J. Pseudo-Differ. Oper. Appl. (2010) 1:3–34 DOI 10.1007/s11868-010-0001-6
Spectral and regularity properties of a pseudo-differential calculus related to Landau quantization Maurice de Gosson · Franz Luef
Received: 24 August 2009 / Revised: 23 October 2009 / Accepted: 30 October 2009 / Published online: 11 March 2010 © Birkhäuser / Springer Basel AG 2010
Abstract The theme of this work is that the theory of charged particles in a uniform magnetic field can be generalized to a large class of operators if one uses an extended a class of Weyl operators which we call “Landau–Weyl pseudodifferential operators”. The link between standard Weyl calculus and Landau–Weyl calculus is made explicit by the use of an infinite family of intertwining “windowed wavepacket transforms”; this makes possible the use of the theory of modulation spaces to study various regularity properties. Our techniques allow us not only to recover easily the eigenvalues and eigenfunctions of the Hamiltonian operator of a charged particle in a uniform magnetic field, but also to prove global hypoellipticity results and to study the regularity of the solutions to Schrödinger equations. Contents 1 Introduction . . . . . . . . . . . . . . 2 Modulation spaces: a short review . . . 2.1 Main definitions . . . . . . . . . . 2.2 Two examples . . . . . . . . . . . 2.3 Duality and kernel theorems . . . 3 Landau–Weyl calculus . . . . . . . . . 3.1 Review of standard Weyl calculus
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M. de Gosson was financed by the Austrian Research Agency FWF (Projektnummer P20442-N13). F. Luef was financed by the Marie Curie Outgoing Fellowship PIOF 220464. M. de Gosson (B) Fakultät für Mathematik, NuHAG, Universität Wien, 1090 Vienna, Austria e-mail:
[email protected] F. Luef Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA
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M. de Gosson, F. Luef 3.2 Landau–Weyl operators . . . . . . . . . . . . . . . 3.3 Symplectic covariance and metaplectic operators . Windowed wavepacket transforms . . . . . . . . . . . . 4.1 Definition and functional properties . . . . . . . . 4.2 The intertwining property . . . . . . . . . . . . . . 4.3 WPT and modulation spaces . . . . . . . . . . . . Spectral properties . . . . . . . . . . . . . . . . . . . . 5.1 General results . . . . . . . . . . . . . . . . . . . . 5.2 Shubin classes . . . . . . . . . . . . . . . . . . . . 5.3 Gelfand triples . . . . . . . . . . . . . . . . . . . . 5.4 Application to the Landau levels . . . . . . . . . . Regularity and hypoellipticity results . . . . . . . . . . 6.1 Global hypoellipticity . . . . . . . . . . . . . . . . 6.2 Regularity results for the Schrödinger equation . . . Generalization: application to deformation quantization γ ,µ . . . . . . . . . . . . . . . . . 7.1 The operators A 7.2 The Moyal product and deformation quantization . Concluding remarks . . . . . . . . . . . . . . . . . . .
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1 Introduction The aim of this paper is to compare the properties of a partial differential operator (or, more generally, a Weyl pseudodifferential operator) A = a w (x, −i h¯ ∂x ) with those of = a(X γ ,µ , Y γ ,µ ) obtained by replacing formally x and −i h¯ ∂x by the the operator A vector fields X γ ,µ =
γ µ i h¯ i h¯ x + ∂ y , Y γ ,µ = y − ∂x 2 µ 2 γ
(1)
where γ and µ are real scalars such that γ µ = 0. A typical situation of physical interest is the following: choose for A the harmonic oscillator Hamiltonian Hhar = −
h¯ 2 ∂ 2 mω2 2 x + 2m ∂ x 2 2
(2)
and γ = 1, µ = mω; defining the Larmor frequency ω L = ω/2 we obtain Hhar (X 1,mω , Y 1,mω ) = Hsym where Hsym = −
h¯ 2 x,y − i h¯ ω L 2m
y
∂ ∂ −x ∂x ∂y
+
mω2L 2 (x + y 2 ) 2
(3)
is the Hamiltonian operator in the symmetric gauge of a charged particle moving in the x, y plane under the influence of constant magnetic field orthogonal to the x, y plane. Another interesting situation occurs if one takes γ = 2, µ = 1. In this case Hhar (X 2,1 , Y 2,1 ) = Hhar h¯ where h¯ is the Moyal product familiar from deformation quantization.
Spectral and regularity properties of a pseudo-differential calculus
5
Of course these observations are not of earthshaking importance unless we can find In fact, as a rule, a procedure for comparing the properties of both operators A and A. the initial operator A is less complicated that its counterpart A so one would like to deduce the properties of the second from those of the first. For this we first have to find a procedure allowing us to associate to a function ψ ∈ L 2 (Rn ) a function ∈ L 2 (R2n ); that correspondence should be linear, and intertwine in some way the operators A and notice that our request for linearity excludes the choice = W ψ (the Wigner A; transform of ψ). It turns out that there exist many procedures for transforming a function of, say, x into a function of twice as many variables; the Bargmann transform is an archetypical (and probably the oldest) example of such a procedure. However, the Bargmann transform (and its variants) is not sufficient to recover all the spectral prop from those of A. For example, it is well known that the “Landau levels” erties of A of the magnetic operator Hsym are infinitely degenerate, so it is illusory to attempt to recover the corresponding eigenvectors from those of Hhar (the rescaled Hermite functions) using one single transform! This difficulty is of course related to the fact that no isometry from L 2 (Rn ) to L 2 (R2n ) can take a basis of the first space to a basis of the other, (intuitively L 2 (Rn ) is “much smaller” than L 2 (R2n )). We will overcome this difficulty by constructing an infinity of isometries Uφ parametrized by the Schwartz space S(Rn ); these isometries are defined in terms of the cross-Wigner transform, or equivalently, in terms of the windowed Fourier transform familiar from time-frequency and Gabor analysis. We will therefore call them windowed wavepacket transforms. For instance, in the case γ = 2, µ = 1 corresponding to the Moyal product, these isometries are (up to a normalization factor) just the mappings ψ −→ = W (ψ, φ) where W (ψ, φ) is the cross-Wigner transform. One of the goals of this paper is to emphasize the great potential of a particular class of function spaces, namely Feichtinger’s modulation spaces, whose elements can be defined in terms of decay properties of their windowed Wigner transform; these spaces have turned out to be a proper setting for the discussion of pseudodifferential operators in the last decade, and have allowed to prove (or to recover) in an elementary way many results which would otherwise requires the use of “hard analysis”. Although modulation spaces and their usefulness is well-known in time–frequency and Gabor analysis they have not received a lot of attention in quantum mechanics (they have however found some applications in the study of Schrödinger operators). This article is structured as follows: in Sects. 2–6 we develop the theory in the case λ = µ = 1, that is we work with the quantization rules X=
1 1 x + i h¯ ∂ y , Y = y − i h¯ ∂x 2 2
and their higher-dimensional generalizations Xj =
1 1 x j + i h¯ ∂ y j , Y j = y j − i h¯ ∂x j . 2 2
Furthermore we introduce the class of modulation spaces and recall some of their basic properties that are of relevance in the present investigation. In Sect. 7 we show how the general case (1) and its multi-dimensional generalization
6
M. de Gosson, F. Luef γ ,µ
Xj
=
γ i h¯ µ i h¯ γ ,µ x j + ∂ y j , Y j = y j − ∂x j 2 µ 2 γ
can be reduced to this one. We thereafter apply the previous study to deformation quantization. Closely related papers are Dasgupta and Wong [11] and Gramchev et al. [29,30]; our interest in these questions was triggered by Wong’s paper [70]. Notation Functions (or distributions) on Rn will usually be denoted by lower-case Greek letters ψ, φ, . . . while functions (or distributions) on R2n will be denoted by upper-case Greek letters , , . . .). We denote the inner product on L 2 (Rn ) by (ψ|φ) and the inner product on L 2 (R2n ) by ((| )); the associated norms are denoted by ||ψ|| and ||||||, respectively. Distributional brackets are denoted ·, · in every dimension. The standard symplectic form on the vector space Rn × Rn ≡ R2n is denoted by σ ; it is given, for z = (x, y), z = (x , y ), by the formula σ (z, z ) = J z · z where 0 I J= . The symplectic group of (R2n , σ ) is denoted by Sp(2n, R). −I 0 2 Modulation spaces: a short review For a rather compete treatment of the theory of modulation spaces we refer to Gröchenig’s book [33]. The main point want to do in this section is that, in contrast to the standard treatment of Weyl calculus, one can use with profit Feichtinger’s algebra M 1 (Rn ) and its weighted variants Mv1s (Rn ) as spaces of test functions instead of the Schwartz class S(Rn ). We also remark that a good class of pseudodifferen(R2n ); it has recently been tial symbols is provided by the modulation spaces Mv∞,1 s proved that they coincide with the so-called Sjöstrand classes [59,60]. In particular 0 (R2n ). they contain the Hörmander class S0,0 2.1 Main definitions Roughly speaking, modulation spaces are characterized by the matrix coefficients of the Schrödinger representation of the Heisenberg group. Recall that the Heisenberg– Weyl operators T (z 0 ) are defined, for z 0 = (x0 , y0 ) ∈ R2n by: i
1
T (z 0 )ψ(x) = e h¯ (y0 ·x− 2 y0 ·x0 ) ψ(x − x0 ).
