Gaussian Distributions and Phase Space Weyl--Heisenberg Frames
aa r X i v : . [ m a t h - ph ] J un GAUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERGFRAMES
MARKUS FAULHUBER, MAURICE A. DE GOSSON, AND DAVID ROTTENSTEINER
Abstract.
Gaussian states are at the heart of quantum mechanics and play an essentialrole in quantum information processing. In this paper we provide approximation formulasfor the expansion of a general Gaussian symbol in terms of elementary Gaussian functions.For this purpose we introduce the notion of a “phase space frame” associated with a Weyl-Heisenberg frame. Our results give explicit formulas for approximating general Gaussiansymbols in phase space by phase space shifted standard Gaussians as well as explicit errorestimates and the asymptotic behavior of the approximation. Introduction and Main Result
Gaussian distribution functions play a central role in many areas of mathematics rangingfrom statistics to signal theory. This is not only because of their relative simplicity, but alsobecause of their usefullness in many applied areas. For instance, in signal processing Gaussiansmoothing, the blurring of an image by a Gaussian function, is used to reduce noise (theoryof low-pass filters). In quantum mechanics the so-called Gaussian states are fundamental inquantum communication protocols [1], one of the reasons being that such states and theirevolution are more accessible in the laboratory than their non-Gaussian counterparts [33].The present paper is motivated by finding convenient expansions of distributions of the type(1) ρ ( z ) = √ det Σ − e − π Σ − z where Σ is a real symmetric positive matrix of order 2 n × n and the vector z = ( x, p ) is anelement of R n ≡ R n × R n . Since the function ρ is even and L -normalized, i.e. Z R n ρ ( z ) d n z = 1 , the matrix Σ is the covariance matrix of ρ viewed as a centered normal probability distribution:Σ = Z R n zρ ( z ) z T d n z for z = (cid:18) xp (cid:19) . In the quantum case, ρ has the following interpretation: assume that the eigenvalues of theHermitian matrix Σ + i ~ J are all non-negative ( J is the standard symplectic matrix); then (M.F., M.G., D.R.) NuHAG, Faculty of Mathematics, University of ViennaOskar-Morgenstern-Platz 1, 1090 Vienna, Austria (M.F.)
Analysis Group, Department of Mathematical Sciences, NTNU TrondheimSentralbygg 2, Gløshaugen, Trondheim, Norway
E-mail addresses : [email protected], [email protected],[email protected] .2010 Mathematics Subject Classification. primary: 42C15, 81S30, secondary: 35S05.
Key words and phrases.
Density Matrices, Gaussians, Phase Space Frame, Weyl–Heisenberg Frame, WignerDistribution. the Weyl operator with symbol (2 π ~ ) n ρ is the quantum mechanical density operator [16] b ρ = X j λ j b ρ j , where ( b ρ j ) j is a sequence of mutually orthogonal projectors on normalized vectors ψ j ∈ L ( R n )and ( λ j ) a sequence of positive numbers summing up to one; it follows that we have(2) ρ = X j λ j W ψ j , where W ψ j is the usual Wigner transform of ψ j defined by W ψ j ( x, p ) = (cid:0) π ~ (cid:1) n Z R n e − i ~ py ψ j ( x + y ) ψ j ( x − y ) d n y. Now, formula (2) raises the following non-trivial question: given a Gaussian distribution (1),is it possible to write it as a linear combination of elementary Gaussians? In the quantumcase this amounts to asking whether the functions ψ j could themselves be chosen as simplerGaussians. In general, this problem is still open, however for the special cases of 2 × × ℓ -norm bythe frame inequality (5) given further down. We reformulate the notion of Weyl–Heisenbergframes as in [12] using the Weyl–Wigner–Moyal formalism. This has the advantage of makingthe underlying symplectic covariance properties, which play an essential role in the studyof Gaussians, more obvious. We will give explicit formulas; these turn out to be rathercomplicated, but are certainly of use in applications, both theoretical and numerical. Ourmain results are as follows. Theorem 1.
For x ∈ R n we denote the n -dimensional standard Gaussian by ϕ ~ ( x ) = ( π ~ ) n/ e − ~ x Let
Λ = δ − QS Z n ⊂ R n be a lattice with δ > , Q ∈ SO (2 n, R ) ∩ Sp ( n ) and S = (cid:18) L L − T P L − T (cid:19) ∈ Sp ( n ) , for which L is invertible and P = P T , whose associated Weyl-Heisenberg system G ( ϕ ~ , Λ) isa frame for L ( R n ) . Consider the proper subspace H ϕ ~ ⊂ L (cid:0) R n (cid:1) given as the image of thelinear operator U ϕ ~ : L ( R n ) → L (cid:0) R n (cid:1) ,ψ Ψ = (2 π ~ ) n/ W ( ψ, ϕ ~ ) , where W ( ψ, ϕ ~ ) is the cross-Wigner transform. Furthermore, we define e T ( z λ )Φ ~ = (2 π ~ ) n/ W ( b T ( z λ ) ϕ ~ , ϕ ~ ) , AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 3 for the usual Heisenberg operator b T . Then, the system e G (Φ ~ , Λ) = { e T ( z λ )Φ ~ : z λ ∈ Λ } is a (so-called phase space) frame for H ϕ ~ . For the associated (phase space) frame operator e A G we obtain the stable approximation (3) || Id − δ − n e A G || op = O (cid:16) e − δ ~ ( || L || R n × n + || L || − R n × n ) (cid:17) , as the parameter δ → ∞ . Theorem 2.
