Hamiltonian deformations of Gabor frames: First steps.

Gabor frames can advantageously be redefined using the Heisenberg-Weyl operators familiar from harmonic analysis and quantum mechanics. Not only does this redefinition allow us to recover in a very simple way known results of symplectic covariance, but it immediately leads to the consideration of a general deformation scheme by Hamiltonian isotopies (i.e. arbitrary paths of non-linear symplectic mappings passing through the identity). We will study in some detail an associated weak notion of Hamiltonian deformation of Gabor frames, using ideas from semiclassical physics involving coherent states and Gaussian approximations. We will thereafter discuss possible applications and extensions of our method, which can be viewed - as the title suggests - as the very first steps towards a general deformation theory for Gabor frames.

Introduction

The theory of Gabor frames (or Weyl–Heisenberg frames as they are also called) is a rich and expanding topic of applied harmonic analysis. It has numerous applications in time–frequency analysis, signal theory, and mathematical physics. The aim of this article is to initiate a systematic study of the symplectic transformation properties of Gabor frames, both in the linear and nonlinear cases. Strangely enough, the use of symplectic techniques in the theory of Gabor frames is often ignored; one example (among many others) being Casazza's seminal paper [7] on modern tools for Weyl–Heisenberg frame theory, where the word “symplectic” does not appear a single time in the 127 pages of this paper! This is of course very unfortunate: it is a thumb-rule in mathematics and physics that when symmetries are present in a theory their use always leads to new insights in the mechanisms underlying that theory. To name just one single example, the study of fractional Fourier transforms belongs to the area of symplectic and metaplectic analysis and geometry (see Section 3.4); remarking this would avoid to many authors unnecessary efforts and complicated calculations. On the positive side, there are however (a few) exceptions to this refusal to include symplectic techniques in applied harmonic analysis: for instance, in Gröchenig's treatise [26] the metaplectic representation is used to study various symmetries in time frequency analysis, and the recent paper by Pfander et al. [50] elaborates on earlier work [29] by Han and Wang, where symplectic transformations are exploited to study various properties of Gabor frames.

In this paper we consider deformations of Gabor systems using Hamiltonian isotopies. A Hamiltonian isotopy is a curve of diffeomorphisms of phase space starting at the identity, and such that there exists a (usually time-dependent) Hamiltonian function H such that the (generalized) phase flow determined by the Hamilton equations consists of the mappings for . In particular Hamiltonian isotopies consist of symplectomorphisms (or canonical transformations, as they are called in physics). Given a Gabor system with window (or atom) ϕ and lattice Λ we want to find a working definition of the deformation of by a Hamiltonian isotopy . While it is clear that the deformed lattice should be the image of the original lattice Λ, it is less clear what the deformation of the window ϕ should be. A clue is however given by the linear case: assume that the mappings are linear, i.e. symplectic matrices ; assume in addition that there exists an infinitesimal symplectic transformation X such that for . Then is the flow determined by the Hamiltonian function where J is the standard symplectic matrix. It is well-known that in this case there exists a one-parameter group of unitary operators satisfying the operator Schrödinger equation where the formally self-adjoint operator is obtained by replacing formally p with in (2); the matrices and the operators correspond to each other via the metaplectic representation of the symplectic group. This suggests that we define the deformation of the initial window ϕ by . It turns out that this definition is satisfactory, because it allows to recover, setting , known results on the image of Gabor frames by linear symplectic transformations. This example is thus a good guideline; however one encounters difficulties as soon as one want to extend it to more general situations. While it is “reasonably” easy to see what one should do when the Hamiltonian isotopy consists of an arbitrary path of symplectic matrices (this will be done in Section 4), it is not clear at all what a “good” definition should be in the general nonlinear case: this is discussed in Section 4.3, where we suggest that a natural choice would be to extend the linear case by requiring that should be the solution of the Schrödinger equation associated with the Hamiltonian function H determined by the equality ; the Hamiltonian operator would then be associated with the function H by using, for instance, the Weyl correspondence. Since the method seems to be difficult to study theoretically and to implement numerically, we propose what we call a notion of weak deformation, where the exact definition of the transformation of the window ϕ is replaced with a correspondence used in semiclassical mechanics, and which consists in propagating the “center” of a sufficiently sharply peaked initial window ϕ (for instance a coherent state, or a more general Gaussian) along the Hamiltonian trajectory. This definition coincides with the definition already given in the linear case, and has the advantage of being easily computable using the method of symplectic integrators (which we review in Section 2.3) since all what is needed is the knowledge of the phase flow determined by a certain Hamiltonian function. Finally we discuss possible extensions of our method.

