choice of argument for the determinant of B:

argdet B = mn mod 4. (7)

Thus, if det B > 0 then m is either 0 or 2 and if det B < 0 it is 1 or 3 (formula (7) is just there to allow us to write concisely the two opposed square roots of det B-1). For instance, if A = J these operators are just ±i-d/2F. For this reason we will call the operators ¡im(AW) and Mm+2(AW) = -Mm(AW) the quadratic Fourier transforms associated with AW (or W). It is easy to see that these operators belong to Mp(d). In fact, definition (1) of the generating function W immediately yields the factorization

Mm(Aw ) = VDB-1 Ms-i,mJ:MB-iA. (8)

Using the factorization (8) together with obvious intertwining formulae for the operators Vp, ML,m, and J one proves that the inverse of Mm(AW) also is a quadratic Fourier transform and that the following generalization of the usual Fourier inversion formula

(Mm (Aw )) 1 = Md-m{AW}) (9)

holds. One proves that the operators Mm(Aw) generate the metaplectic group Mp(d); more precisely, we have the following analogue of Proposition 1, which we glorify as a theorem because of its importance:

Theorem 3. Every operator A € Mp(d) can be written as the product of exactly two quadratic Fourier transforms Mm(AW) and Mm'(Aw').

Proof. See [11,13,17]. □

An immediate observation is that any metaplectic operator can be factored in infinitely many ways as a product Mm(AW)Mm' (Aw'); for instance the identity in Mp(d) can be written as Mm(AW)Md-m(AW~) for all generating functions W. We will moreover see later on (Proposition 6) that one can impose extra conditions on AW and AW'.

The following result connects the metaplectic group Mp(d) to the symplectic group Sp(d): Theorem 4. (i) The mapping nMp : ¡im(Aw) -—^ Aw extends into a surjective homomorphism

nMp : Mp(d) Sp(d)

whose kernel is |+/d, -Id}.

(ii) The group Mp(d) is connected, hence nMp is hence a covering projection which identifies Mp(d) with the double cover of Sp(d).

(iii) We have

nMp(VP) = Vp , nMp(ML,m) = Ml.

Proof. See again [11,13,17]. □

There is a simple - and very important - relation between metaplectic operators and the Wigner transform; recall that if f € S(Rd) (or, more generally, f € S'(Rd)) then its Wigner transform Wf is defined by

Wf(x,u) = J e-2ni"•y f(x + 2^f(x - 1 ^ dy

(the integral must be interpreted in a distributional sense if f G S'(Rd), see [14,25]). For every A G Sp(d) we have the "symplectic covariance formula"

Wf (A-1 (x,w)) = W (Af )(x,w) (10)

where A is any of the two metaplectic operators corresponding to A (this is one instance where one does not need to bother subtleties like the Maslov index: complex factors with modulus one cancel when calculating

W (Af)).

2.3. The Maslov index

There is an important modulo 4 integer invariant attached to each of the factorizations A = lm(Aw)Mm'(Aw'): one proves [9,11] that if A = ¡im(Aw)Mm'(Aw) then

m(A) = [m + m — Inert(D'B'-1 + B-1 A)]4

does not depend on the choice of operators factoring A (the notation [-]4 means "equivalence class modulo 4"), and Inert P is the index of inertia of the symmetric matrix P (both D'B'-1 and B-1A are indeed symmetric because Aw and Aw' are symplectic matrices). The integer m(A) is called the Maslov index of A (see [9,11]); it plays a crucial role in the theory of partial differential equations and its applications to quantum mechanics in its semiclassical formulation, where it appears as a count of "caustics": see Leray [17], de Gosson [11]. In the latter, the interested reader will find various formulae for the calculation of the Maslov index of the product of two metaplectic operators, as well as generalizations; let us mention without proof that following formula holds for the Maslov index of the product of two metaplectic operators:

m(AA') = m(A)+m(A') + [c(A, A')]4 (11)

where c is a group cocycle:

c(A, A') + c(AA', A'') = c(A, A'A'') + c(A', A'');

that cocycle can be expressed in terms of the Wall-Kashiwara index of a triple of Lagrangian planes (see [13] for a detailed discussion of that index and of the formula above). Explicitly c(A, A') is calculated as follows (see [9,10]): consider the quadratic form Qa,A' defined by

Qaa (z1,Z2 ,Z3) = [Z1,Z2] + [z2 ,Z3] + [Z3 ,Z1]

for z1 = (0,^1), z2 = A(0,w2), z3 = A'(0,w3). That quadratic form has a signature signQa,A' (it is the difference between the number of positive and negative eigenvalues of the matrix of Qa,A' ). With these definitions,

c(A, A') = —1 (sign Qa,A' + d + dimker B — dimker B' + dimker B")

where B, B', B" are the upper right d x d blocks in the matrix representations of A, A' and A'' = AA'.