(4)
We have i
T (z 1 + z 2 ) = e− 2h¯ σ (z 1 ,z 2 ) T (z 1 )T (z 2 )
(5)
where z 1 , z 2 ∈ R2n , hence z −→ T (z) is a projective representation of R2n , it is called the Schrödinger representation of the Heisenberg group. Let ψ, g be in L 2 (Rn ). Then
Spectral and regularity properties of a pseudo-differential calculus
7
the matrix coefficient of the Schrödinger representation is given by i i Vφh¯ ψ(z 0 ) = ψ, T (z 0 )φ = e 2h¯ y0 ·x0 e h¯ y0 ·x ψ(x)φ(x − x0 )d x.
(6)
Rn
When h¯ = 1/2π it is, up to the exponential in front of the integral, the short-time Fourier transform (STFT) Vφ ψ familiar from time-frequency analysis:
e2πi y0 ·x ψ(x)φ(x − x0 )d x
Vφ ψ(z 0 ) =
(7)
Rn
(Vφ ψ is also called the “voice transform” or “sliding transform” in signal theory). In what follows p is a non-negative real number and m a real weight function on p R2n . By definition a distribution ψ ∈ S (Rn ) is in the modulation space Mm (Rn ) if p n 2n there exists φ ∈ S(R ), φ = 0, such that Vφ ψ ∈ L m (R ), that is if
1/ p
||ψ|| Mmp =
|Vφ ψ(z)m(z)| dz
< ∞.
p
(8)
The essential point is that that the definition above is independent of the choice of the window φ: if it holds for one such window, it holds for all. Formula (8) defines a norm p on Mm (Rn ) and the replacement of φ by another window leads to an equivalent norm. Modulation spaces are Banach spaces for the topology defined by (8). (See [22,33] for the general theory). 2.2 Two examples In what follows ·s denotes the weight function defined by zs = (1 + |z|2 )s/2 .
(9)
• The modulation spaces Mv1s (Rn ). They are defined as follows: for s ≥ 0 we consider the weighted L 1 (R2n ) space L 1vs (R2n ) = { ∈ S (R2n ) : |||| L 1v < ∞} s
(10)
where || · || L 1v is the norm given by s
|(z)| zs dz.
|||| L 1v = s
(11)
R2n
The corresponding modulation space is Mv1s (Rn ) = {ψ ∈ S (Rn ) : Vφ ψ ∈ L 1vs (R2n )}
(12)
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M. de Gosson, F. Luef
where Vφ is the STFT (7) and φ is an arbitrary element of the Schwartz space S(Rn ); the formula φ
||ψ|| M 1 = ||Vφ ψ|| L 1v vs
s
(13)
defines a norm on Mv1s (Rn ) and if we change φ into another element of S(Rn ) we obtain an equivalent norm (Proposition 12.1.2 (p. 246) in [33]). The space Mv1s (Rn ) can also be very simply described in terms of the Wigner transform: we have Mv1s (Rn ) = {ψ ∈ S (Rn ) : W ψ ∈ L 1vs (R2n )}.
(14)
In the case s = 0 we obtain the Feichtinger algebra M 1 (Rn ) = S(Rn ). We have the inclusions S(Rn ) ⊂ M 1 (Rn ) ⊂ C 0 (Rn ) ∩ L 1 (Rn ) ∩ L 2 (Rn ).