We notice that the notion of general deformations of Gabor frames is an ongoing topic in Gabor analysis; see for instance the recent contribution by Gröchenig et al. [28], also Feichtinger and Kaiblinger [16] where lattice deformations are studied.

Notation and terminology The generic point of the phase space is denoted by where we have set , . The scalar product of two vectors, say p and x, is denoted by or simply px. When matrix calculations are performed, z, x, p are viewed as column vectors.

We will write where and . The scalar product on is defined by and the associated norm is denoted by . The Schwartz space of rapidly decreasing functions is denoted by and its dual (the space of tempered distributions) by .

Hamiltonian isotopies and symplectic integrators

We review the basics of the modern theory of Hamiltonian mechanics from the symplectic point of view; for details we refer to [2,11,38,51]; we are following here the elementary accounts we have given in [21,23].

Hamiltonian flows

We will equip with the standard symplectic structure in matrix notation where (0 and I are here the zero and identity matrices). The symplectic group of is denoted by ; it consists of all linear automorphisms of such that for all . Working in the canonical basis is identified with the group of all real matrices S such that (or, equivalently, ). A diffeomorphism is called a symplectomorphism if the Jacobian matrix is symplectic at every : (Symplectomorphisms are often called “canonical transformations” in physics.) The symplectomorphisms of form a subgroup of the group of all diffeomorphisms of (this follows from formula (3) above, using the chain rule). Of course is a subgroup of .

Let be real-valued; we will call H a Hamiltonian function. The associated Hamilton equations with initial data at time are (cf. Eqs. (1)). Assuming existence and uniqueness of the solution for every choice of the time-dependent flow is the family of mapping which associates to every initial the value of the solution of (4). The importance of symplectic geometry in Hamiltonian mechanics comes from the following result:

Each diffeomorphism is a symplectomorphism of : . Equivalently where is the Jacobian matrix of calculated at z.

See for instance [2,21,23].  □

It is common practice to write . Obviously and the satisfy the groupoid property for all t, and . Notice that it follows in particular that .

A remarkable fact is that composition and inversion of Hamiltonian flows also yield Hamiltonian flows:

Let and be the phase flows determined by two Hamiltonian functions and . We have

It is based on the transformation properties of the Hamiltonian fields under diffeomorphisms; see [21,38,51] for detailed proofs.  □

We notice that even if H and K are time-independent Hamiltonians the functions H#K and are generically time-dependent.

Hamiltonian isotopies

We will call a symplectomorphism f such that for some Hamiltonian function H and time a Hamiltonian symplectomorphism. The choice of time in this definition is of course arbitrary, and can be replaced with any other choice noting that we have where .

Hamiltonian symplectomorphisms form a subgroup of the group of all symplectomorphisms; it is in fact a normal subgroup of as follows from the conjugation formula valid for every symplectomorphism g of (see [38,21,23]). This formula is often expressed in Hamiltonian mechanics by saying that “Hamilton's equations are covariant under canonical transformations”. That is a group follows from the two formulas (8) and (9) in Proposition 2 above.

The following result is, in spite of its simplicity, a deep statement about the structure of the group . It says that every continuous path of Hamiltonian transformations passing through the identity is itself the phase flow determined by a certain Hamiltonian function.

Let be a smooth one-parameter family of Hamiltonian transformations such that . There exists a Hamiltonian function such that . More precisely, is the phase flow determined by the Hamiltonian function where .

See Banyaga [4]; Wang [57] gives an elementary proof of formula (11).  □

The idea is already present in Arnold [2, p. 269] who uses the apparatus of generating functions to produce related results.