Formula (11) has a quite interesting consequence. Consider the set Sp(d) x Z4 equipped with the product

(A,m)(AV) = (AA',m + M + [c(A, A')]4).

It is easy to verify, using the cocycle property ofc that this is a group law; one proves [10,11] that the mapping Mp(d) —> Sp(d) x Z4 which to A associates (A,m(A)) is an isomorphism of Mp(d) onto a subgroup of Sp(d) x Z4; equipping the latter with a carefully chosen topology (not the product topology!) one can then identify Mp(d) with that subgroup (this result has useful consequences in semiclassical mechanics, see [11]).

3. The Weyl representation of metaplectic operators

Schwartz' kernel theorem (see Grochenig [14, Chapter 14]) says that every linear operator L : S(Rd) —> S'(Rd) which is continuous in the weak *-topology can be written in the Weyl form

L = J nL(x,^)eniu xTxMu dxdw (12)

where nL (the spreading function) is a tempered distribution, which is in fact the symplectic Fourier transform of the Kohn-Nirenberg symbol of that operator, when defined:

i = FaVL = FVL oJ-1 (13)

where F is the usual Fourier transform. Recall that i is related to L by

Lf (x) = J e2niw• xi(x, w)Ff (w) dw; (14)

we make the following convention: when this operator is written in the Weyl form (12) we write L = Op(i) = L. The integral (12) has generally to be understood in the distributional sense. Since every operator A € Mp(d) maps continuously S(Rd) C S'(Rd) into itself, a natural question which deserves to be posed is:

What is the spreading function of a metaplectic operator?

We are going to answer this question in detail; we will actually see that Mp(d) is generated by operators having (up to a complex factor) spreading functions which are chirps. Let us first introduce a very convenient tool, the symplectic Cayley transform of a symplectic matrix. It is a variant of a Cayley transform for matrices considered by Howe [15] in his study of the oscillator group (also see Folland [8, Chapter 5]). It will play a crucial role in the forthcoming part of this paper, because it allows to calculate correctly the Weyl symbol and the spreading function of metaplectic operators.

3.1. The symplectic Cayley transform

Let A € Sp(d); we assume that A has no eigenvalue equal to one: det(A — I) = 0. We will denote the set of all such 2d x 2d symplectic matrices by Sp*(d). By definition, the symplectic Cayley transform of A € Sp*(d) is the matrix

Ma = 1J (A + I )(A — I )-1. (15)

The following result summarizes the properties of the symplectic Fourier transform:

Proposition 5. (i) Let A € Sp* (d). We have Ma = (Ma)T and the symplectic Cayley transform is an injection Sp*(d) —> Sym(2d, R) (the symmetric 2d x 2d matrices); its inverse is given by the formula

A = M — 2J) \MA + J .

(ii) We have Ma-i = —Ma and

Ma + Ma' = J (A —I )-1(AA' — I)(A' — I )-1. (16)

(iii) Assume that A, A!, and AA are in Sp*(d). The symplectic Cayley transform of AA is then given by the formula:

Maa' = Ma + (AT — I)'1 J (Ma + Ma )-1J (A —I )-1. (17)

Proof. The properties (i) and (ii) are elementary, using the relations ATJA = AJAT = J. Let us prove (17); equivalently

Ma + M = Maa' (18)

where M is the matrix defined by

M = (AT — I )-1J (Ma + Ma' )-1 J (A —I )-1

that is, in view of (16),

M = (AT — I )-1J (A — I )(AA' — I)-1. Since ST = — JA-1 J and (—A-1 + I)-1 = A(A — I)-1 we have

M = (AT — I)-1J (A! — I)(AA — I)-1 = —J (—A-1 + I )-1(A' — I )(AA — I )-1 = —JA(A — I )-1 (A' — I )(AA/ — I)-1 = —J (A' — I)(AA' — I)-1 — J(A — I)-1(A' — I)(AA' — I)-1.

