Stability of Gabor frames under small time Hamiltonian evolutions
aa r X i v : . [ m a t h - ph ] A p r STABILITY OF GABOR FRAMES UNDER SMALL TIMEHAMILTONIAN EVOLUTIONS
MAURICE A. DE GOSSON, KARLHEINZ GR ¨OCHENIG, AND JOS´E LUIS ROMERO
Abstract.
We consider Hamiltonian deformations of Gabor systems, wherethe window evolves according to the action of a Schr¨odinger propagator andthe phase-space nodes evolve according to the corresponding Hamiltonian flow.We prove the stability of the frame property for small times and Hamiltoniansconsisting of a quadratic polynomial plus a potential in the Sj¨ostrand classwith bounded second order derivatives. This answers a question raised in [deGosson, M. Symplectic and Hamiltonian Deformations of Gabor Frames. Appl.Comput. Harmon. Anal. Vol. 38 No.2, (2015) p.196–221.] Introduction
Let H ( x, p ) be a Hamiltonian on R d and H := H w its Weyl quantization. Thesolution to the Schr¨odinger equation i∂ t u ( t, · ) = Hu ( t, · ) , t ∈ R ,u (0 , · ) = f, is given by the propagation formula u ( t, · ) = e − itH f . The model case is theone of a real quadratic (homogeneous) Hamiltonian: H ( x, p ) = h M ( x, p ) , ( x, p ) i ,with M ∈ R d × d symmetric. In this case, the evolution operator e − itH and the(symmetric) time-frequency shift operators ρ ( z ) f := e − πixξ e πiξ f ( · − x ) , z = ( x, ξ ) ∈ R d , (1)satisfy the symplectic covariance relation e − itH ρ ( z ) = ρ ( e tJM z ) e − itH , (2)where J = (cid:0) I − I (cid:1) is the standard symplectic form (see for example [11, Chapter15]). Thanks to (2), the action of the evolution operator on a state f , can be Mathematics Subject Classification.
Key words and phrases.
Time-frequency analysis, Gabor frame, Hamiltonian flow,Schr¨odinger equation, Weyl quantization, Phase space, Wigner distribution.M. d. G. was supported by the grant P 23902 from the Austrian Science Fund (FWF). K. G.was supported in part by the project P26273-N25 of the Austrian Science Fund (FWF). J. L. R.gratefully acknowledges support from a Marie Curie fellowship, within the 7th. European Com-munity Framework program, under grant PIIF-GA-2012-327063. understood by considering an expansion into coherent states: f = X λ ∈ Λ c λ ρ ( λ ) g, (3)where g is a smooth, fast decaying function, Λ ⊆ R d is a set of phase-space nodesand c λ ∈ C . Such an expansion is a discrete version of the continuous coherentstate representation [2], and the canonical choice for g is a Gaussian function. Theevolution generated by the quadratic Hamiltonian H is then given by e − itH f = X λ ∈ Λ c λ ρ ( e tJM λ ) e − itH g, (4)and therefore the description of the evolution of an arbitrary state f is reduced tothe one of g . If H ( x, p ) = x + p is the harmonic oscillator and g is chosen to be anadequate Gaussian function, then e − itH g = g , and (4) amounts to a rearrangementof the time-frequency content of f . The case of higher order Hermite functions isalso important since these correspond to higher energy Landau levels (see [1]).The collection of coherent states G ( g, Λ) := (cid:8) ρ ( λ ) g : λ ∈ Λ (cid:9) is called a Gabor system, and it is a Gabor frame if every f ∈ L ( R d ) admitsan expansion as in (3) with k c k ≍ k f k . In this case, several properties of f can be read from the coefficients c . The theory of Gabor frames - also calledWeyl–Heisenberg frames - plays an increasingly important role in physics; see forinstance [9, 13, 11] and the references therein.Recently one of us started the investigation of the relation between the theoryof Gabor frames and Hamiltonian and quantum mechanics [12] and introduced thenotion of a Hamiltonian deformation of a Gabor system . For a (time-independent)Hamiltonian H ( x, p ) we let Φ t ( x, p ) be the flow given by the Hamilton equations (cid:26) ˙ x = H p ( x, p ) , ˙ p = −H x ( x, p ) , and let H := H w be the Weyl quantization of H . Given a Gabor system G ( g, Λ),we consider the time-evolved systems G t ( g, Λ) := G ( e − itH g, Φ t Λ) , (5)and investigate the stability of the frame property under the evolution G ( g, Λ) t ( g, Λ).When H is a quadratic form H ( x, p ) = h M ( x, p ) , ( x, p ) i , its flow is given bythe linear map Φ t ( x, p ) = e tJM ( x, p ) and (2) expresses the fact that the evolutionoperator e − itH is the metaplectic operator associated with the linear map e tJM .As a consequence, G t ( g, Λ) is the image of G ( g, Λ) under the unitary map e − itH andhence it enjoys the same spanning properties (in particular, the frame property ispreserved). This observation is called the symplectic covariance of Gabor frames[12]. TABILITY OF GABOR FRAMES UNDER SMALL TIME HAMILTONIAN EVOLUTIONS 3