(15)
The Feichtinger algebra can be used with profit as a space of test functions; its dual Banach space M ∞ (Rn ) = {ψ ∈ S (Rn ) : sup |Vφ ψ(z)| < ∞}
(16)
z∈R2n
contains the Dirac distribution δ. Furthermore M 1 (Rn ) is the smallest Banach space invariant under Heisenberg–Weyl operators (it is the space of integrable vectors of the Heisenberg–Weyl representation). Note that step functions are not in M 1 (Rn ) but triangle functions (which are the convolutions of two step functions) are. For these reasons the modulation spaces Mv1s (Rn ) are considerably larger classes of test functions than the Schwartz space S(Rn ). The spaces Mv1s (Rn ) are invariant under Fourier transform and more generally, under the action of the metaplectic group. They are also preserved by rescaling: Lemma 1 Let λ be a real number different from zero and set ψλ (x) = ψ(λx). We p p have ψλ ∈ Mvs (Rn ) if and only if ψ ∈ Mvs (Rn ). Proof It immediately follows from definition (7) of the STFT that we have Vφ ψλ (z) = λ−n Vφ1/λ ψ(z/λ) hence, performing a simple change of variable,
|Vφ ψλ (z)| p zs dz = λn R2n
|Vφ1/λ ψ(z)| p λzs dz. R2n
(17)
Spectral and regularity properties of a pseudo-differential calculus
9
Since λzs ≤ (1 + λ2 )s/2 zs it follows that there exists a constant Cλ > 0 such that |Vφ ψλ (z)| p zs dz ≤ Cλ |Vφ1/λ ψ(z)| p zs dz; R2n
R2n p
the integral in the right hand side is convergent if and only if ψ ∈ Mvs (Rn ), proving the necessity of the condition. That the condition is sufficient follows replacing ψλ by ψ in the argument above. For more general results about rescaling, see [48]; the result above can actually be viewed as a consequence of the proof of Theorem 1.1 in this work. (R2n ). • The modulation spaces Mv∞,1 0,s (R2n ) if and only if we have Let us set v0,s (z, ζ ) = ζ s . We have a ∈ Mv∞,1 0,s ||a|| ∞,1 = Mv0,s
R2n
sup |V a(z, ζ )| ζ −s dζ < ∞
z∈R2n
for one (and hence all) ∈ S(R2n ); here V is the STFT of functions on R2n . The use ∞,1 (R2n ) suggests that this space could be used of the letter a for the elements of Mv0, s as a symbol class. This is indeed the case; we mention that when s = 0 one recovers the so-called “Sjöstrand classes” [59,60] which have been studied and developed in [5,34,35,47] and [49,64–66] from the point of view of modulation space theory. As a symbol class is rather large; for instance C 2n+1 (R2n ) ⊂ M ∞,1 (R2n ) = M ∞,1 (R2n )
(18)
where C 2n+1 (R2n ) is the vector space of all bounded complex functions on R2n with continuous and bounded derivatives up to order 2n + 1. The following result, follows from Theorem 4.1 and its Corollary 4.2 in [35]; it (R2n ) in the theory of pseudoclearly demonstrates the usefulness of the spaces Mv∞,1 s differential operators: (R2n ) is bounded on every Proposition 2 An operator A with Weyl symbol a ∈ Mv∞,1 s modulation space Mv1s (Rn ). The result above can be viewed as a particular case of results in Toft [65]; also see p,q [35] for a proof of the property that A is bounded on any modulation space Mm (Rn ) when m is an arbitrary moderate weight). 2.3 Duality and kernel theorems ∞ (Rn ) with the norm The dual of Mv1s (Rn ) is the Banach space M1/v s
s ∞ = sup ψ M1/v z∈R2n Vφ ψ(z)| z < ∞. s
(19)
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M. de Gosson, F. Luef
Note that the Schwartz space S(Rn ) is the projective limit of the spaces {Mv1s (Rn ) : s ≥ 0} and consequently S (Rn ) has a description as inductive limit of the spaces ∞ (Rn ) : s ≥ 0}, i.e., {M1/v s S(Rn ) =
Mv1s (Rn ) and S (Rn ) =
s≥0
∞ M1/v (Rn ) s
(20)
s≥0
∞ )s≥0 as family of seminorms for S(Rn ) and S (Rn ), with (. Mv1 )s≥0 and (. M1/v s s respectively. Therefore results about the class of modulation spaces Mv1s (Rn ) and its dual ∞ M1/vs (Rn ) translate into corresponding results about the Schwartz class S(Rn ) and the tempered distributions S (Rn ). The great relevance of the pair S(Rn ) and S (Rn ) comes from the kernel theorem and Feichtinger showed that for the pair of modulation ∞ (Rn ) there also exists a kernel theorem (see [33], Sect. 14.4, spaces Mv1s (Rn ) and M1/v s for a short proof):
Theorem 3 (Kernel theorem) Let A be a continuous operator Mv1s (Rn ) −→ ∞ (Rn ). There exists a distribution K ∈ M ∞ (R2n ) such that M1/v A 1/vs s Aψ, φ = K A , ψ ⊗ φ
for φ, ψ ∈ Mv1s (Rn ).
(21)
As a consequence, using the intersections (20) we get the following version of the Schwartz kernel theorem, which was proven by Feichtinger in the case s = 0 and extended by Gröchenig [33] to submultiplicative weights vs , s ≥ 0. It was extended by Toft [66] to arbitrary weights. Theorem 4 Let A be a continuous linear operator from S(Rn ) to S (Rn ). Then A ∞ (Rn ) for some s ≥ 0. extends to a bounded operator from Mv1s (Rn ) to M1/v s Therefore our framework covers the traditional setting of pseudo-differential calculus.
3 Landau–Weyl calculus For the reader’s convenience we begin by quickly reviewing the basics of standard Weyl calculus. See for instance [43,61] for details and proofs.
3.1 Review of standard Weyl calculus In view of the kernel theorem above, there exists, for every linear continuous oper∞ (Rn ), a distribution K ∈ M ∞ (Rn × Rn ) such that ator A : Mv1s (Rn ) −→ M1/v A 1/vs s 1 n K Aψ(x) = A (x, ·), ψ for every ψ ∈ Mvs (R ).
Spectral and regularity properties of a pseudo-differential calculus
11
By definition the contravariant symbol of A is the distribution a defined by the Fourier transform i a(x, y) = e− h¯ y·η K A (x + 21 η, x − 21 η)dη (22) Rn
(the integral is interpreted in the distributional sense) and we can thus write formally
n/2 i 1 Aψ(x) = 2π h¯ e h¯ y(x−x ) a( 21 (x + x ), y)ψ(x )d x dy; (23) R n ×R n
strictly speaking this formula only makes sense when a ∈ Mv1s (R2n ); when this is not the case the double integral has to be reinterpreted in some way, for instance as a repeated or “oscillatory” integral; see for instance Trèves’ book [67] for an exposition of various techniques which are useful in this context. By definition the covariant symbol aσ of A is the symplectic Fourier transform aσ (z) = Fσ a(z) =
1 2π h¯
n
i e− h¯ σ (z,·) , a(·) .
(24)
Using the covariant symbol we can rewrite (23) as an operator-valued (Bochner) integral
n aσ (z)T (z)dz (25) A = 2π1 h¯ R2n
where T (z) is the Heisenberg–Weyl operator (4). Note that Weyl operators are composed in the following way. Assume that A and B are mappings on Mv1s (Rn ); then C = B A is defined and its contravariant and covariant symbols are given by the formulae
2n i c(z) = 4π1 h¯ e 2h¯ σ (u,v) a(z + 21 u)b(z − 21 v)dudv (26) cσ (z) =
R2n ×R2n
n 1 2π h¯
i
e 2h¯ σ (z,z ) aσ (z − z )bσ (z )dz .
(27)
R2n
The last two equations have a natural interpretation in terms of involutive represeni tations of the twisted group algebra L 1 (R2n , χ ) for the 2-cocycle χ (z, z ) = e 2h¯ σ (z,z ) . Namely, the unitary representation of the Heisenberg group by the Heisenberg–Weyl operators T (z) gives an involutive faithful representation of L 1 (R2n , χ ) via πint (a) = a(z)T (z)dz (28) R2n
12
M. de Gosson, F. Luef
for a ∈ L 1 (R2n ). In representation theory πint is called the integrated representation of the representation T (z). The product of πint (a) and πint (b) for a, b ∈ L 1 (R2n ) yields another element πint (c) of L 1 (R2n , χ ), where c is obtained from a and b by twisted convolution: c(z) = ab(z) =
1 2π h¯
n
i
e 2h¯ σ (z,z ) aσ (z − z )bσ (z )dz
(29)
R2n
(see for instance [69]). Consequently, the composition of two operators in the Weyl calculus is actually the twisted convolution of their covariant symbols in the twisted group algebra L 1 (R2n , χ ). Two particularly nice features of the Weyl pseudodifferential calculus are the following: ∞ (Rn ). Then the contravariant symbol of A∗ • Assume that A : Mv1s (Rn ) −→ M1/v s is complex conjugate to that of A. In particular, the Weyl operator A is self-adjoint if and only if its contravariant symbol a is real; • For every s ∈ Sp(2n, R) there exists a unitary operator S, uniquely defined up to a complex factor, such that the Weyl operator B with contravariant symbol b = a ◦ s −1 is given by the formula
B = S AS −1 ;
(30)
S can be chosen as a multiple of any of the two metaplectic operators covering s. We recall that the metaplectic group Mp(2n, R) is a faithful unitary representation of the twofold connected covering of Sp(2n, R); for a detailed account of the wellknown (and less well-known) properties of Mp(2n, R) see de Gosson [14], Chap. 7.