Since MA = 1J + J (A —I )-1 we have

Ma + M = J ^ 11 + (A — I )-1 — (A — I )(AA! — I )-1 — (A — I )-1 (A — I )(AA — I )-1^; noting that

(A —I)-1 — (A —I )-1(A' — I)(AA' — I)-1 = (A —I )-1(AA' — I — A' + I)(AA' — I)-1

(A —I)-1 (AA' — A) (AA! — I) A/(AA — I)-1

we finally obtain

MA + M = J^21 - (A' -1)(AA' -I) 1 + A'(AA' - I) ^

j( 11 + (AA' - I)-1)

which we set out to prove. □

The following result improves Proposition 1:

Proposition 6. Every A € Sp(d) can be written as a product Aw Aw ' of two free symplectic matrices belonging to the set Sp*(d).

Proof. Choose two free symplectic matrices AW and AW' such that A = AWAW' (this is always possible in view of Proposition 1). We have, using the factorization formula (3)

If det(AW - I) = 0 and det(AW' - I) = 0 then the proposition is proven. If not, let now A be any real number and consider the matrices

One immediately verifies that these matrices are symplectic (using the relations BTD = DTB and ATB = BTA); they are trivially free and we have AwAw' = A!wA!w'; choose now A such that det(AW — I) = 0 and det(A'w' — I) = 0. □

Let us mention that the properties proven above also play an important role in the understanding of the properties of the Conley-Zehnder index appearing in the theory of periodic Hamiltonian orbits (see our discussion in [13, Chapter 4]), and in its applications to semiclassical mechanics (in particular the quantization of classically chaotic systems).

3.2. The operators A(^)

Let us introduce the shorthand notation

with z = (x,w) (p(z) is the Heisenberg-Weyl operator of quantum mechanics). Let A G Sp*(d) and v G R. To the pair (A, v) we associate an operator A(^) by the formula

with dz = dxdw; the integral is interpreted as taking its value in a Banach space ("Bochner integral"). Obviously A(v) maps continuously S(Rd) into itself (and hence S'(Rd) into S'(Rd)). The operators A(v) can easily be rewritten in Weyl form using the notion of symplectic Cayley transform introduced in the previous subsection:

a = v-db-1 mb-1 jv-(b-1a+d'b'-1)mb'-1 jvb'-1a' ■

p(z) = e™'xTxMu = Mw/2TxMl

Lemma 7. For every (A, v) € Sp* (d) x R we have

~ V C 2

A(v) = , = einMAZ p(z) dz. (20)

(V) v/|det(A— I)| J

Proof. We begin by noting that

Maz2 = Q J + J (A —I z • z

= J (A —I )-1z • z = [(A—I )-1z,z].

Performing the change of variables z ——> (A — I)-1z in the integral in (20) we thus get

J einMAz2p(z) dz = iv|det(A — I)\ j ein[z'Az]p((A — I)z) dz.

R2d R2d

Since p(z + z') = e-in[z'z ]p(z)p(z') we have

ein[z, Az\p((A— I) z) = p(Az)p(—z)

from which (20) follows in view of definition (19) of A(v). □

The operators A(v) - which will be identified with the metaplectic operators ¡im (A) provided that v is conveniently chosen - are unitary. More precisely, we have the following composition and inversion result for the operators AW ,v. Let us first recall the generalized Fresnel formula

J e-2nix• eeinMx2 dx = ein sisn M|det M\-1!2e-nM-1(? (21)

valid for M real, det M = 0; sign M is the signature of M (see e.g. Folland [8, Appendix A]); the branch of the square root of det M is determined by the requirement that (det M)-1/2 > 0 when M is real.

Proposition 8. (i) The operator A(v) is invertible and we have (A(v))-1 = (A-1)(-v). (ii) Assume that the symplectic matrices A, A, and AA are in Sp*(d). Then

A(v)A'(v') = [AA!)(v+v'+2 sign(MA+MA')). (22)

(iii) The operator A(v) is unitary: A* = A-1.

Proof. Let us begin by proving the composition formula in (ii). Writing for notational simplicity M = Ma and M' = Ma' , the spreading functions of A(v) and A(y) are, respectively,

n(z) = %V einMz2 /(z) ^ %V einM'z2

/V y v/|det(A — I)| ' IK ' v/|det(A' — I)| and that of the product operator A(V)A(V') is thus given by (see e.g. [25])

that is

where the constant K and the function & are given by

i v+v'

r/"(z) = f ein[z'z\(z — z,)r/'(z,) dz' (23)

= K/e"|z"'le-(z"') dZ

v/|det(A— I)(A' - I)| &(Z,Z') = MZ2 — 2MZ • Z' + (M + M')Z'2.