We will call a smooth path in joining the identity to some element a Hamiltonian isotopy. Proposition 3 above says that every Hamiltonian isotopy is a Hamiltonian flow restricted to some time interval.

Consider in particular the case of the symplectic group . We claim that every path in joining an element to the identity is a Hamiltonian isotopy. Since is connected there exists a path , (in fact infinitely many) joining the identity to S in . In view of Proposition 3 above there exists a Hamiltonian function H such that . The following result gives an explicit description of that Hamiltonian without using formula (11):

Let , , be a Hamiltonian isotopy in . There exists a Hamiltonian function such that is the phase flow determined by the Hamilton equations . Writing the Hamiltonian function is the quadratic form where , etc.

The matrices being symplectic we have . Differentiating both sides of this equality with respect to t we get or, equivalently, This equality can be rewritten hence the matrix is symmetric. Set ; then (these relations reduce to when is time-independent: see (27) below). Define now using (14) one verifies that the phase flow determined by H consists precisely of the symplectic matrices and that H is given by formula (13).  □

Symplectic algorithms

Symplectic integrators are designed for the numerical solution of Hamilton's equations; they are algorithms which preserve the symplectic character of Hamiltonian flows. The literature on the topic is immense; a well-cited paper is Channel and Scovel [8]. Among many recent contributions, a highlight is the recent treatise [40] by Kang Feng and Mengzhao Qin; also see the comprehensive paper by Xue-Shen Liu et al. [60], and Marsden's online lecture notes [47, Chapter 9].

Let be a Hamiltonian flow; let us first assume that H is time-independent so that we have the one-parameter group property . Choose an initial value at time . A mapping on is an algorithm with time step-size Δt for if we have the number k (usually an integer ⩾1) is called the order of the algorithm. In the theory of Hamiltonian systems one requires that be a symplectomorphism; is then called a symplectic integrator. One of the basic properties one is interested in is convergence: setting (N an integer) when do we have ? One important requirement is stability, i.e. must remain close to z for small t (Chorin et al. [9]).

Here are two elementary examples of symplectic integrators. We assume that the Hamiltonian H has the physical form

First order algorithm. One defines by

Second order algorithm. Setting we take

One can show, using Proposition 3 that both schemes are not only symplectic, but also Hamiltonian (Wang [57]). For instance, for the first order algorithm described above, we have where K is the now time-dependent Hamiltonian

When the Hamiltonian H is itself time-dependent its flow does no longer enjoy the group property , so one has to redefine the notion of algorithm in some way. This can be done by considering the time-dependent flow defined by (6): . One then uses the following trick: define the suspended flow by the formula one verifies that the mappings : (the “extended phase space”) satisfy the one-parameter group law and one may then define a notion of algorithm approximating (see Struckmeier [55] for a detailed study of the extended phase space approach to Hamiltonian dynamics). For details and related topics see the paper [9] by Chorin et al. where a general Lie–Trotter is developed.

Gabor frames: the symplectic point of view

Gabor frames are a generalization of the usual notion of basis; see for instance Gröchenig [26], Feichtinger and Gröchenig [15], Balan et al. [3], Heil [33], Casazza [7] for a detailed treatment of this topic. In what follows we give a slightly modified version of the usual definition, better adapted to the study of symplectic symmetries.

Definition and elementary properties

Let ϕ be a non-zero square integrable function (hereafter called window) on , and a lattice Λ in , i.e. a discrete subset of . Observe that we do not require that Λ be regular (i.e. a subgroup of ). The associated ħ-Gabor system is the set of square-integrable functions where is the Heisenberg operator. The action of this operator is explicitly given by the formula (see e.g. [21,23,44]; it will be justified in Section 4.1). We will call the Gabor system a ħ-frame for , if there exist constants (the frame bounds) such that for every square integrable function ψ on . When the ħ-frame is said to be tight.

The product is, up to the factor , Woodward's cross-ambiguity function [59]; its symplectic Fourier transform is the cross-Wigner distribution as was already observed by Klauder [42]; see [21,23] for a detailed study of this relationship.