3.2 Landau–Weyl operators In our discussion of the Weyl calculus we stressed its interpretation in terms of the integrated representation of the Heisenberg–Weyl operators. A basic result about the representation theory of groups says that there is a one-to-one correspondence between projective representations of a group and integrated representations of the twisted group algebra of the group. Therefore a new representation of the Heisenberg group yields a new kind of calculus for pseudo-differential operators. In a series of papers on phase space Schrödinger equations the first author has implicitly made use of this fact. In the following we want to present these results in terms of integrated representation of a representation of the Heisenberg group on L 2 (R2n ). (z) on L 2 (R2n ) by the formula We define unitary operators T i
(z 0 )(z) = e− 2h¯ σ (z,z 0 ) (z − z 0 ). T
(31)
Spectral and regularity properties of a pseudo-differential calculus
13
We point out that these operators satisfy the relation i
(z 1 ) (z 0 )T (z 0 + z 1 ) = e− 2h¯ σ (z,z 0 ) T T which is formally similar to the relation (5) for the Heisenberg–Weyl operators (4). In fact, in [14, Chapter 10], one of us has shown that these operators can be used to construct an unitary irreducible representation of the Heisenberg group Hn on (infinitely many) closed subspace(s) of L 2 (R2n ). This is achieved using the wavepacket transforms we define in Sect. 4. (z) : S(R2n ) −→ S (R2n ) by replacing T (z) with T We now define the operator A in formula (25): = A
1 2π h¯
n
(z)dz, aσ (z)T
(32)
R2n
(z). i.e. as integrated representation of T defined by (32) the Landau–Weyl (for short: Definition 5 We will call the operator A LW) operator with symbol a (or: associated with the Weyl operator A). This terminology is motivated by the fact that the magnetic operator (3) appears as a particular case of these operators, choosing for a the harmonic oscillator Hamilto viewed as Weyl operator nian. Let us in fact determine the contravariant symbol of A, ∞ 1 2n 2n Mvs (R ) −→ M1/vs (R ). Because of the importance of this result for the rest of this paper we give it the status of a theorem: Theorem 6 Let a be the contravariant symbol of the Weyl operator A. Let (z, ζ ) ∈ viewed as R2n × R2n and z = (x, y), ζ = ( px , p y ). The contravariant symbol of A, ∞ 1 2n 2n a Weyl operator Mvs (R ) −→ M1/vs (R ) is given by the formula a (z, ζ ) = a
1 2z
− J ζ = a( 21 x − p y , 21 y + px ).
(33)
is given by the formula Proof The kernel of A K A(z, u) =
1 2π h¯
n/2
i
e 2h¯ σ (z,u) aσ (z − u)
(34)
as is easily seen by performing the change of variables u = z − z 0 in definition (32) and noting that σ (z, z − u) = −σ (z, u). We have [cf. formula (22)] a (z, ζ ) = R2n
i e− h¯ ζ ·η K A z + 21 η, z − 21 η dη
14
M. de Gosson, F. Luef
hence, using the identity σ (z + 21 η, z − 21 η) = −σ (z, η), a (z, ζ ) =
1 2π h¯
n
i
i
e− h¯ ζ ·η e− 2h¯ σ (z,η) aσ (η)dη
(35)
R2n
where aσ is the covariant symbol of A. By definition (24) of the symplectic Fourier transform we have
n i i i e− h¯ ζ ·η e− h¯ σ (η,z) a(z)dz e− h¯ ζ ·η aσ (η) = 2π1 h¯ R2n
hence, observing that σ (η, z) + ζ · η = σ (η, z + J ζ ), e
− h¯i ζ ·η
aσ (η) =
1 2π h¯
n
i
e− h¯ σ (η,z) T J ζ a(z)dz
R2n
where T J ζ a(z) = a(z − J ζ ), that is i
e− h¯ ζ ·η aσ (η) = Fσ (T J ζ a)(η). Formula (35) can thus be rewritten as
n i 1 a (2z, ζ ) = 2π h¯ e− h¯ σ (z,η) Fσ (T J ζ a)(η)dη R2n
hence a (2z, ζ ) = T J ζ a(z) (the symplectic Fourier transform is involutive); formula (33) follows. is Here are two immediate consequences of the result above. The first says that A self-adjoint if and only A is; the second says that the LW operators compose as the usual Weyl operators. is self-adjoint if and only a is real; (ii) The contraCorollary 7 (i) The operator A = A variant symbol of C B is given by c(z) = c( 21 z − J ζ ) where c is the contravariant symbol of C = AB. Proof To prove property (i) it suffices to note that a is real if and only a is. Property (ii) immediately follows from (26) and (33). Another consequence of these results is a statement about Landau–Weyl operators on Mv1s (Rn ). A well-known result due to Feichtinger (see Gröchenig [31] for a proof) asserts that a Weyl pseudodifferential operator bounded on Mv1s (Rn ) is of trace-class. Consequently the same results holds for operators in the Landau–Weyl calculus. Therefore we can compute the trace of these operators by integrating their kernel along the diagonal.
Spectral and regularity properties of a pseudo-differential calculus
15
be a bounded selfadjoint operator on Mv1 (Rn ) with kernel K A. Corollary 8 Let A s is given by The trace of A = Tr( A)
K A(z, z)dz.
(36)
R2n
Remark 9 We emphasize that trace formulas of the type (36) are usually not true for arbitrary trace-class operators (see the very relevant discussion of “trace formulas” in Reed and Simon [55]). 3.3 Symplectic covariance and metaplectic operators The symplectic covariance property (30) carries over to the LW calculus: for every S ∈ Mp(4n, R) with projection s we have (z) (sz) , ST S −1 = T SA S −1 = B
(37)
where B corresponds to b = a ◦ s −1 . Metaplectic operators are Weyl operators in their own right (see [13,14]). Let us determine the corresponding LW operators. Proposition 10 Let S ∈ Mp(2n, R) have projection s ∈ Sp(2n, R). If det(s − I ) = 0 then
n 1 (z)dz aσS (z)T (38) S = 2π h¯ R2n
where the function aσS is given by aσS (z) = √
i ν(S) i Ms z · z exp 2h¯ | det(s − I )|
(39)
with Ms = 21 J (s + I )(s − I )−1 MsT . The integer ν(S) is the class modulo 4 of the Conley–Zehnder index [15,16] of a path joining the identity to S in Sp(2n, R). Proof In [13] one of us showed that every S ∈ Mp(2n, R) with det(s − I ) = 0 can be written in the form
n S = 2π1 h¯ aσS (z)T (z)dz R2n
where aσS is given by (39). The proposition follows.
16
M. de Gosson, F. Luef
The operators S are metaplectic operators belonging to Mp(4n, R); we will not prove this fact here, but rather focus on a class of elementary operators which will be very useful for defining the general parameter dependent LW calculus in Sect. 7: Lemma 11 For (γ , µ) ∈ R2 , γ µ = 0, let S γ ,µ be the unitary operator on L 2 (R2n ) defined by S γ ,µ (x, y) = |γ µ|n (γ x, µy).