Observing that

[Z, Z/] — 2MZ • Z' = (J — 2M )z • z' = —2 J (A — I )-1Z • z'

we have the equality

[z,z/] + &(z,z!) = — 2 J (A—I )-1z • z' + Mz2 + (M + M r)z'2

and hence

V(z) = KeinMz" j e-2niJ(A-1 )-1z•z'ein(M+M')z'2 dz'. (24)

Applying the Fresnel formula (21) to the integral in the right-hand side, and replacing K with its value, this is simply

rj"(z) = |det [(Ma + Ma')(A —I)(A' — I)] |-1/2e^ ssnMe2nio(z) (25)

where 0 is the quadratic form

o(z) = (M + (AT — I)-1 J (ma + ma' )-1 J (A — I)-1)Z2 that is 0(z) = maa' in view of Proposition 5. Noting that we have

ma + ma' = J (I + (A — I )-1 + (A' — I )-1)

and det J = 1 we get, after a straightforward calculation,

det [(Ma + Ma' )(A — I) (A' — I)] = det [(A — I)(Ma + Ma' ) (A' — I)]

= det(AA' — I)

which concludes the proof of (ii). Let us prove (i). Since det(A — I) = 0 we also have det(A-1 — I) = 0. Formula (24) in the proof of part (ii) shows that the spreading function of A(v)(A-1)(-v) is

£(Z) = KeinMA"2 j e-2niJ(A-I)-1z• z'ein(MA + MA-i )z' 2 dZ' R2d

where the constant K is this time given by

K =_1_=_1_

y/\det(A-I)(A-1 - I)| \det(A — I)|

since det(A-1 — I) = det(I — A). We have MA + MA-i = 0, hence, setting z" = (AT — I)-1 J z',

C(7) = 1 inMAZ2 [ p-2ni(J(A-I)-1z,z') d r

^(z)= \det(A— I )\e Je

= einMAZ'2 J e-2niz•z" dz" R2d

= S(z)

where we have used the Fourier inversion formula J e-2niz'z dz" = S(z) in the last step; 5 is precisely the Weyl symbol of the identity operator. (iii) The product formula (23) allows us to prove that the operators A(v) are unitary. The Weyl symbol of the adjoint of a pseudodifferential operator is the complex conjugate of the Weyl symbol of that operator. Since the Weyl symbol and spreading function are symplectic Fourier transforms of each other, the symbol a of A(v) is therefore given by

a(z) = , - / e-2ni[z'z'^einM^z'2 dz',

y/\det(A — I)\ J

a(z) = , ' - e2ni[z'z'^e-inM^z'2 dz'.

y/\det(A — I)\ J

Since Ma-i = —Ma and \det(A — I)\ = \det(A 1 — I)\ we can rewrite this as

V\det(A-1 — I)

(z) ^ i _ e-2ni[z'z 1 einMA-iz 2 dz'

V \ det(A-1 — I) \

e2ni[z'z'] einMA-1 z' dz',

hence a(z) is the symbol of (A(v))-1 and this concludes the proof. □

3.3. The spreading function of a metaplectic operator

We are next going to apply the results above to the calculation of the spreading function of metaplectic operators.

We begin by proving that the operators A(v) are scalar multiples of metaplectic operators: Lemma 9. Let (A, v) G Sp*(d) x R. There exists a constant c = c(A, v) with \c\ = 1 such that A(v) = c^m(A). Proof. Since A(v) is unitary it is sufficient to show that it satisfies the intertwining relation

A(v)p(z') = p(Az')A(„)

for every z' G R (see the discussion in Grochenig [14, Chapter 9]). Now,

A(v)p(z') = iv^J|det(A — I)| J p(Az)p(—z)p(z') dz,

p(Az')A(v) = |det(A —I)l j p(Az')p(Az)p(—z) dz

and we have p(Az)p(—z)p(z') = p(Az')p(Az)p(—z) by repeated use of the formula p(z + z') = e-in\z,A p(z)p(z'). □

The following result is the link between the operators A(v) and the metaplectic operators; because of its importance it deserves the status of theorem:

Theorem 10. (i) Let Aw be a free symplectic matrix in Sp*(d) and |m(Aw) be an element of the fiber of Aw in Mp(d). Then