Under the assumptions of Theorem 1, for Ψ ∈ H ϕ ~ the (phase space) frameexpansion approximates the element Ψ in the sense of Ψ ≈ δ − n e A G Ψ = δ − n X z λ ∈ Λ ((Ψ | e T ( z λ )Φ ~ )) e T ( z λ )Φ ~ , where the accuracy of the approximation is determined by (3) .In particular, for a generalized n -dimensional Gaussian defined by ϕ ~ M ( x ) = ( π ~ ) n/ det( Re ( M )) / e − ~ Mx , with M = M ∗ and Re ( M ) > , we have Φ ~ F ( z ) = U ϕ ~ ϕ ~ M ( z ) = det( Re ( M )) / det( ( M + I )) − / e − ~ F z ∈ H ϕ ~ , with F = (cid:18) M + I ) − M − i ( M − I )( M + I ) − − i ( M + I ) − ( M − I ) − M + I ) − (cid:19) . We obtain the numerically stable approximation Φ ~ F ≈ δ − n X z λ ∈ Λ c z λ e T ( z λ )Φ ~ with convergence rate (3) from Theorem 1 and coefficients c z λ = (cid:0) π ~ (cid:1) n/ det( F + I ) − / e − ~ z λ e − ~ ( F + I ) − (cid:0) ( J − iI ) z λ (cid:1) . The paper is structured as follows: • In Section 2 we first review the main results we will need from the theory of Weyl–Heisenberg frames using the notation and approach in [12]; we thereafter propose anextension of this notion to phase space, where we make use of the notion of “Boppquantization”, which one of us has introduced and studied in [13, 14, 16] (also see deGosson and Luef [19]). The resulting frames, called phase space frames, are relatedto the usual Weyl–Heisenberg frames by a family of partial linear isometries U φ : L ( R n ) −→ L ( R n ) defined in terms of the cross-Wigner transform. • In Section 3 we study n -dimensional Gaussian mixed states. We begin by recallingresults about the cross-Wigner and cross-ambiguity transforms of pairs of Gaussians.These formulas (which have their own intrinsic interest) are necessary for computingthe Weyl–Heisenberg coefficients in a Gaussian frame expansion. We successivelyderive the results stated in Theorems 1 and 2. M. FAULHUBER, M.A. DE GOSSON, AND D. ROTTENSTEINER • In Section 4 we restrict our study to Gaussian states in the case n = 1. The reasonfor treating this case separately is that a full characterization of Gaussian Weyl–Heisenberg frames is known and Sp (1) = SL (2 , R ), which means that all lattices aresymplectic. Also, the blocks appearing in the symplectic matrix S are scalars in thiscase. Hence, some of the formulas simplify substantially.Our notation and results are meant for mathematical physicists as well as for people workingin time-frequency analysis. For the latter group, the dependence on ~ might seem irritatingat the beginning, but by setting ~ = π one obtains the classical setting for time-frequencyanalysis.1.1. Notation and Terminology.
The generic point in phase space R n ≡ R n × R n isdenoted by z = ( x, p ), where we have set x = ( x , . . . , x n ), p = ( p , . . . , p n ). The scalarproduct of two vectors p and x is denoted by px . When matrix calculations are performed, z, x, p are viewed as column vectors. We write M x for the quadratic form x T M x , and
M xp for p T M x . For an invertible matrix M we write M − T for its transposed inverse. Moreover,we equip R n with the standard symplectic structure σ ( z, z ′ ) = px ′ − p ′ x, in matrix notation σ ( z, z ′ ) = ( z ′ ) T J z , where J = (cid:18) I − I (cid:19) is the standard symplectic ma-trix. The symplectic group of R n is denoted by Sp( n ); it consists of all linear automorphismsof R n such that σ ( Sz, Sz ′ ) = σ ( z, z ′ ) for all z, z ′ ∈ R n . Working in the canonical basis,Sp( n ) is identified with the group of all real 2 n × n matrices S such that S T J S = J (or,equivalently, SJ S T = J ).We will write d n z = d n x d n p , where d n x = dx . . . dx n and d n p = dp . . . dp n . The scalarproduct in L ( R n ) is denoted by ( ψ | φ ) = Z R n ψ ( x ) φ ( x ) d n x and the associated norm by || . || (in physicist’s bra-ket notation we thus have ( ψ | φ ) = h φ | ψ i ).In phase space we denote the inner product by((Ψ | Φ)) = Z R n Ψ( z )Φ( z ) d n z and the induced norm by ||| . ||| .The Schwartz space of rapidly decreasing functions is denoted by S ( R n ) and its dual spaceby S ′ ( R n ). The Fourier transform on R n is formally defined by F ψ ( p ) = (cid:0) π ~ (cid:1) n/ Z R n ψ ( x ) e − i ~ px d n x. Weyl–Heisenberg Frames
Definition and Terminology.
It is customary in frame theory to introduce Gaborframes, which we prefer to call Weyl–Heisenberg frames in this work because of their very closerelationship with the theory of the Heisenberg group and Weyl pseudodifferential calculus.
AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 5
Let φ be a (non-zero) square integrable function (the frame window ) on R n , and let Λa discrete (countable) subset of R n (the index set of the frame). The associated Weyl–Heisenberg system is the family of square-integrable functions G ( φ, Λ) = { b T ( z λ ) φ : z λ ∈ Λ } , where b T ( z ) = e − iσ (ˆ z,z ) / ~ is the Heisenberg operator and b z = ( b x, b p ) is the formal position-momentum operator with b xψ = xψ and b pψ = i ~ ∂ t ψ . The action of b T ( z ) in L ( R n ) is givenby(4) b T ( z ) φ ( x ) = e i ~ ( p x − p x ) φ ( x − x ) , where z = ( x , p ). They satisfy the commutation and addition properties b T ( z ) b T ( z ) = e i ~ σ ( z ,z ) b T ( z ) b T ( z )and b T ( z + z ) = e − i ~ σ ( z ,z ) b T ( z ) b T ( z ) . See [15, 16, 28] for detailed studies of these operators.We will call the system G ( φ, Λ) a
Weyl–Heisenberg frame if there exist constants a, b > frame bounds ) such that(5) a || ψ || ≤ X z λ ∈ Λ | ( ψ | b T ( z λ ) φ ) | ≤ b || ψ || for every ψ ∈ L ( R n ). If a = b , then the frame G ( φ, Λ) is said to be tight . Note that a tightframe is an orthonormal basis if all frame elements of G ( φ, Λ) are normalized and a = b = 1(see e.g. [20, Lemma 5.1.6.]). An immediate example of a Weyl–Heisenberg frame is given bythe well-known Fourier (orthogonal) basis G ( χ [0 , π ~ ) , Z × Z ).Given a Weyl–Heisenberg frame G ( φ, Λ), the associated frame operator b A G is given by b A G ψ = X z λ ∈ Λ ( ψ | b T ( z λ ) φ ) b T ( z λ ) φ. It is a positive, bounded, self-adjoint, invertible operator on L ( R n ) with bounded inverse,and we have(6) ψ = X z λ ∈ Λ ( ψ | b T ( z λ ) b A − G φ ) b T ( z λ ) φ = X z λ ∈ Λ ( ψ | b T ( z λ ) φ ) b T ( z λ ) b A − G φ. We will see that Weyl–Heisenberg frames can be expressed, both, in terms of the cross-Wigner transform and the cross-ambiguity function.The usefulness of Weyl–Heisenberg frames comes from the fact that they serve as “gener-alized bases” in the Hilbert space L ( R n ). The Weyl–Heisenberg expansion of an element ψ in the Hilbert space L ( R n ) with respect to the window φ is given by(7) ψ = X z λ ∈ Λ c z λ b T ( z λ ) φ. Due to the possible over-completeness of the system (see Section 2.2), this expansion is notalways unique and in general it is difficult to determine the coefficients c z λ . One possibility M. FAULHUBER, M.A. DE GOSSON, AND D. ROTTENSTEINER to do so, is to introduce the canonical dual window to φ , φ ◦ = b A − G φ (see equation (6) above).For every Weyl–Heisenberg expansion of type (7) we have X z λ ∈ Λ | c z λ | ≥ X z λ ∈ Λ | ( ψ | b T ( z λ ) φ ◦ ) | with equality only if c z λ = ( ψ | b T ( z λ ) φ ◦ ) . Lattices, the Frame Set and an Approximation Formula.