For more general Hamiltonians H , no strict covariance property holds, and theanalysis of the deformation t
7→ G t ( g, Λ) is difficult. In [12], one of us analyzeda linearized version of this problem and established some stability estimates (seealso [6] for a higher-order approximation to the deformation problem). Based onthese results, [12] conjectured that the evolution G ( g, Λ)
7→ G t ( g, Λ) preserves theframe property for more general Hamiltonians. In particular, one would expectperturbations of quadratic Hamiltonians to exhibit a certain approximate sym-plectic covariance, in the form of stability of the frame property of G t ( g, Λ) for acertain range of time.In this article we solve the deformation problem for small times. More pre-cisely, we consider a perturbation of a quadratic Hamiltonian by an element of the
Sj¨ostrand class M ∞ , ( R d ) with bounded second order derivatives. We also con-sider a Gabor frame with window in the Feichtinger algebra M ( R d ) of functionswith integrable Wigner distribution. (See Section 2 and 2.1 for precise definitions).The following is our main result. Theorem 1.1.
Let a be a real-valued, quadratic, homogeneous polynomial on R d and let σ ∈ M ∞ , ( R d ) ∩ C ( R d ) have bounded second order derivatives. Considerthe Hamiltonian H ( t, x, p ) := a ( x, p ) + σ ( x, p ) . Let H := H w ( x, D ) be the Weylquantization of H and let (Φ t ) t ∈ R be the flow of H .Let g ∈ M ( R d ) and Λ ⊆ R d , such that G ( g, Λ) is a Gabor frame. Then thereexists t > such that for all t ∈ [ − t , t ] , G ( e − itH g, Φ t (Λ)) is a Gabor frame. To see what is at stake, we consider once more the symplectic covariance prop-erty (2). It links the classical Hamiltonian flow e tJM on phase space to the quan-tum mechanical evolution. If H is not quadratic, then the flow Φ t is no longerlinear, and, in general, there is no explicit and exact formula for the quantummechanical evolution. We therefore have to understand the classical evolution ofthe set Λ under Φ t separately from the quantum mechanical evolution of the state(window) g under e − itH .The stability of the frame property of G ( g, Φ t (Λ)) is part of the deformationtheory of Gabor frames. While there is a significant literature on the stabilityof Gabor frames under linear distortions of the time-frequency nodes Λ (coveringperturbation of lattice parameters [4, 18] on the one hand, and general pointsets [3]), only recently a fully non-linear deformation theory of Gabor systemswas developed in [23]. It turns out that the concept of Lipschitz deformation isprecisely the right tool to treat non-linear Hamiltonian flows, and we will use themain result of [23] in a decisive manner.The second ingredient in Theorem 1.1 is the assumption g ∈ M ( R d ). This is anessential assumption for Gabor frames to be useful in phase space analysis. In par-ticular, most stability results for Gabor frames under perturbations of the windowrequire that g ∈ M ( R d ). Outside M one encounters quickly pathologies [10]. Inregard to our problem it is therefore important to understand whether M ( R d ) isinvariant under the evolution of the Schr¨odinger equation. This is indeed the case MAURICE A. DE GOSSON, KARLHEINZ GR ¨OCHENIG, AND JOS´E LUIS ROMERO for certain classes of Hamiltonians [5, 8], and will be the second important toolused to prove Theorem 1.1.The rest of the article is organized as follows. In Section 2 we provide somedefinitions and background results. Section 3 collects the essential tools and de-rives some auxiliary estimates. Finally, the proof of Theorem 1.1 is presented inSection 4. 2.