1m (Aw) = ±(Aw)( m — Inert Wxx) (26)

where Inert Wxx is the index of inertia of the Hessian matrix of the function x i—> W(x, x). (ii) Every A G Mp(d) can be written as a product (Aw)(v)(Aw')(v') with

v = m — Inert Wxx, v' = m' — Inert W'xx (27)

where m and m' are defined by (7), and we have in this case

A = A(v''), v" = v + v + 1 sign(MAw + MaW ). Proof. Applying (Aw)(v) to the Dirac delta and setting thereafter x = 0 a straightforward calculation yields

K W>(v) W v/|det(Aw — I)IJ

Since Maw = 2 J + J (Aw — I )-1 we have

Maw (0,w') • (0,w') = [(Aw — I)-1(0,W), (0,w')].

To evaluate the right-hand side of this equality let us set (x, w) = (Aw — I)-1(0,w'). This is equivalent to Aw(x,w) = (x,w + w') and since W generates Aw this is the same thing as w + w' = (dxW)(x,x) and w = —(dx'W)(x,x) (cf. (2)). Using the explicit formula (1) for W we have

x = (DB-1 + B-1A — B-1 — (BT )-1)-1w' = W^w'

and hence Maw (0,w') • (0,w') = —W-w'2. We thus have

iv r -1 ,2

(Aw)(v) ¿(0) ^ = e-inW-x - dw'

y ' v/|det(Aw — I)| J

and applying Fresnel's' formula to the integral, this is

(Aw)(v)S(0) = e^114 sign Wxx = V|det Wxx|

y |det(Aw — I )|

where sign Wxx is the signature of the symmetric matrix Wxx. Using the identity (4) in Lemma 2 the formula above can be rewritten in the simple form

(Aw)M5(0) = e-i4 sign w-iv^|det B-1|. (28)

On the other hand, we have, by a straightforward calculation,

Mm (Aw )S(0) = im-d/2^ | det B-1|

hence v — 1 sign Wxx = m — 1 d that is v = m — Inert Wxx since Wxx is non-degenerate. Formula (26) follows in view of Lemma 9. Let us prove (ii). In view of Proposition 6 we can find Aw and Aw' in Sp*(d) such that A = (Aw)(v)(Aw)(v'); the conclusion now follows from Proposition 8. □

Here is a simple example:

Example 11. Consider the usual Fourier transform F on L2(Rd). We have F = id/2 mo (J). It is easily verified that MJ = — 11, Inert MJ = d, det(J — I) = 2d, and hence, by formula (27),

F =(4i)-d/2 f i-z2p(z) dz

where, by definition, iv = e12v for p G R. 4. Applications

We will now apply the metaplectic machinery developed above to two important topics, namely the Schrodinger equation for quadratic Hamiltonians and the fractional Fourier transform (these topics are actually closely related, as we will see).

4.1. The Schrodinger equation

We recall that the symplectic algebra sp(d) (i.e. the Lie algebra of Sp(d)) consists of all 2d x 2d matrices such that JX is symmetric. In what follows a will denote a real parameter; in quantum mechanics that parameter is identified with time, while it is usually an angle in time-frequency analysis.

Consider a one-parameter subgroup (Aa) of the symplectic group Sp(d): AaAa' = Aa+a' and A0 = I. The mapping a ——> Aa being differentiable (it is even analytical) it makes sense to ask which differential equation this mapping satisfies. The elementary theory of Lie groups tells us that there exists X G sp(d) such that Aa = eaX, hence Aa satisfies the differential equation daAa = XAa. This equation is just Hamilton's equation from classical mechanics in disguise: introducing the quadratic form H(z) = — 2 JXz2 and setting za = Aaz0 we have

^ = J dz H (Za) da

that is, setting za = (xa,pa),

= dpH (xa,Pa), da = —dxH (xa,Pa)

which are Hamilton's equations for H. The second, related, question is "what differential equation does the one-parameter subgroup (Aa) of Mp(d) defined by ^Mp(Aa) = Aa satisfy?" The answer is that Aa satisfies the "Schrodinger equation"

where H is the operator with Weyl symbol H. This result is well-known in quantum mechanics, and can be proven by examining in detail the Lie algebra of the metaplectic group (which is isomorphic to the Lie algebra sp(d)); see for instance [13,8]. What is however perhaps a little bit less well-known, is that one does not have to assume that a ——> Aa is a one-parameter group to obtain a Schrodinger equation. Let in fact a ——> Aa be any continuously differentiable path in Sp(d) such that Ao = I. We can, exactly as in the one-parameter group case, again lift that path to a (differentiable) path a ——> Aa in Mp(d) such that A0 = I. One proves, with slightly more work than in the one-parameter case, that Aa again satisfies a Schrodinger equation, namely

id Aa = H (a)Aa da

where H (a) is the operator with Weyl symbol

H (z) = - 2 J X (a)z2,

X (a) = ( -f-Ala IA da

(it is easy to verify that X(a) G sp(d) hence JX(a) is symmetric). For a detailed proof of this property see de Gosson [13, Chapter 7, Corollary 7.20]; this result is also implicit in Folland's [8] discussion of the metaplectic group.