(40)
1 1
, We have S γ ,µ ∈ Mp(4n, R), ( S γ ,µ )−1 = S γ µ , and the projection of S γ ,µ onto Sp(4n, R) is the diagonal matrix
S γ ,µ = diag(γ −1 I, µ−1 I, γ I, µI )
(41)
(I the n × n identity). Proof That S γ ,µ ∈ Mp(4n, R) and formula (41) are standard results from the theory of metaplectic operators [14,39,40]. 4 Windowed wavepacket transforms In this section we treat resolutions of identity from a representation theoretic point of view. This approach has been of great relevance in various works in mathematics and physics (see e.g., [24,58] for a very general discussion of the topic). In the terminology of [58] we investigate in the present section coherent vectors and coherent projections generated by the square integrable representations T (z) and T˜ (z). The square-integrability of the representation T (z) of the Heisenberg group on L 2 (Rn ) is the Moyal identity: ((Vφ1 ψ1 |Vφ2 ψ2 )) = (ψ1 |ψ2 )(φ1 |φ2 )
(42)
for φ1 , φ2 , ψ1 , ψ2 in L 2 (Rn ). Note that Moyal’s identity is equivalent to the equality: ⎛ ⎜ (ψ1 |ψ2 )(φ1 |φ2 ) = ⎝
⎞ ⎟ f, T (z)ψ1 T (z)ψ2 dz|ψ2 ⎠ .
(43)
R2n
Setting ψ1 = ψ and assuming that (φ1 |φ2 ) = 0 this equality becomes a resolution of the identity: ψ = (φ1 |φ2 )−1
(ψ|T (z)φ1 )T (z)φ2 dz.
(44)
R2n
In the language of frames in Hilbert spaces, this resolution of the identity amounts to the statement that the set {T (z)ψ : z ∈ R2n } is a tight frame for L 2 (Rn ) (see
Spectral and regularity properties of a pseudo-differential calculus
17
[33] for a thorough discussion of frames in time–frequency analysis). In the present setting we can always find φ1 in Mv1s (Rn ) or in S(Rn ), i.e. there exist tight frames {T (z)ψ1 : z ∈ R2n } for L 2 (Rn ) with good phase space localization. The main purpose of this section is to discuss the consequences of the square-integrability of T˜ (z) on L 2 (R2n ). Unless otherwise specified φ will denote a function in S(Rn ) such that ||φ|| = 1; we will call φ a “window”. We note that a precursor to our notion of wavepacket transforms was introduced in Nazaikiinskii et al. [53]. 4.1 Definition and functional properties By definition the wavepacket transform Uφ on L 2 (Rn ) with window φ is defined by Uφ ψ(z) =
π h¯ n/2 2
W (ψ, φ)( 21 z) ,
ψ ∈ S (Rn );
(45)
here W (ψ, φ) is the cross-Wigner distribution, defined for ψ, φ ∈ L 2 (Rn ) by W (ψ, φ)(z) =
1 2π h¯
n e
i − h y·η ¯ ψ(x
+ 21 η)φ(x − 21 η)dη.
(46)
Rn
We observe for further use that Uφ ψ can be written Uφ ψ(z) =
1 2π h¯
n/2
( ( 21 z)ψ|φ)
(47)
where (z 0 ) is the Grossmann–Royer operator [38,56] defined by 2i
(z 0 )ψ(x) = e h¯ y0 (x−x0 ) ψ(2x0 − x).
(48)
It is useful to have a result showing how the windowed wavepacket transform behaves under the action of symplectic linear automorphisms. Proposition 12 Let s ∈ Sp(2n, R) and ψ ∈ L 2 (Rn ). We have Uφ ψ(s −1 z) = U Sφ (Sψ)(z)
(49)
where S is any of the two operators in the metaplectic group Mp(2n, R) covering s. Proof It immediately follows from the well-known covariance formula W (ψ, φ) ◦ s −1 = W (Sψ, Sφ) satisfied by the cross-Wigner distribution (see for instance de Gosson [14, Proposition 7.14, p. 207]).
18
M. de Gosson, F. Luef
The following theorem, part of which was proven in [14,16], summarizes the main functional analytical properties of the wavepacket transform. Theorem 13 The wavepacket transform Uφ is a partial isometry from L 2 (Rn ) into L 2 (R2n ). More explicitly, the wavepacket transform has the following properties: (i) Uφ is a linear isometry of L 2 (Rn ) onto a closed subspace Hφ of L 2 (R2n ); (ii) Let Uφ∗ : L 2 (R2n ) −→ L 2 (Rn ) be the adjoint of Uφ . We have Uφ∗ Uφ = I on L 2 (Rn ) and the operator Pφ = Uφ Uφ∗ is the orthogonal projection in L 2 (R2n ) onto the space Hφ ; (iii) The inverse Uφ−1 : Hφ −→ L 2 (Rn ) is given by the formula (2π h¯ )n/2 ψ(x) = (γ |φ)
Uφ ψ(z 0 ) ( 21 z 0 )γ (x)dz 0
(50)
Rn
where γ ∈ L 2 (Rn ) is such that (γ |φ) = 0; (iv) The adjoint Uφ∗ of Uφ is given by Uφ∗ (z)
=
2 π h¯
n/2
2i
e h¯
p·(x−y)
φ(2y − x)(y, p)dpdy.
(51)
R n ×R n
Proof Properties (i) and (ii) are proven in de Gosson [16] and detailed in de Gosson [14, Chap. 10, Sect. 2]. note that the fact that Uφ is an isometry immediately follows from Moyal’s identity (42): ((W (ψ, φ)|W (ψ , φ ))) =
1 2π h¯
n
(ψ|ψ )(φ|φ ).
(52)
Let us prove the inversion formula (50). Set
ψ (x) = Cγ
(z 0 ) ( 21 z 0 )γ (x)dz 0
Rn
where Cγ is a constant. For every θ ∈ L 2 (Rn ) we have, using successively (47) and the Moyal identity (52 ), (ψ |θ ) = Cγ
Uφ ψ(z 0 )(( 21 z 0 )γ |θ )dz 0
Rn
= Cγ 2−n/2 (π h¯ )3n/2 = Cγ (2π h¯ )3n/2
W (ψ, φ)
1 2 z0
W (γ , θ )
R2n
W (ψ, φ)(z 0 )W (γ , θ )(z 0 )dz 0
R2n
= Cγ (2π h¯ )n/2 (ψ|θ )(φ|γ ).
1 2 z0
dz 0
Spectral and regularity properties of a pseudo-differential calculus
19
It follows that ψ = ψ if we choose the constant Cγ so that Cγ (2π h¯ )n/2 (φ|γ ) = 1, which proves (50). Formula (51) for Uφ∗ is obtained by a straightforward calculation using the identity (Uφ ψ|) L 2 (R2n ) = (ψ|Uφ∗ ) L 2 (Rn ) and the definition of Uφ in terms of the cross-Wigner distribution. 4.2 The intertwining property are linked by the Here is the key result which shows how the operators A and A wavepacket transforms: Proposition 14 Let Uφ be an arbitrary wavepacket transform. The following intertwining formula hold: φ = Uφ A (z)Uφ = Uφ T (z) and AU T ∗ φ = A. Uφ T (z)Uφ = T (z) and Uφ∗ AU
(53) (54)
Proof The proof of formula (53) is purely computational (see [14, Theorem 10.10, and A are denoted there by Tph and Aph , respectively). Formulae (54) p.317], where T immediately follow since Uφ∗ Uφ = I on L 2 (Rn ). For instance, if Hhar = −
h¯ 2 ∂ 2 mω2 2 x + 2m ∂ x 2 2
is the harmonic oscillator Hamiltonian operator, and Hsym = −
h¯ 2 x,y − i h¯ ω L 2m
y
∂ ∂ −x ∂x ∂y
+
mω2L 2 (x + y 2 ) 2
is the magnetic operator considered in the introduction we have Hsym Uφ = Uφ Hhar
(55)
as an immediate consequence of the second formula (53). We will use this intertwining relation in Sect. 5 to recover the Landau levels and the corresponding Landau eigenfunctions. 4.3 WPT and modulation spaces The cross-Wigner transform (46) is related to the STFT by the formula W (ψ, φ)(z) =
2 π h¯
n/2
2i
e h¯ y·x Vφ√∨
2π h¯
ψ√2π h¯
√1 z 2π h¯
(56)
20
M. de Gosson, F. Luef
where we set as usual ψλ (x) = ψ(λx) for a non-zero λ ∈ R, φ ∨ (x) = φ(−x). It follows that there is a simple relationship between the windowed wavepacket transform Uφ and the short-time Fourier transform Vφ . In fact, using formula (56) relating Vφ to the cross-Wigner transform together with definition (45) of Uφ we have: i
Uφ ψ(z) = e 2h¯ y·x Vφ√∨
2π h¯
√
ψ√2π h¯ z/2 2π h¯
(57)
and hence also Vφ ψ(z) = e−4πi y·x Uφ ∨ √
1/ 2π h¯
√
ψ1/√2π h¯ 2 2π h¯ z .