Background
Time-frequency analysis.
Given a function g ∈ L ( R d ), with k g k = 1,the short-time Fourier transform of a function f ∈ L ( R d ) with respect to thewindow g is defined as V g f ( x, ξ ) := (cid:10) f, e πixξ ρ ( x, ξ ) g (cid:11) , ( x, ξ ) ∈ R d , (6)where ρ ( x, ξ ) is the (symmetric) time-frequency shift defined in (1). The function g is often called window and the normalization k g k = 1 implies that k V g f k L ( R d ) = k f k L ( R d ) , f ∈ L ( R d ) . (7)The standard choice for g is the Gaussian φ ( x ) := 2 d/ e − π | x | . Analogously, theFeichtinger algebra, originally introduced in [14], is defined to be M ( R d ) := (cid:8) f ∈ L ( R d ) : k f k M := k V φ f k L ( R d ) < + ∞ (cid:9) , and is used as a standard reservoir for windows g . Equivalently, f ∈ M ( R d ), if theWigner distribution W f ( x, ξ ) = R f ( x + t/ f ( x − t/ e − πiξ · t dt of f is integrableon R d . When g ∈ M ( R d ), the map f V g f can be extended beyond L ( R d ).We define the modulation spaces as follows: fix a non-zero g ∈ S ( R d ) and let1 ≤ p, q ≤ ∞ . Then M p,q ( R d ) is the class of all distributions f ∈ S ′ ( R d ) such that k f k M p,q ( R d ) := Z R d (cid:18)Z R d | V g f ( x, ξ ) | p dx (cid:19) q/p dξ ! /q < ∞ , (8)with the usual modification when p or q is ∞ . Different choices of non-zerowindows g ∈ S ( R d ) yield the same space with equivalent norms, see [17] and [20,Chapter 11]. In addition, for g ∈ M ( R d ), the short-time Fourier transform is well-defined on all M p,q ( R d ). Originally introduced by Feichtinger in [15], modulationspaces combine smoothness and integrability conditions. In this article, we will bemainly concerned with Feichtinger’s algebra M ( R d ), as a window class for Gaborsystems, and M ∞ , ( R d ) - also known as Sj¨ostrand’s class, as a symbol class forpseudodifferential operators.2.2. Sampling the short-time Fourier transform.
A set Λ ⊆ R d is called relatively separated ifrel(Λ) := sup { ∩ ( { x } + [0 , d )) : x ∈ R d } < ∞ . (9) TABILITY OF GABOR FRAMES UNDER SMALL TIME HAMILTONIAN EVOLUTIONS 5
The assumption that g ∈ M ( R d ) implies certain sampling estimates for the short-time Fourier transform. We quote the following standard result (see for example[20, Chapter 13].) Proposition 2.1.
Let g ∈ M ( R d ) and let Λ ⊆ R d . Then X λ ∈ Λ | V g f ( λ ) | ! / ≤ C rel(Λ) k g k M k f k , f ∈ L ( R d ) , where the constant C depends only on the dimension d . Gabor frames.