For the choice the notion of ħ-Gabor frame coincides with the usual notion of Gabor frame as found in the literature. In fact, in this case, writing and , we have where is the time–frequency shift operator defined by for . The two following elementary results can be used to toggle between both definitions:

Let . The system is a Gabor frame if and only if is a ħ-Gabor frame.

We have where . By definition is a Gabor frame if and only if for every that is this inequality is equivalent to that is to hence the result since is equivalent to the condition .  □

We can also rescale simultaneously the lattice and the window (which amounts to a “change of Planck's constant”):

Let be a Gabor system, and set Then is a frame if and only if is a ħ-frame.

We have where has projection on (see Appendix A). The Gabor system is a ħ-frame if and only for every , that is, taking the symplectic covariance formula (21) into account, if and only if But this is inequality is equivalent to and one concludes using Proposition 7.  □

In Appendix A (formula (65)) we state a rescaling property for the covering projection of metaplectic group onto .

Symplectic covariance of Gabor frames

Gabor frames behave well under symplectic transformations of the lattice (or, equivalently, under metaplectic transformations of the window). Formula (21) below will play a fundamental role in our study; it relates Heisenberg–Weyl operators, linear symplectic transformations, and metaplectic operators (we refer to Appendix A for a concise review of the metaplectic group and its properties). Let have projection . Then (see e.g. [21,23,44]); one easy way to derive this intertwining relation is to prove it separately for each generator , , of the metaplectic group described in formulae (61), (62), (63). We remark the time–frequency shift operators do not satisfy any simple analogue of property (21). As a consequence, the covariance properties we will study below do not appear in any “obvious” way when using the standard tools of Gabor analysis.

The following result is well-known, and appears in many places in the literature (see e.g. Gröchenig [26], Pfander et al. [50], Luo [45]). Our proof is somewhat simpler since it exploits the symplectic covariance property of the Heisenberg–Weyl operators, which we explain now.

Let (or ). A Gabor system is a ħ-frame if and only if is a ħ-frame; when this is the case both frames have the same bounds. In particular, is a tight ħ-frame if and only if is.

Using formula (21) intertwining metaplectic and Heisenberg–Weyl operators we have and hence, since is a ħ-frame, The result follows since because metaplectic operators are unitary; the case is similar since metaplectic operators are linear automorphisms of .  □

The result above still holds when one assumes that the window ϕ belongs to the Feichtinger algebra (see Appendix B and the discussion at the end of the paper).

Application to Gaussian frames

The problem of constructing Gabor frames in with an arbitrary window ϕ and lattice Λ is difficult and has been tackled by many authors (see for instance the comments in [27], also [50]). Very little is known about the existence of frames in the general case. We however have the following characterization of Gaussian frames which extends a classical result of Lyubarskii [46] and Seip and Wallstén [53]:

Let (the standard centered Gaussian) and with and . Then is a frame if and only if for .

Bourouihiya [5] proves this for ; the result for arbitrary follows using Proposition 8.  □

It turns out that using the result above one can construct infinitely many symplectic Gaussian frames using the theory of metaplectic operators:

Let be the standard Gaussian. The Gabor system is a frame if and only if is a frame (with same bounds) for every . Writing S in block-matrix form the window is the Gaussian where are symmetric matrices, and .

That is a frame if and only if is a frame follows from Proposition 10. To calculate it suffices to apply formulas (72) and (73) in Appendix A.  □

An example: fractional Fourier transforms

Let us choose and consider the rotations (we assume ). The matrices form a one-parameter subgroup of the symplectic group . To corresponds a unique one-parameter subgroup of the metaplectic group such that . It follows from formula (67) in Appendix A that is explicitly given for (k integer) by where is an integer (the “Maslov index”) and

The metaplectic operators are the “fractional Fourier transforms” familiar from time–frequency analysis (see e.g. Almeida [1], Namias [48]). The argumentation above clearly shows that the study of these fractional Fourier transforms belong to the area of symplectic and metaplectic analysis and geometry.

Applying Proposition 10 we recover without any calculation the results of Kaiser [39, Theorem 1 and Corollary 2] about rotations of Gabor frames; in our notation:

Let be a frame; then is a frame for every .