(58)
Proposition 15 (i) We have ψ ∈ Mv1s (Rn ) if and only if Uφ ψ ∈ L 1vs (R2n ) for one (and hence for all) window(s) φ ∈ S(Rn ). (ii) For φ ∈ S(Rn ), φ = 0, the formula ||ψ|| Mv1 ,φ = |||Uφ ψ||| L 1v s
s
(59) φ
defines a family of norms on Mv1s (Rn ) which are equivalent to the norms ||ψ|| M 1
vs
defined by (13). (iii) The operator Uφ∗ maps L 1vs (R2n ) into Mv1s (Rn ) and the inversion formula (50) in Theorem 13 holds in Mv1s (Rn ). (iv) Mv1s (Rn ) is invariant under the action of the metaplectic group Mp(2n, R). Proof It is more or less a consequence of diverse results in [33]. (i) Immediately follows from formula (57) using Lemma 1. The statement (ii) follows from Proposition 11.3.2 in [33]; (iii) follows from Corollary 11.3.4 in [33]. Properties (iv) and (v) have been established in Proposition 11.3.2 of [33]. 5 Spectral properties We are going to use the results above to compare the spectral properties of A and A. are defined on some dense subspace We assume throughout that the operators A and A of L 2 (Rn ) and L 2 (R2n ), respectively (for instance S(Rn ) and S(R2n )). 5.1 General results The following result is very useful for the study of the eigenvectors of the LW operators; it shows that the wavepacket transforms can be used to construct orthonormal bases in L 2 (R2n ) starting from an orthonormal basis in L 2 (Rn ). Lemma 16 Let (φ j ) j∈F be an arbitrary orthonormal basis of L 2 (Rn ); setting j,k = Uφ j φk the family { j,k : ( j, k) ∈ F × F} forms an orthonormal basis of L 2 (R2n ), i.e. L 2 (R2n ) = j Hφ j (Hilbert sum), and we have j,k ∈ H j ∩ Hk .
Spectral and regularity properties of a pseudo-differential calculus
21
Proof Since the Uφ j are isometries the vectors j,k form an orthonormal system. It is thus sufficient to show that if ∈ L 2 (R2n ) is orthogonal to the family ( j,k ) j,k (and hence to all the spaces Hφ j ) then = 0. Assume that ((| jk )) = 0 for all j, k. Since ((| jk )) = ((|Uφ j φk )) = (Uφ∗ j |φk ) it follows that Uφ∗ j = 0 for all j since (φ j ) j is a basis; using the anti-linearity of Uφ
in φ we have in fact Uφ∗ = 0 for all φ ∈ L 2 (Rn ). Let us show that this implies that = 0. Fixing the variable x we have, using formula (51), Uφ∗ (x) =
2 π h¯
n/2
2i
e h¯
p·(x−y)
φ(2y − x)(y, p)dpdy
R n ×R n
F(2) (x, 2(y − x))φ(2y − x)dy
= 2n Rn
where F(2) is the inverse partial Fourier transform of in the second set of variables. The condition Uφ∗ = 0 for all φ ∈ S(Rn ) thus implies that F(2) = 0 and hence = 0. That j,k ∈ H j ∩ Hk is clear by definition of the Hilbert spaces H j . are the same; (ii) Let ψ be Theorem 17 (i) The eigenvalues of the operators A and A an eigenvector of A: Aψ = λψ. Then = Uφ ψ is an eigenvector of A corresponding = λ. (ii) Conversely, if is an eigenvector of A then to the same eigenvalue: A ∗ ψ = Uφ is an eigenvector of A corresponding to the same eigenvalue. is clear: if Aψ = λψ Proof (i) That every eigenvalue of A also is an eigenvalue of A for some ψ = 0 then φ ψ) = Uφ Aψ = λUφ ψ A(U and = Uφ ψ = 0 ; this proves at the same time that Uφ ψ is an eigenvector of A = λ for ∈ L 2 (R2n ), because Uφ has kernel {0}. (ii) Assume conversely that A = 0, and λ ∈ R. For every φ we have = λUφ∗ AUφ∗ = Uφ∗ A hence λ is an eigenvalue of A and ψ an eigenvector if ψ = Uφ∗ = 0. We have Uφ ψ = Uφ Uφ∗ = Pφ where Pφ is the orthogonal projection on the range Hφ of Uφ . Assume that ψ = 0; then Pφ = 0 for every φ ∈ S(Rn ), and hence = 0 in view of Lemma 16. The reader is urged to remark that the result above is quite general: it doses not make any particular assumptions on the operator A (in particular it is not assumed that A is self-adjoint), and the multiplicity of the eigenvalues can be arbitrary.
22
M. de Gosson, F. Luef
is (essentially) self-adjoint: Let us specialize the results above to the case where A Corollary 18 Suppose that A is a self-adjoint operator on L 2 (Rn ) and that each of the eigenvalues λ0 , λ1 , . . . , λ j , . . . has multiplicity one. Let ψ0 , ψ1 , . . . , ψ j , . . . be a corresponding sequence of orthonormal eigenvectors. Let j be an eigenvector of corresponding to the eigenvalue λ j . There exists a sequence (α j,k )k of complex H numbers such that j = α j, j, with j, = Uψ ψ j ∈ H j ∩ H . (60)
have same eigenvalues and that Proof We know from Theorem 17 above that A and A j,k = λ j j,k . Since A is self-adjoint its eigenvectors ψ j j,k = Wψk ψ j satisfies H form an orthonormal basis of L 2 (Rn ); it follows from Lemma 16 that the j,k form an orthonormal basis of L 2 (R2n ), hence there exist non-zero scalars α j,k, such that k, = λk k, , j = k, α j,k, k, . We have, by linearity and using the fact that A j = A
k, = α j,k, A
k,
α j,k, λk k, .