Given a window g ∈ M ( R d ) and a relatively separated setΛ ⊆ R d , the collection of functions G ( g, Λ) := { ρ ( λ ) g : λ ∈ Λ } is called the Gabor system generated by g and Λ. It is a Gabor frame, if thereexist constants A, B > A k f k ≤ X λ ∈ Λ |h f, ρ ( λ ) g i| ≤ B k f k , f ∈ L ( R d ) . (10)The constants A, B are called frame bounds for G ( g, Λ). We remark that the def-inition of Gabor system given here is slightly non-standard. In signal processing,it is more common to define the time-frequency shifts by π ( z ) f ( t ) := e πiξt f ( t − x ) , z = ( x, ξ ) ∈ R d × R d , t ∈ R d . Since π ( x, ξ ) = e πixξ ρ ( x, ξ ), the choice ρ has no impact on the frame inequality in(10). Note that the sum in (10) is the same as k V g f | Λ k . The use of ρ instead of π in this article is motivated by the symplectic covariance property in (2), whichwould require additional phase factors if π was used instead of ρ .The following basic fact can be found for example in [7, Theorem 1.1]. Lemma 2.2. If G ( g, Λ) is a frame, then Λ is relatively separated. The essential tools
Schr¨odinger operators on modulation spaces.
The
Weyl transform of adistribution σ ∈ S ′ ( R d × R d ) is an operator σ w that is formally defined on functions f : R d → C as σ w ( f )( x ) := Z R d × R d σ (cid:18) x + y , ξ (cid:19) e πi ( x − y ) ξ f ( y ) dydξ, x ∈ R d . The fundamental results in the theory of pseudodifferential operators provide con-ditions on σ for the operator σ w to be well-defined and bounded on various functionspaces. In particular, Sj¨ostrand proved that if σ ∈ M ∞ , ( R d ), then σ w is boundedon L ( R d ) [25, 26]. See also [21, 22] for extensions of these results to weightedsymbol classes and modulations spaces. MAURICE A. DE GOSSON, KARLHEINZ GR ¨OCHENIG, AND JOS´E LUIS ROMERO
The following result is one of our main tools. It shows that perturbing a qua-dratic Hamiltonian with a potential in the Sj¨ostrand’s class M ∞ , ( R d ) gives riseto propagators that are strongly continuous on M ( R d ). Theorem 3.1 ([8, Theorems 1.5 and 4.1]) . Let a be a real-valued, quadratic,homogeneous polynomial on R d and let σ ∈ M ∞ , ( R d ) . Let H := a w ( x, D ) + σ w ( x, D ) . Then e itH is a strongly continuous one-parameter group of operators on M ( R d ) . In other words: (a) for all t ∈ R , e itH : M ( R d ) → M ( R d ) , (b) for each g ∈ M ( R d ) , e itH g −→ g in M ( R d ) , as t −→ . (11)3.2. Deformation of Gabor frames.
Our second essential tool is a descriptionof the stability of the frame property of a Gabor frame G ( g, Λ) under small de-formations of Λ. Our general assumption is that g ∈ M ( R d ). (Without thisassumption the frame property might be very unstable under perturbation of Λ,even for lattices [24, 10]).The classical results in signal processing describe the stability of the frameproperty under the so-called jitter perturbations: if G ( g, Λ) sup λ ∈ Λ inf λ ′ ∈ Λ ′ | λ − λ ′ | < ǫ and sup λ ′ ∈ Λ ′ inf λ ∈ Λ | λ − λ ′ | < ǫ , for sufficiently small ε , then G ( g, Λ ′ ) is alsoa frame. A much deeper property is the stability of the frame condition underlinear maps Λ A Λ, where A is a matrix that is sufficiently close to the identity(but possibly not symplectic!). Such results have been derived first for lattices[4, 18] and then for general sets [3]. In order to deal with Hamiltonian flows, wewill resort to a recent fully non-linear stability theory [23].Let Λ ⊆ R d be a set. We consider a sequence { Λ n : n ≥ } of subsets of R d produced in the following way. For each n ≥
1, let τ n : Λ → R d be a map and letΛ n := τ n (Λ) = { τ n ( λ ) : λ ∈ Λ } . We assume that τ n ( λ ) −→ λ , as n −→ ∞ , for all λ ∈ Λ. The sequence of sets { Λ n : n ≥ } together with the maps { τ n : n ≥ } iscalled a deformation of Λ. We think of each sequence of points { τ n ( λ ) : n ≥ } asa (discrete) path moving towards the endpoint λ .We will say that { Λ n : n ≥ } is a deformation of Λ, with the understandingthat a sequence of underlying maps { τ n : n ≥ } is also given.We now describe a special class of deformations. Definition 3.2.