Notice that fractional Fourier transforms (and their higher-dimensional generalizations) are closely related to the theory of the quantum mechanical harmonic oscillator: the metaplectic operators are solutions of the operator Schrödinger equation

Hamiltonian deformations of Gabor frames

The symplectic covariance property of Gabor frames studied above can be interpreted as a first result on Hamiltonian deformations of frames because, as we will see, every symplectic matrix is the value of the flow (at some time t) of a Hamiltonian function which is a homogeneous quadratic polynomial (with time-depending coefficients) in the variables , . We will in fact extend this result to deformations by affine flows corresponding to the case where the Hamiltonian is an arbitrary quadratic function of these coordinates.

The case of linear isotopies

The first example in Section 3.4 (the fractional Fourier transform) can be interpreted as a statement about continuous deformations of Gabor frames. For instance, assume that , X in the Lie algebra of the symplectic group (it is the algebra of all matrices X such that ; when this condition reduces to ; see e.g. [18,21]). The family can be identified with the flow determined by the Hamilton equations where is a quadratic polynomial in the variables , (cf. formula (15)). That flow satisfies the matrix differential equation We now make the following fundamental observation: in view of the unique lifting property of covering spaces (see Appendix A), to the path of symplectic matrices , , corresponds a unique path , , of metaplectic operators such that and , and it can be shown that this path satisfies the operator Schrödinger equation where is the Weyl quantization of the function H (for a detailed discussion of the correspondence between symplectic and metaplectic paths see de Gosson [21,23], Leray [43]; it is also hinted at in Folland [18]). Collecting these facts, one sees that is obtained from the initial Gabor frame by a smooth deformation More generally, let S be an arbitrary element of the symplectic group . Such an element can in general no longer be written as an exponential , , so we cannot define an isotopy joining to S by the formula . However, in view of Proposition 5, such an isotopy exists (but it does not satisfy the group property as in the case ). Exactly as above, to this isotopy corresponds a path of metaplectic operators such that and , and this path again satisfies a Schrödinger equation (28) where the explicit form of the Hamiltonian function is given by formula (13) in Proposition 5. Thus, it makes sense to consider smooth deformations (29) for arbitrary symplectic isotopies. This situation will be generalized to the nonlinear case later in this paper.

Heisenberg–Weyl operators and affine isotopies

A particular simple example of transformation is that of the translations in . On the operator level they correspond to the Heisenberg–Weyl operators . This correspondence is very easy to understand in terms of “quantization”: for fixed consider the Hamiltonian function The associated Hamilton equations are just , whose solutions are and , that is . Let now be the “quantization” of H, and consider the Schrödinger equation Its solution is given by (the second equality can be verified by a direct calculation, or using the Campbell–Hausdorff formula [18,21,23,44]).

Translations act in a particularly simple way on Gabor frames; writing we have:

Let . A Gabor system is a ħ-frame if and only if is a ħ-frame; the frame bounds are in this case the same for all values of , .

We will need the following well-known [18,21,23,44] properties of the Heisenberg–Weyl operators: Assume first and let us prove that is a ħ-frame if and only is. We have, using formula (30) and the unitarity of , it follows that the inequality is equivalent to hence our claim in the case . We next assume that ; we have, using this time formula (31), and one concludes as in the case . The case of arbitrary immediately follows.  □

Identifying the group of translations with the inhomogeneous (or affine) symplectic group is the semi-direct product (see [6,18,21]); the group law is given by

Using the conjugation relation (cf. (21)) one checks that is isomorphic to the group of all affine transformations of of the type (or ) where .

The group appears in a natural way when one considers Hamiltonians of the type where is symmetric and is a vector. In fact, the phase flow determined by the Hamilton equations for (33) consists of elements of . Assume for instance that the coefficients M and m are time-independent; the solution of Hamilton's equations is provided that . When the solution (34) is still formally valid and depends on the nilpotency degree of . Since we have ; setting the flow is thus given by

The metaplectic group is a unitary representation of the double cover of (see Appendix A). There is an analogue when is replaced with : it is the Weyl-metaplectic group , which consists of all products ; notice that formula (21), which we can rewrite is the operator version of formula (32).