One of the most chal-lenging questions in time-frequency analysis is to determine whether a Weyl–Heisenberg sys-tem already constitutes a frame. Only in this case we have a stable Weyl–Heisenberg expan-sion of every element in our Hilbert space with respect to the elements of the Weyl–Heisenbergsystem. This question seems to be far too general to be answered. At this point the frameset enters the scene. For a fixed window φ , the frame set consists of all discrete point setsΛ ⊂ R n which together with φ give a frame, that is F ( φ ) = { Λ ⊂ R n : G ( φ, Λ) is a frame } . For the 1-dimensional standard Gaussian function ϕ ~ ( x ) = ( π ~ ) / e − ~ x the system G ( ϕ ~ , Λ)is a frame for L ( R ) whenever the lower Beurling density of Λ is greater than (2 π ~ ) − , i.e., if(8) δ ∗ (Λ) = lim inf r →∞ r min ( x,p ) ∈ R { z ∈ Λ : z ∈ ( x, p ) + [0 , r ] } > (2 π ~ ) − . In this case, the necessary conditions imposed by the Balian-Low theorem and the densitytheorem are already sufficient as proved by Lyubarskii [29], Seip [31] and Seip and Wallst´en[32] (see also e.g. [20, chap. 8.4]).Condition (8) is one of the manifestations of the classical uncertainty principle and it impliesthat we cannot construct an orthonormal basis consisting of quantum displaced Gaussians.Allowing arbitrary point sets in phase space is too general for this work as we want to comeup with explicit formulas. Therefore, we focus on lattices and for the rest of this work Λ willbe a lattice, unless otherwise mentioned. We recall that a lattice is a discrete co-compactsubgroup of ( R n , +) and that one can write a given lattice asΛ = M Z n for some, not uniquely determined, invertible matrix M ∈ GL (2 n, R ). The columns of M serve as a basis for Λ and its non-uniqueness results from the fact that we can choose fromcountably many bases. The volume and density of the lattice are given byvol(Λ) = | det( M ) | and δ (Λ) = 1vol(Λ)respectively. For a lattice the Beurling density δ ∗ coincides with the density δ of the lattice.A lattice is called symplectic if the generating matrix is a multiple of a symplectic matrix,i.e., Λ = cS Z n with S ∈ Sp ( n ) and c = 0 (we may assume, without loss of generality, that c > − Λ). Note that for n = 1 every lattice is symplectic, whereas this is not true inhigher dimensions. The adjoint lattice to Λ = M Z n is defined byΛ ◦ = J M − T Z n . AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 7
If Λ is symplectic, we simply have Λ ◦ = /n Λ. This follows from the definition of asymplectic matrix, SJ = J S − T , and J Z n = Z n . In this work we will exclusively considersymplectic lattices, although some of the results are valid for general index sets Λ, which wewill point out in those cases.There are some canonical subsets of F ( φ ) which we introduce now. We define the latticeframe set of φ by F Λ ( φ ) = { Λ ⊂ R n , Λ lattice : G ( φ, Λ) is a frame } . A special class of lattices are the so-called separable lattices, which are of the form Λ α,β = α Z n × β Z n . A generating matrix is given by S = (cid:18) αI βI (cid:19) , and the density of the lattice is δ (Λ α,β ) = ( αβ ) − n . The reduced frame set of φ is defined as F ( α,β ) ( φ ) = { ( α, β ) ∈ R : G ( φ, α Z n × β Z n ) is a frame } . Clearly, if ( α, β ) ∈ F ( α,β ) ( φ ) then α Z n × β Z n ∈ F Λ ( φ ). Hitherto, the only windows for which acomplete characterization of the general and the lattice frame sets are known are generalized1-dimensional Gaussians. There are some classes of functions for which the reduced frame setis fully known, we refer to the surveys by Gr¨ochenig [21] and Heil [22] for more details. Notethat for n = 1 we have Sp (1) = SL (2 , R ) and that 1-dimensional generalized Gaussians canbe written as the composition of a metaplectic operator and the standard Gaussian.The metaplectic group Mp( n ) is a unitary representation of the double cover Sp ( n ) of thesymplectic group Sp( n ). The simplest (but not necessarily the most useful) way of describingMp( n ) is to use its elementary generators b J , b V − P , and c M L,m ; denoting by π Mp the coveringprojection Mp( n ) −→ Sp( n ) these operators and their projections are given by b J ψ ( x ) = e − inπ/ F ψ ( x ) , π Mp ( b J ) = J b V − P ψ ( x ) = e i ~ P x ψ ( x ) , π Mp ( b V − P ) = V − P c M L,m ψ ( x ) = i m p | det L | ψ ( Lx ) , π Mp ( c M L,m ) = M L,m . Here F is the unitary ~ -Fourier transform and V − P ( P = P T ), M L,m (det L = 0) are thesymplectic generator matrices V − P = (cid:18) I P I (cid:19) , M L,m = (cid:18) L − L T (cid:19) . The index m in c M L,m is the Maslov index, an integer corresponding to a choice of arg det L : m is even if det L >
L < b T ( Sz ) = b S b T ( z ) b S − for every b S ∈ Mp( n ), S = π Mp ( b S ).It is always possible to construct frames with non-separable lattices from a given framewith separable lattice using the property of symplectic/metaplectic covariance as one of ushas shown in [12] (see also [5]): M. FAULHUBER, M.A. DE GOSSON, AND D. ROTTENSTEINER
Proposition 3.