A deformation { Λ n : n ≥ } of Λ is called Lipschitz , denoted by Λ n Lip −−→ Λ , if the following two conditions hold: (L1) Given
R > , sup λ,λ ′∈ Λ | λ − λ ′ |≤ R | ( τ n ( λ ) − τ n ( λ ′ )) − ( λ − λ ′ ) | → , as n −→ ∞ . (L2) Given
R > , there exists R ′ > and n ∈ N such that if | τ n ( λ ) − τ n ( λ ′ ) | ≤ R for some n ≥ n and some λ, λ ′ ∈ Λ , then | λ − λ ′ | ≤ R ′ . TABILITY OF GABOR FRAMES UNDER SMALL TIME HAMILTONIAN EVOLUTIONS 7
The following results shows that the frame property of a Gabor system is stableunder Lipschitz deformations.
Theorem 3.3 ([23, Thm. 7.1 and Rem. 7.3]) . Let g ∈ M ( R d ) and Λ ⊆ R d .Assume that G ( g, Λ) is a (Gabor) frame and that Λ n Lip −−→ Λ . Then there exist A, B > and n ∈ N such that G ( g, Λ n ) is a frame with uniform bounds A, B forall n ≥ n . We will also need the following technical lemma concerning Lipschitz conver-gence and relative separation.
Lemma 3.4 ([23, Lemma 6.7]) . Let Λ n Lip −−→ Λ and assume that Λ is relativelyseparated. Then lim sup n rel(Λ n ) < ∞ . The following corollary enables us to combine the stability of Gabor framesunder deformations of Λ with small perturbations of the window g on M -norm. Corollary 3.5.
Assume that g n −→ g in M ( R d ) and that Λ n Lip −−→ Λ . Then G ( g n , Λ n ) is a frame for all sufficiently large n . (Moreover, the correspondingframe bounds can be taken to be uniform in n ).Proof. By Theorem 3.3, there exist
A, B > n ∈ N such that for all n ≥ n A k f k ≤ k V g f | Λ n k ≤ B k f k , f ∈ L ( R d ) . (12)(Here A, B are the square roots of the frame bounds.) By Lemma 2.2, Λ and allΛ n with n ≫ f ∈ L ( R d ) (cid:12)(cid:12)(cid:12) k V g f | Λ n k − k V g n f | Λ n k (cid:12)(cid:12)(cid:12) ≤ k V g − g n f | Λ n k ≤ C k g − g n k M rel(Λ n ) k f k . Letting A n := A − C k g − g n k M rel(Λ n ) ,B n := B + C k g − g n k M rel(Λ n ) , we deduce from (12) and the triangle inequality that A n k f k ≤ k V g n f | Λ n k ≤ B n k f k , f ∈ L ( R d ) . (13)By Lemma 3.4 and the fact that g n −→ g in M it follows that A n −→ A and B n −→ B . Combining this with (13) we conclude that for all sufficiently large nA/ k f k ≤ k V g n f | Λ n k ≤ B/ k f k , f ∈ L ( R d ) . Hence, for n ≫ G ( g n , Λ n ) is a frame with bounds A / , B / (cid:3) MAURICE A. DE GOSSON, KARLHEINZ GR ¨OCHENIG, AND JOS´E LUIS ROMERO
Flows and Lipschitz convergence.
A function F : R × R d → R d is Lip-schitz in the second variable if there exists L > | F ( t, x ) − F ( t, y ) | ≤ L | x − y | , for all ( t, x ) ∈ R × R d . Under this assumption, we let (Φ t ) t ∈ R denote the flow of F (associated with time0). This means that for each x ∈ R d , R ∋ t Φ t ( x ) ∈ R d is a C function andthat(a) Φ ( x ) = x ,(b) ddt Φ t ( x ) = F ( t, Φ t ( x )).The theory of ODEs implies that the flow exists and it is uniquely determined byproperties ( a ) and ( b ) above. Moreover, the flow satisfies the following distortionestimate: given T >
0, there exist constants c t , C T > c T | x − y | ≤ | Φ t ( x ) − Φ t ( y ) | ≤ C T | x − y | , x, y ∈ R , t ∈ [ − T, T ] . (14)The previous estimate is normally proved using the following useful lemma. Lemma 3.6 (Gronwall) . Let I ⊆ R be an interval and a ∈ I . Let g : I → [0 , + ∞ ) be a continuous function that satisfies g ( t ) ≤ A + B (cid:12)(cid:12)(cid:12)(cid:12)Z ta g ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) , t ∈ I, for some constants A, B ∈ R . Then g ( t ) ≤ Ae B | t − a | , t ∈ I. (The reason for the absolute value outside the integral is that t − a can be negative.) We now show that the flows of ODEs provide examples of Lipschitz deforma-tions.