Weak Hamiltonian deformations

We now turn to the central topic of this paper, which is to propose and study “reasonable” definitions of the notion of deformation of a Gabor frame by a Hamiltonian isotopy. We begin by briefly recalling the notion of Weyl quantization.

Let H be a Hamiltonian which we assume to be well-behaved at infinity; more specifically we impose, for fixed t, the condition We will call such a Hamiltonian function admissible. We denote by the pseudo-differential operator on associated to H by the Weyl rule. Formally, for , more rigorously (that is avoiding convergence problems in the integral above) where is the symplectic Fourier transform of H and is the Heisenberg–Weyl operator defined by formula (18). An essential observation is that the operator is (formally) self-adjoint (because a Hamiltonian is a real function). We refer to the standard literature on pseudo-differential calculus for details (see for instance [18,21,23,49,54,58]); a nice review accessible to non-specialists is given by Littlejohn in [44]. Our choice of this particular type of quantization – among all others available on the market – is not arbitrary; it is due to the fact that the Weyl rule is the only [58] quantization procedure which is symplectically covariant in the following sense: let be an arbitrary element of the metaplectic group (see Appendix A); if has projection then This property, which easily follows from the intertwining relation (21) for Heisenberg–Weyl operators, is essential in our context, since our aim is precisely to show how symplectic covariance properties provide a powerful tool for the study of transformations of Gabor frames.

It is usually to consider the Schrödinger equation associated with an admissible Hamiltonian function H: it is the linear partial differential equation where the initial function is usually chosen in the Schwartz space . Every solution ψ can be written and is called the evolution operator (or “propagator”) for the Schrödinger equation (37). An essential property is that the are unitary operators on . To see this, set where ψ is in the domain of (for instance ); differentiating with respect to t and using the product rule we have since is (formally) self-adjoint; it follows that hence is unitary as claimed.

We now turn to the description of the problem. Let and be a Hamiltonian isotopy joining the identity to f; in view of Proposition 3 there exists a Hamiltonian function H such that for . We want to study the deformation of a ħ-Gabor frame by ; that is we want to define a deformation here is an (unknown) operator associated in some (yet unknown) way with . We will proceed by analogy with the case where we defined the deformation by where , . This suggests that we require that:

The operators should be unitary in ;

The deformation (38) should reduce to (39) when the isotopy lies in .

The following property of the metaplectic representation gives us a clue. Let be a Hamiltonian isotopy in . We have seen in Proposition 5 that there exists a Hamiltonian function with associated phase flow precisely . Consider now the Schrödinger equation where is the Weyl quantization of H (recall that is a formally self-adjoint operator). It is well-known [21,23,18] that where is the unique path in passing through the identity and covering . This suggests that we should choose in the following way: let H be the Hamiltonian function determined by the Hamiltonian isotopy : . Then quantize H into an operator using the Weyl correspondence, and let be the solution of Schrödinger's equation

It can actually be shown (de Gosson and Hiley [24]) that this procedure can, under certain conditions, be reversed: given a one-parameter family of unitary operators on one can find a Hamiltonian isotopy such that (40) holds.

While definition (39) of a Hamiltonian deformation of a Gabor system is “reasonable”, its practical implementation is difficult because it requires the solution of a Schrödinger equation. We will therefore try to find a weaker, more tractable definition of the correspondence (38), which is easier to implement numerically.

The “thawed Gaussian approximation”

The “weak Hamiltonian deformation” scheme method we are going to use is the so-called Gaussian wavepacket method which comes from semiclassical mechanics and is widely used in chemistry; it is due to Heller and his collaborators (Heller [34,35], Davis and Heller [10]) and Littlejohn [44]. (For a rather up to date discussion of various Gaussian wavepacket methods see Heller [36].)