k,
j = λjj, On the other hand we also have A j = λ j j = H
α j,k, λ j k,
j,k
and this is only possible if α j,k, = 0 for k = j; setting α j, = α j, j, formula (60) follows. (That j, ∈ H j ∩ H is clear using the definition of H and the sesquilinearity of the cross-Wigner transform.) 5.2 Shubin classes Shubin has introduced in [61] very convenient symbol classes ρm 1 ,m 0 for studying global hypoellipticity. These symbol classes moreover contain the following interesting subclasses, particularly well adapted to the study of global hypoellipticity questions: H ρm 1 ,m 0 (R2n ) (m 0 , m 1 ∈ R and 0 < ρ ≤ 1) is the complex vector space of all functions a ∈ C ∞ (R2n ) for which there exists a number R ≥ 0 such that for |z| ≥ R we have C0 |z|m 0 ≤ |a(z)| ≤ C1 |z|m 1 , |∂zα a(z)| ≤ Cα |a(z)||z|−ρ|α|
(61)
for some constants C0 , C1 , Cα ≥ 0; we are using here multi-index notation α = α (α1 , . . . , α2n ) ∈ Nn , |α| = α1 + · · · + α2n , and ∂zα = ∂xα11 . . . ∂xαnn ∂ y1n+1 . . . ∂ yαn2n . We m 1 ,m 0 are invariant under linear changes of variables: notice that the Shubin classes H ρ if f ∈ G L(2n, R) and a ∈ H ρm 1 ,m 0 (R2n ) then a ◦ f ∈ H ρm 1 ,m 0 (R2n ). In particular they are invariant under linear symplectic transformations.
Spectral and regularity properties of a pseudo-differential calculus
23
1 ,m 0 We denote by H G m (Rn ) the class of operators A with τ -symbols aτ belonging ρ m 1 ,m 0 2n (R ); this means that for every τ ∈ R there exists aτ ∈ H ρm 1 ,m 0 (R2n ) to H ρ such that 1 n Au(x) = 2π ei(x−y)·ξ aτ ((1 − τ )x + τ y, ξ )u(y)dydξ ;
R2n
choosing τ = 21 this means, in particular, that every operator with Weyl symbol a ∈ 1 ,m 0 H ρm 1 ,m 0 (R2n ) is in H G m (R2n ). Conversely, the condition a ∈ H ρm 1 ,m 0 (R2n ) ρ is also sufficient, because if aτ ∈ H ρm 1 ,m 0 (R2n ) is true for some τ then it is true for all τ . Shubin [61] (Chapter 4) proves the following result: 1 ,m 0 (R2n ) with m 0 > 0. If A is formally selfProposition 19 (Shubin) Let A ∈ H G m ρ adjoint, that is if (Aψ|φ) = (ψ|Aφ) for all ψ, φ ∈ C0∞ (Rn ), then: (i) A is essentially self-adjoint and has discrete spectrum in L 2 (Rn ); (ii) There exists an orthonormal basis of eigenfunctions φ j ∈ S(Rn ) ( j = 1, 2, . . .) with eigenvalues λ j ∈ R such that lim j→∞ |λ j | = ∞.
This result has the following consequence for LW operators: 1 ,m 0 (R2n ) be formally self-adjoint. Then the LW operator Corollary 20 Let A ∈ H G m ρ A has discrete spectrum (λ j ) j∈N and lim j→∞ |λ j | = ∞ and the eigenfunctions of A are in this case given by jk = Uφ j φk where the φ j are the eigenfunctions of A; (iv) We have jk ∈ S(R2n ) and the jk form an orthonormal basis of φ j ∈ S(Rn ).
Proof It is an immediate consequence of Theorem 17 using the proposition above. 5.3 Gelfand triples Dirac already emphasized in his fundamental work [18] the relevance of rigged Hilbert spaces for quantum mechanics. Later Schwartz provided an instance of rigged Hilbert spaces based on his class of test functions and on tempered distributions. Later Gelfand and Shilov formalized the construction of Schwartz and Dirac and introduced what is nowadays known as Gelfand triples. The prototypical (and probably the oldest) example of a Gelfand triple is (S(Rn ), L 2 (Rn ), S (Rn )). In the last decade Feichtinger and some of his collaborators (see [7,19,25]) have emphasized the relevance of the Gelfand triple ∞ (Mv1s (Rn ), L 2 (Rn ), M1/v (Rn )) s
in time-frequency analysis. An important feature of Gelfand triples is the existence of a kernel theorem, as we explained in Sect. 2.3. In the present investigation these
24
M. de Gosson, F. Luef
classes of Gelfand triples will allow us to treat the case of the continuous spectrum of selfadjoint operators. The main idea underlying the notion of Gelfand triple is the observation, that a triple of spaces—consisting of the Hilbert space itself, a small (topological vector) space contained in the Hilbert space, and its dual—allows a much better description of the spectrum. The main appeal of the notion of Banach triple is, in our context, the fact that we can even take a Banach space, namely the modulation space Mv1s (Rn ). Definition 21 A (Banach) Gelfand triple (B, H, B ) consists of a Banach space B which is continuously and densely embedded into a Hilbert space H, which in turn is w ∗ -continuously and densely embedded into the dual Banach space B . In this setting the inner product on H extends in a natural way to a pairing between B and B producing an anti-linear functional F of the same norm. The framework of the Gelfand triple (S(Rn , L 2 (Rn ), S (Rn )) or more generally of ∞ (Rn )) allows one to formulate a spectral theorem for selfad(Mv1s (Rn ), L 2 (Rn ), M1/v s n joint operators on S(R ) or Mv1s (Rn ). If F(Aψ) = λF(ψ) holds for all ψ ∈ Mv1s (Rn ) or in S(Rn ) in the distributional sense, then λ is called a generalized eigenvalue to the generalized eigenvector of the selfadjoint operator A. For a given generalized eigen∞ (Rn ) value λ ∈ C we denote by E λ be the set of all generalized eigenvectors F in M1/v s n or S(R ), respectively. The set of all generalized eigenvalues ∪λ E λ is called complete, ∞ (Rn ) or S(Rn ) such that F(ψ) = F(φ) for all F ∈ ∪ E , if for any ψ, φ in M1/v λ λ s then ψ = φ. Theorem 22 Let T be a selfadjoint operator on Mv1s (Rn ) or S(Rn ). Then all generalized eigenvalues λ are real numbers and L 2 (Rn ) can be written as a direct sum of Hilbert spaces H(λ) such that E λ ⊂ Hλ , and such that the λ-component of T f is given by (A f )λ = λ f for all Mv1s (Rn ) or S(Rn ). Moreover, the set of generalized eigenvectors ∪λ E λ is complete. As an illustration we treat generalized eigenvectors of the translation operator Tx f (y) = f (y − x). We interpret the characters χω (x) = e−2πiω·x as generalized eigenvectors for the translation operator Tx on Mv1s (Rn ). Furthermore the set of generalized eigenvectors {χω : ω ∈ Rn } is complete by Plancherel’s theorem, i.e., if the Fourier transform fˆ(ω) = χω , f vanishes for all ω ∈ Rn implies f ≡ 0. This suggests to think of the Fourier transform of f at frequency ω as the evaluation of the linear functional χω , f . Therefore the preceding theorem allows us to deal with the continuous spectrum as treated in Theorem 17 for the discrete spectrum. 5.4 Application to the Landau levels As an illustration consider the harmonic oscillator Hamiltonian (2) of the Introduction. To simplify notation we take m = ω = 1 (this corresponds to the choice γ = µ = 1 γ ,µ ); in addition we choose units in which h¯ = 1 In view of the results above the for A spectra of the harmonic oscillator Hamiltonian (2) and of the magnetic operator (3)
Spectral and regularity properties of a pseudo-differential calculus
25
are identical. The eigenvalues of the first are the numbers λk = k + 21 (k an integer). These are the well-known Landau energy levels [50]. The harmonic oscillator operator (2) satisfies the assumptions of Corollary 18. The normalized eigenvectors are the rescaled Hermite functions √ 1 1 2 φk (x) = (2k k! π )− 2 e− 2 x Hk (x). where Hk (x) = (−1)km e x
2
d k −x 2 e dx
is the k-th Hermite polynomial. Using definition (45) of the wavepacket transform together with known formulae for the cross-Wigner transform of Hermite functions (Thangavelu [63, Chapter 1], Wong [69, Chap. 24, Theorem 24.1]) one finds that the eigenvectors of the magnetic operator are linear superpositions of the functions
j+k,k (z) = (−1) j √1
2π
j! ( j+k)!