Let φ ∈ L ( R n ) . A Weyl–Heisenberg system G ( φ, Λ) is a frame if and only if G ( b Sφ, S Λ) is a frame; when this is the case both frames have the same bounds. In particular, G ( φ, Λ) is a tight frame if and only if G ( b Sφ, S Λ) is a tight frame. From [3, 30] we recall the following generalization of the results originating from the workof Lyubarskii [29], Seip [31], and Seip and Wallst´en [32]:
Proposition 4.
For the multi-indices α = ( α , . . . , α n ) , β = ( β , . . . , β n ) ∈ Z n let Λ α,β = α Z n × β Z n . Let ϕ ~ j ( x j ) = ( π ~ ) − / e − ~ x j be the 1-dimensional standard Gaussian and let ϕ ~ ( x ) = ( π ~ ) − n/ e − ~ x = Q nj =1 ϕ ~ j ( x j ) be the n -dimensional standard Gaussian. Then thefollowing are equivalent: (i) G ( ϕ ~ , Λ αβ ) is a frame. (ii) G ( ϕ ~ j , Λ α j β j ) is a frame for all j = 1 , . . . , n . (iii) α j β j < π ~ for all j = 1 , . . . , n . Combining these two results we get the following statement, whose proof can be found inde Gosson [12]:
Corollary 5.
Let b S ∈ M p ( n ) have projection S ∈ Sp ( n ) . The Weyl–Heisenberg system G ( b Sϕ ~ , S Λ αβ ) is a frame if and only if α j β j < π ~ for ≤ j ≤ n . In this case the framebounds of G ( b Sϕ ~ , S Λ αβ ) are the same as those of G ( ϕ ~ , Λ αβ ) . Note that for n > Sp ( n ) is a proper subgroup of SL (2 n, R ) and, hence, the above resultsdo not necessarily carry over to arbitrary lattices.It was already observed by Folland [11, Chap. 4] that for n = 1 the only family of functionswhose Wigner transforms are rotation invariant are the Hermite functions (which include thestandard Gaussian). This is usually referred to as the “rotational invariance” of Hermitefunctions; heuristically it gives an explanation for the fact that all systems G ( ϕ ~ , R Λ) with R ∈ SO (2 n, R ) give a frame whenever δ ∗ > (2 π ~ ) − (in this case Λ need not to be a lattice).Moreover, the frame bounds are the same regardless of R . See Faulhuber [5] and de Gosson[17] for an extension of this result to arbitrary Gaussian and Hermitian frames. This has ledto the following conjecture in one of the author’s doctoral thesis [6]: Conjecture 6.
For φ ∈ L ( R ) the following are equivalent:(i) φ is a Hermite function.(ii) W φ is rotation-invariant.(iii) The frames G ( φ, R Λ) possess the same frame bounds for all R ∈ SO (2 , R ) . For n >
1, Conjecture 6 in combination with Folland’s result suggests the somewhatrestricted conjecture.
Conjecture 7.
Let n > . Then for φ ∈ L ( R n ) the following are equivalent:(i) φ is a Gaussian.(ii) W φ is rotation-invariant.(iii) The frames G ( φ, R Λ) possess the same frame bounds for all R ∈ O (2 n, R ) . Let us note that for both conjectures the relations ( i ) ⇔ ( ii ) ⇒ ( iii ) are well-known (see,e.g., [11]) in contrast to ( iii ) ⇒ ( i ). AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 9
We will discuss some properties of the Weyl–Heisenberg frame operator now. Regardingthe approximation we have the following result (for more details see [10]). For φ ∈ L ( R n )with k φ k = 1 we have(9) k Id − vol(Λ) b A G k op ≤ X z ◦ λ ∈ Λ ◦ \{ } | ( φ | b T ( z ◦ λ ) φ ) | . This result follows from Janssen’s representation of the Weyl–Heisenberg frame operator b A G [25]. The idea behind it is the following. If the lattice Λ is quite dense, then the adjoint latticeΛ ◦ is rather sparse and since (cid:0) φ | b T ( z ◦ λ ) φ (cid:1) ∈ C ( R d ), the sum on the right-hand side of (9)tends to zero as vol(Λ) →
0. The speed of convergence depends on the concrete function aswell as on the lattice (see the works of Faulhuber [7] and Faulhuber and Steinerberger [8]for a study on optimal lattices for Gaussian Weyl–Heisenberg frames). For Gaussians theconvergence is of exponential order with respect to the density δ (Λ) = (see Sections 3and 4 for exact results). It follows that the frame operator satisfieslim volΛ → vol(Λ) b A G = Id.
So, as the density increases, the frame operator converges to the identity operator in theoperator norm. The analogous statement is of course true for the inverse frame operator,meaning that lim vol(Λ) → b A − G = Id in the operator norm. This implies lim vol(Λ) → φ ◦ = φ in the Hilbert space norm. At this point, it seems appropriate to introduce Feichtinger’salgebra S ( R n ) which (densely) contains the Schwartz space S ( R n ). Feichtinger’s algebra,introduced by Feichtinger in the 1980s [9], is easily characterized as follows; φ ∈ S ( R n ) ⇐⇒ W φ ∈ L ( R n ) . It is a Banach space invariant under the Fourier transform and the action of the Heisenbergoperators b T . For more details we refer to the study by Jakobsen [24]. If φ ∈ S ( R n ), then φ ◦ also converges uniformly to the window φ [10]. Hence, for large density (small volume)of the lattice, we have the approximate Weyl–Heisenberg expansion(10) ψ ( x ) ≈ vol(Λ) X z λ ∈ Λ ( ψ | b T ( z λ ) φ ) b T ( z λ ) φ ( x ) . Two Reformulations of the Frame Condition.