For fixed we set and define the new Hamilton function it is the Taylor series of H at with terms of order 0 and >2 suppressed. The corresponding Hamilton equations are We make the following obvious but essential observation: in view of the uniqueness theorem for the solutions of Hamilton's equations, the solution of (42) with initial value is the same as that of the Hamiltonian system with . Denoting by the Hamiltonian flow determined by we thus have . More generally, the flows and are related by a simple formula involving the “linearized flow” :

The solutions of Hamilton's equations (42) and (43) are related by the formula where , and is the phase flow determined by the quadratic time-dependent Hamiltonian Equivalently, where is the translation operator.

Let us set . We have, taking (42) into account, that is, since , It follows that and hence which is precisely (44).  □

The function satisfies the “variational equation” (this relation can be used to show that is symplectic [21,23]; it thus gives a simple proof of the fact that Hamiltonian phase flows consist of symplectomorphisms [21,23]).

The thawed Gaussian approximation (TGA) (also sometimes called the nearby orbit method) consists in making the following Ansatz:

The approximate solution to Schrödinger's equation where is the standard coherent state centered at is given by the formula where the phase is the symmetrized action calculated along the Hamiltonian trajectory leading from at time to at time t. One shows that under suitable conditions on the Hamiltonian H the approximate solution satisfies, for , an estimate of the type where is a positive constant depending only on the initial point and the time interval (Hagedorn [31,32], Nazaikiinskii et al. [49]).

Formula (49) shows that the solution of Schrödinger's equation with initial datum is approximately the Gaussian obtained by propagating along the Hamiltonian trajectory starting from while deforming it using the metaplectic lift of the linearized flow around this point.

Application to Gabor frames

Let us state and prove the main result of this paper.

In what follows we consider a Gaussian Gabor system ; applying the nearby orbit method to yields the approximation where we have set . Let us consider the Gabor system where .

The Gabor system is a Gabor ħ-frame if and only if is a Gabor ħ-frame; when this is the case both frames have the same bounds.

Writing we set out to show that the inequality (for all ) holds for every t if and only if it holds for (for all ). In view of definition (52) we have the commutation formula (30) yields and hence Since is unitary the inequality (53) is thus equivalent to In view of formula (44) we have, since because , hence the inequality (54) can be written In view of the product formula (31) for Heisenberg–Weyl operators we have so that (55) becomes the unitarity of implies that (56) is equivalent to Using the symplectic covariance formula (21) we have so that the inequality (57) can be written since is unitary, this is equivalent to The proposition follows.  □

The fact that we assumed that the window is the centered coherent state is not essential. For instance, Proposition 15 shows that the result remains valid if we replace with a coherent state having arbitrary center, for instance . More generally:

Let be a Gabor system where the window ϕ is the Gaussian where , . Then is a Gabor ħ-frame if and only if it is the case for .

It follows from the properties of the action of the metaplectic group on Gaussians (see Appendix A) that there exists such that . Let be the projection on of ; the Gabor system is a ħ-frame if and only if is a ħ-frame in view of Proposition 10. The result now follows from Proposition 20.  □

We finally remark that the fact that we have been using Gaussian windows (coherent states and their generalizations) is a matter of pure convenience. In fact, the definition of weak Hamiltonian deformations of a Gabor frame as given above is valid for arbitrary windows (or ). It suffices for this to replace the defining formula (52) with One can prove that if ϕ is sufficiently concentrated around the origin, then is again a good semiclassical approximation to the true solution of Schrödinger's equation. This question is related to the uncertainty principle, see [19,20,25]. However, when one wants to impose to the initial window to belong to more sophisticated functional spaces than or one might be confronted to technical difficulties if one wants to prove that the deformed window (59) belongs to the same space. However, there is a very important case where this difficulty does not appear, namely if we assume that the initial window ϕ belongs to Feichtinger's algebra (reviewed in Appendix B). Since our definition of weak transformations of Gabor frames only makes use of phase space translations and of metaplectic operators it follows that if and only if (see de Gosson [22]). This is due to the fact that the Feichtinger algebra is the smallest Banach algebra invariant under these operations, and is thus preserved under the semiclassical propagation scheme used here. It is unknown whether this property is conserved under passage to the general definition (39), that is where is the solution of the Schrödinger equation associated with the Hamiltonian operator corresponding to the Hamiltonian isotopy : one does not know at the time of writing if the solution to Schrödinger equations with initial data in also is in for arbitrary Hamiltonians. The same difficulty appears when one considers other more general functions spaces (e.g. modulation spaces).