Theorem 3.7.
Let F : R × R d → R d be Lipschitz in the second variable and let (Φ t ) t ∈ R be the corresponding flow. Let Λ ⊆ R d be a relatively separated set. Then Φ t (Λ) Lip −−→ Λ , as t −→ .(More precisely, for each sequence t n −→ , Φ t n (Λ) Lip −−→ Λ .)Proof. Let
L > F (in the second variable). We firstcheck condition ( L
1) from Definition 3.2. From the definition of the flow it followsthat Φ t ( x ) = x + Z t F ( s, Φ s ( x )) ds, t ∈ R . Therefore,Φ t ( λ ) − Φ t ( λ ′ ) − ( λ − λ ′ ) = Z t ( F ( s, Φ s ( λ )) − F ( s, Φ s ( λ ′ ))) ds. (15) TABILITY OF GABOR FRAMES UNDER SMALL TIME HAMILTONIAN EVOLUTIONS 9
As a consequence, | Φ t ( λ ) − Φ t ( λ ′ ) − ( λ − λ ′ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t | F ( s, Φ s ( λ )) − F ( s, Φ s ( λ ′ )) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)Z t L | Φ s ( λ ) − Φ s ( λ ′ ) | ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ L (cid:12)(cid:12)(cid:12)(cid:12)Z t | Φ t ( λ ) − Φ t ( λ ′ ) − ( λ − λ ′ ) | ds (cid:12)(cid:12)(cid:12)(cid:12) + L | t | | λ − λ ′ | . Applying Gronwall’s Lemma 3.6 to g ( t ) := | Φ t ( λ ) − Φ t ( λ ′ ) − ( λ − λ ′ ) | we de-duce that | Φ t ( λ ) − Φ t ( λ ′ ) − ( λ − λ ′ ) | ≤ L | t | | λ − λ ′ | e L | t | . Condition ( L
1) follows from here.To check condition ( L t ∈ [ − ,
1] and (14) to obtain aconstant C such that C − | x − y | ≤ | Φ t ( x ) − Φ t ( y ) | ≤ C | x − y | , t ∈ ( − , . Hence, if for some instant t we know that | Φ t ( λ ) − Φ t ( λ ′ ) | ≤ R , then we candeduce that | λ − λ ′ | ≤ R ′ := CR .This completes the proof. (cid:3) Hamiltonian deformations: d´enouement
We finally combine all tools from the previous section and prove the main result.
Proof of Theorem 1.1.
Let us define F : R × R d → R d by F ( t, x, p ) := ( ∂ p H ( x, p ) , − ∂ x H ( x, p )) . Then F is a C function with bounded derivatives and, consequently, F is Lipschitzin the second set of variables ( x, p ). Let t n −→ n := Φ t n (Λ).Theorem 3.7 implies that Λ n Lip −−→ Λ, while Theorem 3.1 implies that e − it n H g −→ g in M . Hence, Corollary 3.5 yields the desired conclusion. (cid:3) Remark 4.1.
The proof shows that, under the conditions of Theorem 1.1, theGabor systems G ( e − ith g, Φ t (Λ)) admit uniform frame bounds for t ∈ [ − t , t ] . Remark 4.2.
We do not know whether the conclusion of Theorem 1.1 remainsvalid for arbitrary times. Moreover, we do not know of any example of a Hamil-tonian deformation that does not preserve the frame property.
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