We will see that Weyl–Heisenbergframes can be expressed both in terms of the cross-Wigner transform and the cross-ambiguityfunction.Recall that the cross-Wigner transform of a pair of square-integrable functions ( ψ, φ ) is W ( ψ, φ )( x, p ) = (cid:0) π ~ (cid:1) n Z R n e − i ~ py ψ ( x + y ) φ ( x − y ) d n y ; the cross-ambiguity function of ( ψ, φ ) is in turn given by A ( ψ, φ )( x, p ) = (cid:0) π ~ (cid:1) n Z R n e − i ~ py ψ ( y + x ) φ ( y − x ) d n y. For the (auto) Wigner transform and the ambiguity function we write
W φ = W ( φ, φ ) and Aφ = A ( φ, φ ) , respectively. It was observed already by Klauder that these functions are (symplectic) Fouriertransforms of each other, that is W ( ψ, φ ) = F σ A ( ψ, φ ) and A ( ψ, φ ) = F σ W ( ψ, φ ) , for F σ ψ ( z ) = F ψ ( J z ). We also have the algebraic relation(11) A ( ψ, φ )( z ) = 2 − n W ( ψ, φ ∨ )( z ) , where φ ∨ ( x ) = φ ( − x ) (see for instance [11, 16]). The cross-Wigner transform satisfies theMoyal identity(12) (cid:0)(cid:0) W ( ψ, φ ) | W ( ψ ′ , φ ′ ) (cid:1)(cid:1) = (cid:0) π ~ (cid:1) n ( ψ | ψ ′ )( φ | φ ′ )for all ψ, φ, ψ ′ , φ ′ in L ( R n ); using Plancherel’s formula together with (11), we also have (cid:0)(cid:0) A ( ψ, φ ) | A ( ψ ′ , φ ′ ) (cid:1)(cid:1) = (cid:0) π ~ (cid:1) n ( ψ | ψ ′ )( φ | φ ′ ) . To reformulate the frame conditions, we will need the following lemma:
Lemma 8.
For all ψ, φ ∈ L ( R n ) we have (13) A ( ψ, φ )( z ) = (cid:0) π ~ (cid:1) n ( ψ | b T ( z ) φ ) . We omit the proof as it is a trivial consequence of (4) (see [15, 16]).
Proposition 9.
The system G ( φ, Λ) is a Weyl–Heisenberg frame with bounds a, b if and onlyif either of the following two equivalent conditions hold for all ψ ∈ L ( R n ) : (2 π ~ ) n a || ψ || ≤ X z λ ∈ Λ | A ( ψ, φ )( z λ ) | ≤ (2 π ~ ) n b || ψ || (14) (4 π ~ ) n a || ψ || ≤ X z λ ∈ Λ | W ( φ, ψ )( z λ ) | ≤ (4 π ~ ) n b || ψ || (15) Proof.
Condition (14) immediately follows from (13) and, by (11), it is equivalent to (15).2.4.
Extension of a Frame to Phase Space.
The cross-Wigner transform satisfies thetranslational property W ( b T ( z ) ψ, b T ( z ) φ )( z ) = e − i ~ [ σ ( z,z − z )+ σ ( z ,z )] W ( ψ, φ )( z − ( z + z ))for all ψ, φ ∈ S ′ ( R n ) (see [11, 16, 18]). In particular, taking z = 0, we have W ( b T ( z ) ψ, φ )( z ) = e − i ~ σ ( z,z ) W ( ψ, φ )( z − z ) , which motivates the notation(16) e T ( z ) W ( ψ, φ ) = W ( b T ( z ) ψ, φ ) . AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 11
This suggests to define, as in [16], the operators e T ( z ) : S ( R n ) → S ( R n ) , e T ( z )Ψ( z ) = e − i ~ σ ( z,z ) Ψ( z − z ) . These operators extend to unitary operators on L ( R n ) and, a fortiori, to continuous auto-morphisms of S ′ ( R n ). We have e T ( z ) ∗ = e T ( z ) − = e T ( − z ) , and it is easily checked that the operators e T ( z ) satisfy the same commutation and additionproperties as the Heisenberg operators, that is e T ( z ) e T ( z ) = e i ℏ σ ( z ,z ) e T ( z ) e T ( z ) e T ( z + z ) = e − i ℏ σ ( z ,z ) e T ( z ) e T ( z ) . One of us has studied these operators in relation with the extension of Weyl operators tophase space [16]. This extension works as follows: let a ∈ S ′ ( R n ) be viewed as a symbol; thecorresponding Weyl operator Op W ( a ) : S ( R n ) −→ S ′ ( R n )is defined by(17) Op W ( a ) ψ ( x ) = (cid:0) π ~ (cid:1) n Z R n a σ ( z ) b T ( z ) ψ ( x ) d n z where a σ is the symplectic Fourier transform of a , formally given by a σ ( z ) = (cid:0) π ~ (cid:1) n Z R n e − i ~ σ ( z,z ′ ) a ( z ′ ) d n z ′ . For example, we have b ρ = Op W ( ρ ) for the density operator b ρ and its Wigner transform ρ mentioned in the introduction.One then defines the “Bopp operator” Op B ( a ) (see [4, 16, 14]) by replacing b T ( z ) in (17)by e T ( z ): for Ψ ∈ S ( R n ) we thus haveOp B ( a )Ψ( z ) = (cid:0) π ~ (cid:1) n Z R n a σ ( z ) e T ( z )Ψ( z ) d n z . This quantization associates to the symbols x and p the operators˜ x = x + i ~ ∂p and ˜ p = p − i ~ ∂x, respectively.2.5. Definition of a Phase Space Frame.
For fixed φ ∈ L ( R n ) we define the operator U φ : L ( R n ) → L ( R n ) ,ψ Ψ = (2 π ~ ) n/ W ( ψ, φ ) . By Moyal’s formula we have (( U φ ψ | U φ ψ ′ )) = ( ψ | ψ ′ ), hence U φ is an isometry of L ( R n ) ontoproper subspace of L ( R n ), which we denote by H φ . Observe that U ∗ φ U φ is the identityoperator on L ( R n ) and that Π φ = U φ U ∗ φ is the orthogonal projection of L ( R n ) onto H φ (we have Π ∗ φ = Π φ , Π φ = Π φ and the range of U ∗ φ is L ( R n )). Let G (Λ , φ ) be a Weyl–Heisenberg frame. We associate to G (Λ , φ ) the phase space frameoperator e A G Ψ( x ) = X z λ ∈ Λ ((Ψ | e T ( z λ )Φ)) e T ( z λ )Φ , where Φ = U φ φ = (2 π ~ ) n/ W φ . Proposition 10.