Discussion and additional remarks

We shortly discuss some future issues that will be studied in forthcoming papers; the list is of course far from being exhaustive, since these “first steps” of a general theory of Hamiltonian deformations of Gabor frames will hopefully become a marathon!

Numerical implementation We briefly indicate here how the weak Hamiltonian deformation method could be practically and numerically implemented; we will come back to this important practical issue in a forthcoming paper where experimental results will be given. The main observation is that a weak deformation of a Gabor frame consists of two objects: a Hamiltonian flow and a family of operators approximating the quantized version of that flow (semiclassical propagator). First, the action of the Hamiltonian isotopy on the Gabor lattice can be computed (to an arbitrary degree of precision) using the symplectic algorithms reviewed in Section 2.3; a host of numerical implementations can be found in the literature, see for instance the already mentioned works [8,40,60], and the references therein. The corresponding deformation of the window should not be more difficult to compute numerically, since the essence of the method consists in replacing the “true” quantum propagation with a linearized operator, expressed in terms of translations and metaplectic operators as in formula (49), which says that (up to an unessential phase factor) the propagated coherent state is an expression of the typeNumerically, this term can be calculated using the symplectic algorithm to evaluate and then calculate by numerical (or explicit) methods using formula (67) for generating metaplectic operators given in Appendix A. Of course, precise error bounds have to be proven, but this should not be particularly difficult, these approximation theories being well-established parts of the toolbox of numerical analysts.

Higher order weak deformations Since our definition of weak deformations was motivated by semiclassical considerations one could perhaps consider refinements of this method using the asymptotic expansions of Hagedorn [31,32] and his followers; this could then lead to “higher order” weak deformations, depending on the number of terms that are retained. The scheme we have been exposing is a standard and robust method; its advantage is its simplicity. In future work we will discuss other interesting possibilities. For instance, in [34,35] Heller proposes a particular simple semiclassical approach which he calls the “frozen Gaussian approximation” (FGA). It is obtained by surrounding the Hamiltonian trajectories by a fixed (“frozen”) Gaussian function (for instance ) and neglecting its “squeezing” by metaplectic operators used in the TGA. Although this method seems to be rather crude, it yields astoundingly accurate numerical results applied to superpositions of infinitely many Gaussians; thus it inherently has a clear relationship with frame expansions. A more sophisticated procedure would be the use of the Kluk–Herman (HK) approximate propagator, which has been widely discussed in the chemical literature (Herman [37] shows that the evolution associated with the HK propagator is unitary, and Swart and Rousse [56] put the method on a firm mathematical footing by relating it with the theory of Fourier integral operators; in [30] Grossmann and Herman discuss questions of terminology relating to the FGA and the HK propagator). Also see the review papers by Heller [36] and Kay [41] where the respective merits of various semiclassical approximation methods are discussed.

“Exact” deformations Still, there remains the question of the general definition (60) where the exact quantum propagator is used. It would indeed be more intellectually (and also probably practically!) satisfying to study this definition in detail. As we said, we preferred in this first approach to consider a weaker version because it is relatively easy to implement numerically using symplectic integrators. The general case (60) is challenging, but probably not out of reach. From a theoretical point of view, it amounts to construct an extension of the metaplectic representation in the non-linear case; that such a representation indeed exists has been shown in our paper with Hiley [24] (a caveat: one sometimes finds in the physical literature a claim following which such an extension could not be constructed, a famous theorem of Groenewold and Van Hove being invoked to sustain this claim. This is merely a misunderstanding of this theorem, which only says that there is no way to extend the metaplectic representation so that the Dirac correspondence between Poisson brackets and commutators is preserved). There remains the problem of how one could prove that the deformation scheme (60) preserves the frame property; a possible approach could consist in using a time-slicing (as one does for symplectic integrators); this would possibly also lead to some insight on whether the Feichtinger algebra is preserved by general quantum evolution. This is an open question which is being actively investigated.