This procedure for generating Schrödinger equations is very general; since the symplectic group is connected we can pick any symplectic matrix A G Sp(d) and join it to the identity by an arbitrary C1 path a -—> Aa; reparameterizing if necessary we may assume without restriction that Ao = I; the corresponding path a -—> Aa in the metaplectic group Mp(d) will then automatically satisfy (30).

Using the results above it is easy to prove regularity results in Feichtinger's algebra for the solutions of Schrödinger equations associated to an operator with quadratic Weyl symbol. Recall [5,14] that the Feichtinger algebra S0(Rd) consists of all distributions f such that for one (and hence all) g G S(Rd) the short-time Fourier transform

^ f (x

-2nit-x

f (t)g(x - t) dt

belongs to the space L1(R2d). Fixing once for all g the formula

WVg f Uli = ff \Vg f (x,u)\dxdu

defines a norm for which S0(Rd) is a Banach space; it is in addition an algebra for pointwise multiplication and convolution. A crucial fact is that if \\Vgf< (x> for one g then \\Vgf < (x> for all g and that the corresponding norms then are equivalent. The Feichtinger algebra S0(Rd) coincides with the Wiener amalgam space W(FL1(Rd),l1(Zd)), which is the Banach space of all functions f : Rd —> C which are locally in FL1(Rd) and have global decay of l1-type (see [14]); its use is therefore particularly well adapted to the study of decay properties of quantum mechanical wavefunctions.

Proposition 12. Let H (z, a) = 1M (a)z ■ z where a i—> M (a) is a Cmapping R —> Sym(2d, R) (the real

symmetric 2d x 2d matrices). Let f be a solution of the partial differential equation

(x,a) = H (a)f (x,a)

where H (a) is the operator with Weyl symbol H (z,a). If f (-,a) G So(Rd) for one a G R then f (•,a/) G S0(Rd) for all a' G R.

Proof. It is elementary: the Feichtinger algebra is invariant under the action of the metaplectic group as follows from the relation

f (x, u) = 2-de-i™xW(/Y, ff)(2x, 1 ^ (32)

where /Y(x) = /(—x), together with the covariance formula (10). □

4-2. Fractional Fourier transform

The idea of the fractional Fourier transform (FRFT) goes back to 1929 (see the historical account and the references in [3]); the first really rigorous account however seems to go back to Kober's 1939 paper [16]. The FRFT has numerous applications, ranging from filter design and signal analysis to phase retrieval, pattern recognition, and optics (see for instance the book [19] by Ozaktas et al., or the articles by Almeida [2] and Alieva and Bastiaans [1], to name only a few recent contributions). It turns out that the theory of the FRFT is actually no more than a particular case of that of the metaplectic representation. In this section we will systematically use this fact, which will allow us to highlight the connections between the FRFT and various objects from symplectic geometry.

The one-dimensional FRFT is an operator depending on a real parameter a which can be interpreted as a rotation angle in the time-frequency plane; roughly speaking it transforms a function to an intermediate domain between time and frequency. For instance, an FRFT with a = n/2 corresponds to the classical Fourier transform, and an FRFT with a = 0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations: as we will see, this only reflects the one-parameter group property of the propagator of Schrodinger's equation for quadratic Hamiltonian operators, whose theory is just an application of that of the metaplectic group. The usual definition found in the literature goes as follows:

Definition 13. The FRFT at an angle a in the time-frequency plane is given, for a G nZ, by

Faf(x) = J Ka(x,x')f (x') dx (33)

where the kernel Ka is defined by the formula

Ka( x,x') = Vl — i cot a exp

. (x + x 2)cosa — 2xx

One proves that (Fa)2n/a is the identity operator; this is usually painfully arrived at using eigenfunction expansions for the Fourier transform in terms of Hermite functions. Closer inspection however reveals that if the argument of the square root of 1 — i cot a is correctly chosen, then Fa is just one of the two metaplectic operators associated to the rotation

cos a sin a

r(—a) =

sin a cos a

with angle —a. This is immediately checked noting that the function

(x2 + x'2) cos a — 2xx'