Let b A G be the frame operator of G (Λ , φ ) and e A G the associated phase spaceframe operator. Then we have the intertwining relation e A G Ψ = U φ ( b A G ψ ) for all Ψ ∈ H φ . Proof.
Since every Ψ ∈ H φ can be written asΨ = U φ ψ = (2 π ~ ) n/ W ( ψ, φ )for some uniquely determined ψ ∈ L ( R n ), it suffices to show(18) e A G W ( ψ, φ ) = W ( b A G ψ, φ ) . for all ψ ∈ L ( R n ). Using (16), we have e A G W ( ψ, φ ) = (2 π ~ ) n X z λ ∈ Λ (( W ( ψ, φ ) | e T ( z λ ) W φ )) e T ( z λ ) W φ = (2 π ~ ) n X z λ ∈ Λ (( W ( ψ, φ ) | W ( b T ( z λ ) φ, φ )) W ( b T ( z λ ) φ, φ ) , on the one hand. On the other hand, by Moyal’s identity (12), we have(( W ( ψ, φ ) | W ( b T ( z λ ) φ, φ ))) = (cid:0) π ~ (cid:1) n ( ψ | b T ( z λ ) φ )since φ is normalized, and hence e A G W ( ψ, φ ) = X z λ ∈ Λ ( ψ | b T ( z λ ) φ ) W (cid:16) b T ( z λ ) φ, φ (cid:17) = X z λ ∈ Λ W (cid:16) ( ψ | b T ( z λ ) φ ) b T ( z λ ) φ, φ (cid:17) . This is precisely (18).
Corollary 11.
Let G ( φ, Λ) be a Weyl–Heisenberg frame and set Φ = U φ φ . Then the system e G (Φ , Λ) = { e T ( z λ )Φ : z λ ∈ Λ } is a frame on the Hilbert space H φ with frame operator e A G andwe have a ||| Ψ ||| ≤ X z λ ∈ Λ | ((Ψ | e T ( z λ )Φ)) | ≤ b ||| Ψ ||| for all Ψ ∈ H φ where a and b are the same bounds as for the system G ( φ, Λ) . Proof.
It is an immediate consequence of (5) by employing the isometry U φ . AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 13 Gaussian Mixed States on R n In this section we derive all results stated in Theorems 1 and 2. We start with a quickcharacterization of Gaussian mixed states, which is followed by a description and explicitformulas of the cross-Wigner transform and the cross-ambiguity function of pairs of Gaussians.We conclude the section with explicit phase space frame expansions in terms of standard phasespace Gaussians.3.1.
Characterization.
For the moment, we go back to using the notation of density oper-ators from the introduction. Let ρ be a Gaussian of the type ρ ( z ) = √ det Σ − e − π Σ − z , where the covariance matrix Σ is a real positive definite symmetric matrix. The function ρ isnormalized such that Z R n ρ ( z ) d n z = 1so that it can be viewed as a probability density. As briefly discussed in the introduction, ρ is the Wigner distribution of a density operator b ρ if and only if Σ satisfies the condition Σ + i ~ J ≥ . The Cross-Wigner and Cross-Ambiguity Transform of Pairs of Gaussians.
Let ϕ ~ M be the centered Gaussian(19) ϕ ~ M ( x ) = (cid:0) π ~ (cid:1) n/ (det X ) / e − ~ Mx , where M = X + iY with positive definite, symmetric X and symmetric Y . The coefficient infront of the exponential is chosen such that ϕ ~ M is L -normalized. If M = I , we simply write ϕ ~ instead of ϕ ~ I .In [16] one of us has shown the following result. Proposition 12.
Let ( ϕ ~ M , ϕ ~ M ′ ) be a pair of Gaussians of the type (19) . Their cross-Wignertransform is given by W ( ϕ ~ M , ϕ ~ M ′ )( z ) = (cid:0) π ~ (cid:1) n C M,M ′ e − ~ F z , where C M,M ′ is the complex constant C M,M ′ = (det XX ′ ) / det (cid:2) ( M + M ′ ) (cid:3) − / and F is the symmetric complex matrix F = (cid:18) M ′ ( M + M ′ ) − M − i ( M − M ′ )( M + M ′ ) − − i ( M + M ′ ) − ( M − M ′ ) 2( M + M ′ ) − (cid:19) . If M = M ′ = X + iY with positive definite, symmetric X and symmetric Y , we write G = F and recover the well-known formula W ϕ ~ M ( z ) = (cid:0) π ~ (cid:1) n e − ~ Gz . (20) Observe that Σ + i ~ J is a self-adjoint complex matrix since J T = − J = J − ; it follows that its eigenvaluesare real, and the condition above is equivalent to saying that these eigenvalues are all non-negative. The matrix G is then a real Gram matrix which can be factored as (21) G = S T S, where the symplectic matrix S is given by S = (cid:18) X / X − / Y X − / (cid:19) . (22)3.3. Gaussian Phase Space Frame Expansions.
In the following we will give a quanti-tative estimate for the approximation error (9) for arbitrary symplectic lattices in R n as thedensity tends to infinity. Recall that for the symplectic lattice Λ = δ − S Z n , S ∈ Sp ( n ), ofdensity δ n = vol(Λ) − , the adjoint lattice is simply given by Λ ◦ = δS Z n = vol(Λ) − /n Λ.Given the Weyl–Heisenberg frame G ( ϕ ~ , Λ), Janssen’s representation [25] of the associatedframe operator is given by(23) ˆ A G = (2 π ~ ) n vol(Λ) X z λ ◦ ∈ Λ ◦ Aϕ ~ ( z λ ◦ ) b T ( z λ ◦ ) = (2 π ~ ) n vol(Λ) X k,l ∈ Z n Aϕ ~ (cid:0) S ( δk, δl ) (cid:1) . The value of the ambiguity function evaluated at a point on the adjoint lattice is given by Aϕ ~ (cid:0) δSz λ (cid:1) = (2 π ~ ) − n e − δ ~ Gz λ , z λ ∈ Λ , where G = S T S is positive definite with S = (cid:18) L L − T (cid:19) (cid:18) P (cid:19) = (cid:18) L L − T P L − T (cid:19) ∈ Sp ( n ) , for some invertible matrix L and real, symmetric matrix P = P T . For our purpose, thisparticular splitting is no restriction (recall (20), (21), (22) as well as [11, Prop.4.76.]); this isdue to the fact that for S ′ = QS with Q ∈ SO (2 n, R ) ∩ Sp ( n ) we have ( S ′ ) T S ′ = S T S . Weobtain the following generalization of [7, Prop. 3.1.], where the following result was provenfor n = 1. Proposition 13.
Let G ( ϕ ~ , Λ) be the Weyl–Heisenberg frame with Λ = δ − QS Λ , with δ > , Q, S and ϕ ~ as above. Then for the Weyl–Heisenberg frame operator b A G we have || Id − δ − n ˆ A G || op = O (cid:16) e − δ ~ ( || L || R n × n + || L || − R n × n ) (cid:17) . Proof.
Let δ >
0. Due to (23), we findvol(Λ) ˆ A G = X k,l ∈ Z n e − δ ~ | S ( k,l ) | = X k,l ∈ Z n e − δ ~ Lk e − δ ~ | L − T ( P k + l ) | ≤ X k,l ∈ Z n e − δ ~ || L || R n × n k e − δ ~ || L || R n × n | P k + l | = X k,m ∈ Z n e − δ ~ || L || R n × n k || L || R n × n √ δ e − δ ~ || L || R n × n m e i ~ kP m , (24) AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 15 where the last step is due to Poisson summation. As our summation goes over all m ∈ Z n ,(24) equals X k,m ∈ Z n e − δ ~ || L || R n × n k || L || R n × n √ δ e − δ ~ || L || R n × n m cos (cid:0) ~ kP m (cid:1) . Finally, since cos(2 x ) = 1 − x ) , we obtainvol(Λ) ˆ A G ≤ X k,m ∈ Z n e − δ ~ || L || R n × n k || L || R n × n √ δ e − δ ~ || L || R n × n m = X k,l ∈ Z n e − δ ~ (cid:0) || L || R n × n k + || L || − R n × n l (cid:1) , where we have applied Poisson summation once more. Definition 14.
For ϕ ~ M , ϕ ~ M ′ as in (19) we set Φ ~ F ( z ) = (2 π ~ ) n/ W ( ϕ ~ M , ϕ ~ M ′ )( z ) = (cid:0) π ~ (cid:1) n/ C M,M ′ e − ~ F z , where C M,M ′ and F are as in Proposition 12. If F = I we simply write Φ ~ instead of Φ ~ I . Corollary 15.
Let
Λ = δ − QS Z n . For the associated phase space frame G (Φ ~ , Λ) for H ϕ ~ ⊂ L (cid:0) R n (cid:1) associated to the Weyl–Heisenberg frame G ( ϕ ~ , Λ) we have the stable approximation || Id − δ − n e A G || op = O (cid:16) e − δ ~ ( || L || R n × n + || L || − R n × n ) (cid:17) . Proof.
Since the according property of the Weyl–Heisenberg frame G (Φ ~ , Λ) holds due toProposition 13, the statement follows directly from an application of the isometry U ~ ϕ asΦ ~ = U ~ ϕ ϕ ~ .This proves Theorem 1. Proposition 16.
Let Λ ⊂ R n be a discrete index set such that G (Φ ~ , Λ) is a phase spaceframe for H ϕ ~ . Then the coefficients of the expansion e A G Φ ~ F = X z λ ∈ Λ c z λ e T ( z λ )Φ ~ are given by c z λ = ((Φ ~ F , e T ( z λ )Φ ~ )) = (cid:0) π ~ (cid:1) n/ det( F + I ) − / e − ~ z λ e − ~ ( F + I ) − (cid:0) ( J − iI ) z λ (cid:1) . Equivalently, by setting H = ( J − iI ) T ( F + I ) − ( J − iI ) , we have ((Φ ~ F , e T ( z λ )Φ ~ )) = (cid:0) π ~ (cid:1) n/ det( F + I ) − / e − ~ ( I + H ) z λ . Proof.
Let z = ( x, p ) ∈ R n and z λ = ( x λ , p λ ) ∈ Λ. Then we compute((Φ ~ F , e T ( z λ )Φ ~ )) = Z R n (cid:0) π ~ (cid:1) n/ e − ~ F z (cid:0) π ~ (cid:1) n/ e i ~ σ ( z,z l ) e − ~ ( z − z λ d n z = (cid:0) π ~ (cid:1) n e − ~ ( x λ + p λ ) Z R n e − ~ ( F + I ) z e i ~ [ x ( p λ − ix λ ) p ( x λ + ip λ )] d n x d n p = (cid:0) π ~ (cid:1) n e − ~ z λ Z R n e − ~ F + I ) z e i ~ ( J − iI ) z λ d n z, where in the last equality we have used ( J − iI ) z λ = ( p λ , x λ ) − i ( x λ , p λ ). The last integralcan be viewed as the 2 n -dimensional Fourier transform of the Gaussian (2 π ~ ) n/ e − ~ F + I ) z at ( iI − J ) z λ (see Appendix A). We have Z R n e − ~ F + I ) z e i ~ ( J − iI ) z λ d n z = (2 π ~ ) n/ det(2( F + I )) / e − ~ ( F + I ) − (( iI − J ) z λ ) . Consequently,((Φ ~ F , e T ( z λ )Φ ~ )) = (cid:0) π ~ (cid:1) n/ det( F + I ) − / e − ~ z λ e − ~ ( F + I ) − (cid:0) ( J − iI ) z λ (cid:1) = (cid:0) π ~ (cid:1) n/ det( F + I ) − / e − ~ z λ e − ~ ( I + H ) z λ . The second identity is evident.This proves Theorem 2.