2 sin a

appearing in the exponent defining Ka in (34) is the generating function (1) of r(—a). It follows that the function a ——> Fa (which can be extended by continuity to all a G R) is then operator solution of Schrodinger's equation

■d r 1 ( d2 2\ ^ i — Fa = - — 3-7 + x )Fa, lim Fa = I

da 2 V dx2 J a^o

where lima^0 Fa = I means that

lim Faf (x) = f (x) (35)

for every f G S(R). From the group property FaFa' = Fa+a' it readily follows that for every real number s we have (Fa)s = Fsa and hence, in particular, (Fa)2n/a = I so that Fa is indeed the FRFT at an angle a.

The easiest way to construct an FRFT in higher dimensions is to generalize the operator Fa briefly discussed above by defining the one-parameter group (Ja) of symplectic matrices by

j =( (cos a)I (sin a)A (36)

a \ — (sin a)I (cos a)I J (I is the d x d identity). These matrices are free for a G nZ and are generated for these values of a by

Wa( x,x') =-((\x\2 + \x'\ ) cos a — 2x • x');

v 7 2sin av v ii/ /

to the one-parameter group (Ja) one can associate canonically the one-parameter group (Ja) of metaplectic operators given by

Jaf (x) = i-d/2-d[a/2] \sin a\-1/2 J e2niW*(x'x,) f [xJ) dx'

where [•] is the integer-part function. The operator J0 = lima^0 Ja (in the sense of (35)) is the identity and Jn/2 = J. By the same argument as above it is easy to show that these operators satisfy ( Ja)2n/a = I so that they are indeed fractional Fourier transforms in their own right.

The results of previous section allow us to write Ja in a particularly simple form: a straightforward calculation shows that the symplectic Cayley transform of the rotation (36) is given, for a G 2nZ, by

1 ( cot(a/2)I 0 Ma = - ' ^ ' '

2 V 0 cot(a/2)I

and the FRFT Ja can therefore be written in Weyl form as

J° ^2s.n(o/2)(1+'cos2(o/2))1/0 / e4W2",W2)P<z)dz. (37)

It follows from the covariance formula (10) that we have the following relation between the FRFT and the Wigner transform:

W/(r(a)(x,u)) = W (Aa/ )(x,u). (38)

It has the following obvious geometric interpretation: if Wf is the Wigner transform of a function / expressed in the x, u coordinates, then W(Aa/) is the expression of W/ in the coordinate system obtained by rotating the x and u axes by an angle a.

Remark 14. Everything that has been said above can (with obvious modifications) be extended to any A G Mp(d) whose projection A on Sp(d) lies in the range of the exponential mapping exp : sp(d) —> Sp(d). Suppose in fact that A = eX and let (Aa) be the one-parameter subgroup of Sp(d) defined by Aa = eaX. In view of general principles (the path-lifting property from the theory of covering spaces) there exists a unique one-parameter subgroup (Alt) of Mp(d) such that nMp(Aa) = Aa for every a G R, and we have

(Aa)1^ = A

4.3. Multiple-angle FRFT

In this last subsection we use the Weyl form of the metaplectic operators to substantially generalize the previous results. Consider a positive-definite quadratic form on time-frequency space R2d; since it is going to play the role of a Hamiltonian function we denote it by H:

H(z) = 1 Mz • z, M = MT > 0.

A classical theorem due to Williamson [24] (see [8,13] for "modern" proofs) tells us that there exists A G Sp(d) such that

M = AT DA, D =( f 0 V 0 A

where A is the diagonal matrix whose non-zero entries are the moduli Xj of the eigenvalues of JM (the eigenvalues of the latter are of the type ±iXj with Xj > 0 because JM is similar to the skew-symmetric matrix M1/2J M1/2). Consider now Hamilton's equations

— = J Mz da

for H; since JAT = A-1 J they are equivalent to the system

—— = J Du, u = Az. da

We have u(a) = eaJDuu(0) and hence z(a) = Aaz(0) where a ——> Aa is the one-parameter group given by

Aa = A-1R(a)A, R(a) =

^ A(a) B(a)

-B (a) A(a)

with A(i) and B(t) the diagonal matrices

/ cos(Aia) 0

0 cos(A2a)