Corollary 17. If F = I , it is quickly checked that ( J − iI ) T ( F + I ) − ( J − iI ) = 0 . In this case we get ((Φ ~ , e T ( z λ )Φ ~ )) = (cid:0) π ~ (cid:1) n/ e − ~ z λ = (2 π ~ ) n/ Aϕ ~ ( z λ ) . Gaussian Mixed States on R We treat the 1-dimensional case separately since this is the only case where for a Gaussianwindow ϕ ~ a complete characterization of the frame set is known. Also, in this case all latticesare symplectic since Sp (1) = SL (2 , R ). We omit all proofs since the results follow alreadyfrom Section 3. Note that the blocks L and P are now scalars. We denote the 1-dimensionalstandard Gaussian by ϕ ~ ( x ) = (cid:0) π ~ (cid:1) / e − ~ x . In time-frequency analysis ~ is usually set to π . In this case we write ϕ ( x ) = ϕ π ( x ) =2 / e − πx . We note that ( ϕ ~ | ϕ ~ ) = k ϕ ~ k = 1 and the (general) frame set is given by F ( ϕ ~ ) = { Λ ⊂ R | δ ∗ (Λ) > π ~ } . Its Wigner transform is given by
W ϕ ~ ( x, p ) = π ~ e − ~ ( x + p ) . and its ambiguity function by Aϕ ~ ( x, p ) = π ~ e − ~ ( x + p ) . A generalized Gaussian in L ( R ) is of the form c M L b V − P ϕ ~ ( x ) = ϕ ~ P,L ( x ) = (cid:0) π ~ (cid:1) / √ Le − ~ ( L + iLP ) x , where b V − P is the metaplectic chirp and c M L is the metaplectic dilation operator (where weignore the Maslov index and set it to m = 0) given by b V − P ψ ( x ) = e i ~ P x ψ ( x ) and c M L ψ ( x ) = √ L ψ ( Lx ) , L > . AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 17
For certain densities of the lattice, it is possible to explicitly compute Weyl–Heisenberg ex-pansions of a function and the canonical dual window of ϕ ~ using results from Baastians [2]and Janssen [26].However, this is a quite cumbersome task. Since we know that the canonical dual windowresembles (up to a factor determined by the density) the original window if the density ishigh enough, we might use the simpler approximation (10) instead of the Weyl–Heisenbergexpansion (6) (see Figure 1 for an example). For a Gaussian window, this approximationbecomes accurate quite quickly. In fact, we have the following property for the frame operator b A G associated to the Weyl–Heisenberg frame G ( ϕ ~ , Λ), whereΛ = δ − (cid:18) L L − (cid:19) (cid:18) − P (cid:19) Z . For our analysis we can focus on lattices generated by lower triangular matrices since anymatrix can be decomposed into an orthogonal matrix and a lower triangular matrix (QR-decomposition). The orthogonal matrix can be ignored since ϕ ~ is an eigenfunction witheigenvalue of modulus 1 of the corresponding metaplectic operator (see, e.g., [5, 12]). Inshort, a rotation of the lattice does not affect our results since the Wigner transform and theambiguity function of the standard Gaussian are invariant under rotations. Computing theambiguity function of a generalized Gaussian yields Aϕ ~ P,L ( x, p ) = π ~ e − ~ (cid:18)(cid:18) L + P L (cid:19) x +2 PL xp + 1 L p (cid:19) . In the 1-dimensional case, it is customary to parametrize a lattice in phase space by thetriple ( α, β, γ ) in the following wayΛ = (cid:18) α βγ β (cid:19) Z = (cid:18) α β (cid:19) (cid:18) γ (cid:19) Z . Sometimes, the shearing and the dilation matrix are interchanged in the literature, however,this has no effect on our results. We also note that all 2-dimensional lattices can be expressedin the above form, up to a rotation which is negligible for the analysis of Gaussian states[5, 12].The following proposition is essentially [7, Prop. 3.1.].
Proposition 18.
For r > and α, β, γ ∈ R with αβ = 1 we have X k,l ∈ Z e − r ( α k +2 αβγ kl + β (1+ γ ) l ) ≤ X k,l ∈ Z e − r (cid:16) α k + 1 β l (cid:17) . This yields the following result.
Proposition 19.
For G ( ϕ ~ , Λ) with Λ = δ − Q (cid:18) α βγ β (cid:19) Z , where δ > , Q ∈ SO (2 , R ) isan orthogonal matrix and αβ = 1 , we have k Id − δ − b A G k op = O e − δ ~ (cid:16) α + 1 β (cid:17) ! . (a) The Gaussian ϕ , .(b) Approximation of ϕ , by using formula (10)for the integer lattice Z × Z . (c) Approximation of ϕ , by using formula (10)for the square lattice Z × Z .(d) The pointwise difference (in absolute values)between the approximated Gaussian and the orig-inal Gaussian. (e) The pointwise difference (in absolute values)between the approximated Gaussian and the orig-inal Gaussian. Figure 1.
Comparison of a dilated Gaussian and its approximation bystandard Gaussians using a truncated expansion of type (10) for squarelattices at critical density δ = 1 (left) and density δ = 4 (right). For the plotswe set ~ = π . Note the different scales! AUSSIAN DISTRIBUTIONS AND PHASE SPACE WEYL–HEISENBERG FRAMES 19
Appendix A. The Fourier Transform of a Gaussian and Poisson’s SummationFormula
Recall that the Fourier transform of a function ψ ∈ S ( R n ) is given by F ψ ( p ) = (cid:0) π ~ (cid:1) n/ Z R n ψ ( x ) e − i ~ px d n x. The corresponding Plancherel’s theorem reads k ψ k L ( R n ) = kF ψ k L ( R n ) . For a Gaussian of type φ ( x ) = e − ~ Mx such that M has positive definite real part and M ∗ = M , the Fourier transform is given by F φ ( p ) = (cid:0) π ~ (cid:1) n/ Z R n e − ~ Mx e − i ~ px d n x = det( M ) − / e − ~ M − p . This result is just a slight variation of Folland’s formula [11, App. A, eq. (1)]. With thesedefinitions Poisson’s summation formula reads X k ∈ Z n ψ ( k + x ) = X l ∈ Z n F ψ ( l ) e i ~ lx . Since we will use the formula only for Gaussians, we omit the technical details for when thisformula holds pointwise.
Acknowledgements.
The authors thank the anonymous referee for a careful reading of themanuscript and pointing out some references. Markus Faulhuber was supported by the Aus-trian Science Fund (FWF) project P27773-N23 and by the Erwin–Schr¨odinger Program ofthe Austrian Science Fund (FWF) J4100-N32. Maurice A. de Gosson was supported by theAustrian Science Fund (FWF) projects P23902-N13 and P27773-N23. David Rottensteinerwas supported by the Austrian Science Fund (FWF) project P27773-N23.
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