( sin(A1a)

0 sin(A2c

cos(Ada) )

sin(Ada) )

In conformity with the general principles previously discussed to that symplectic one-parameter group corresponds a unique metaplectic one-parameter group a -—> Aa satisfying the Schrödinger equation associated with H. We now ask the question "for which values of a is the metaplectic operator Aa a fractional Fourier transform?" A complete answer is given by

Proposition 15. (i) Assume that Aj a 2nZ for j = 1, 2,...,d; the operator Aa is then given by the formulae

Aa = MR/ ^ 4 ^

M(a) = 2dJJ sin(Aja/2)(1+cos2(Aja/2))1/2, j=1

C(z, a) = J^cot(Aja/2) (xj2 + p]). j=1

(ii) Assume that there exists v G R such that vAj = 1 mod 2 for j = 1, 2,...,d. Then (Aa)vn/o

Avn I.

Proof. Let us calculate the symplectic Cayley transform Ma of the symplectic matrix Aa = A 1R(a)A. We have

MAa =2 J (Aa + I )( Aa - I )

2 JA(R(a) + I)(R(a) - I) 1 A-1

Setting

= 2 (AT) J(R(a)+I) (R(a) - I) 1A-1.

M (a) = 2 J (R(a)+I )(R(a) - I) 1

(it is the symplectic Cayley transform of R(a)) we thus have MAa = (AT ) 1M(a)A 1 and thus

v/|det(R(a) - I)|

einM(a)z2p(Az) dz.

A straightforward computation shows that

(R(a) - I )-1 = - i(

-(I - A(a))-1 B(a)

(I - A(a))-1B(a) I

hence the symplectic Cayley transform of R(a) is the block-diagonal matrix

M (a) =

l f (I - A(a))-1B(a)

(I - A(a)) B(a) J '

(I - A(a))-1B(a) = i

/cot(Aia/2)

cot(A2a/2)

cot(Ada/2) J

det(R(a) - I) =4^ sin2(A^a/2)(l +cos2(A^a/2))

formulae (39)-(41) follow. (ii) Assume that there exist integers kj, j = 1, 2,..., d, such that vAj = 2kj + 1 and set a = vn. Then cos(Aja/2) = 0 and since det A = 1,

Avn = p(Az) dz = p(z) dz = I. □

5. Conclusions, discussion, and conjectures

We have shown that every metaplectic operator can be represented in the two equivalent forms

A = iv\ |det(A-I)l / p(Az)p(-z) dz

v/|det(A-I )|

inMAz2

p(z) dz

when det(A-I) = 0, or as the product of two such operators; as a by-product we have been able to determine the spreading function of metaplectic operators: these are just complex Gaussians ("chirps"). It would be interesting to see how the formulae we have obtained apply in the analysis of Gabor frames, or in other topics in time-frequency analysis. In quantum mechanics they can be used with profit to study the semiclassical behavior of chaotic systems (see [12] and the references therein), and probably also in the theory of quantum revival ("Loschmidt echo"); perhaps these topics could be adapted to get new results in time-frequency analysis. In a recent work [7] Feichtinger et al. have analyzed a class of metaplectic operators on Cd. A natural question that arises is whether the techniques we have developed in this paper could be of some use in this context.

We briefly discussed in Section 4.1 the regularity of the solutions to Schrodinger's equation associated to an operator associated to a quadratic Hamiltonian; the fact that the Feichtinger algebra allowed us to prove that if a solution is in that algebra at any time, then it remains so for all times. It would be extremely interesting, for quantum mechanical purposes, to examine whether this property remains true for the solutions of Schrodinger equations associated to an arbitrary Hamiltonian. It is well-known in quantum

mechanics that if the initial wave-function is a Gaussian, then it remains approximately a Gaussian in the course of time. This qualitative property, together with the fact that rather wide classes of functions can be expanded in terms of Gaussians (this is a particular case of the theory of Gabor frames, see [14]) makes us conjecture that it is indeed so; the key to a proof might be the use of known semiclassical approximations together with the use of the Duhamel principle: we hope to come back to this important question in a near future, together with extensions to more general modulation spaces.

Acknowledgments

Maurice de Gosson has been supported by a research grant from the Austrian Research Agency FWF (Project No. P23902-N13). Franz Luef was supported by an APART fellowship from the Austrian Academy of Sciences and large parts of the work were done at UC Berkeley during a long term stay. The authors would like to thank the referee for very useful comments and suggestions.

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