1 2
k
2− 2 z k Lkj ( 21 |z|2 )e−
|z|2 4
and j, j+k = j+k,k for k = 0, 1, 2, . . .; in the right-hand side z is interpreted as x + i y and Lkj (x) =
1 −k x j! x e
d j dx
(e−x x j+k ), x > 0
is the Laguerre polynomial of degree j and order k. In particular we recover the textbook result that the eigenspace of the ground state is spanned by the functions 1
0,k (x, y) = (k!2k+1 π )−1/2 (x − i y)k e− 4 (x
2 +y 2 )
;
notice that this eigenspace is just Hφ0 . Finally we want to mention that the intertwining between the Weyl calculus and the Landau–Weyl calculus allows one to define annihilation and creation operators as in the case of the harmonic oscillator. Therefore our calculus provides us with natural operators that allow us to “move” between the eigenvectors of the Landau levels. We will come back to this issue in a forthcoming work.
6 Regularity and hypoellipticity results We begin by stating a few boundedness results for Weyl and Landau–Weyl operators in modulation spaces. The main result of this section is Theorem 26 where we prove a global hypoellipticity result for Landau–Weyl operators whose symbol belong to the Shubin class H ρm 1 ,m 0 (R2n ).
26
M. de Gosson, F. Luef
6.1 Global hypoellipticity In [61, Corollary 25.1, p. 186] Shubin has introduced the notion of global hypoellipticity (also see Boggiatto et al. [4, p. 70]). This notion is more useful in quantum mechanics than the usual C ∞ hypoellipticity because it incorporates the decay at infinity of the involved distributions. Definition 23 We will say that a linear operator A : S (Rn ) −→ S (Rn ) is “globally hypoelliptic” if we have ψ ∈ S (Rn ) and Aψ ∈ S(Rn ) ⇒ ψ ∈ S(Rn ).
(62)
Let us return to the Shubin classes we used in Sect. 5 when we studied spectral properties of Landau–Weyl operators. Using the properties of these classes Shubin [61, Chap. IV, Sect. 23] constructs a (left) parametrix of A. i.e. a Weyl operator B ∈ Gρ−m 1 ,−m 0 (Rn ) such that B A = I + R where the kernel of R is in S(Rn × Rn ); from the existence of such a parametrix follows readily that: 1 ,m 0 Proposition 24 Any Weyl operator A belonging to the class H G m (R2n ) is globρ ally hypoelliptic.
Note that the previous proposition of Shubin remains true for the case of modulation spaces, because all the arguments of his proof remain valid for this more general class of function spaces. In [8] Fredholm properties of (localization) pseudodifferential operators on modulation spaces have been proved by Cordero and Gröchenig. These results provide natural generalizations of well-known results due to Shubin on global hypoellipticity. We will use the following refinement of Proposition 24, also due to Shubin [61, Chap. IV, Sect. 25]: 1 ,m 0 (R2n ) be such that Ker A = Ker A∗ = {0}. Then there Proposition 25 A ∈ H G m ρ −m 1 ,−m 0 2n exists B ∈ H G ρ (R ) such that B A = AB = I (i.e. B is the inverse of A).
In [8] a class of symbol classes Mv is introduced, which for the weight vs contains 1 ,m 0 (R2n ). Therefore by Theorem 7.1 in [8] we can discuss the global the classes H G m ρ hypoellipticity for Landau–Weyl operators. The main result of this section is: associated to an operator A ∈ Gρm 1 ,m 0 Theorem 26 The Landau–Weyl operator A 2n ∗ (R ) such that Ker A = Ker A = {0} is globally hypoelliptic. Proof In view of Proposition 25 the operator A has an inverse B belonging to 1 ,−m 0 H G −m (R2n ). In view of Corollary 7 the LW operator B is then an inverse of A. ρ 2n Assume now that A = ∈ S(R ); then = B . The result now follows from the observation that B maps S(R2n ) into S(R2n ) and S (R2n ) into S (R2n ). ∈ Gρm 1 ,m 0 (R4n ) Remark 27 The condition A ∈ Gρm 1 ,m 0 (R2n ) does not imply that A as is seen by inspection of formula (33) for the symbol a.
Spectral and regularity properties of a pseudo-differential calculus
27
Let us illustrate this when the symbol a is a non-degenerate quadratic form: Example 28 Let a be a positive-definite quadratic form on R2n : a(z) = 21 M z · z with M = M T > 0. Then A is globally hypoelliptic; in fact a ∈ H 12,2 (R2n ) as is seen is globally hypoelliptic. In using an adequate diagonalization of M. The operator A particular the magnetic operator (3) is globally hypoelliptic. Notice that in this example we have recovered the global hypoellipticity of the magnetic operator obtained by Wong [70] using very different methods (the theory of special functions). 6.2 Regularity results for the Schrödinger equation Let us apply some of the previous results to the study of regularity properties of the Schrödinger equations i h¯
∂ ∂ . ψ = H ψ, i h¯ = H ∂t ∂t
Let the Hamiltonian function be a quadratic form: H (z) = 21 M z · z,
M = MT .
The corresponding Hamiltonian flow consists of the linear symplectic mappings st = et J M and is hence a one-parameter subgroup of Sp(2n, R). It follows from the theory of covering spaces that there is a bijective correspondence between the oneparameter subgroups of the symplectic group Sp(2n, R) and those of the metaplectic group Mp(2n, R); let us denote this correspondence by µ. Thus µ(st ) = St means that if (st ) is a one-parameter subgroup of Sp(2n, R) then (St ) is the only oneparameter subgroup of Mp(2n, R) whose projection is precisely (st ). We will similarly write µ(st ) = St where St ∈ Mp(4n, R) is defined by formula (38). The first part of following result is well-known: Proposition 29 Let (st ) be the Hamiltonian flow determined by the Hamilton equaSt ) defined by tions z˙ = J ∂z H (z) = M z. The one parameter groups (St ) and ( St = µ(st ) satisfy the Schrödinger equations St = µ(st ) and i h¯
∂ ∂ St St = H St , i h¯ St = H ∂t ∂t
(63)
28
M. de Gosson, F. Luef
are the Weyl and LW operators determined by the Hamilwhere H (x, −i h¯ ∂x ) and H tonian function H . Proof That St satisfies the first equation (63) is a classical result (see for instance [14,39,40], for detailed accounts). That St satisfies the second equation immediately follows. We next show that the spreading of wavefunction and its evolution in time can be controlled in terms of the spaces L 1vs (R2n ). Proposition 30 Let ∈ S (R2n ) be a solution of the Schrödinger equation i h¯
∂ , (·, 0) = 0 . =H ∂t
(64)
If 0 ∈ Hφ ∩ L 1vs (R2n ) for some φ then (·, t) ∈ L 1vs (R2n ) for every t ∈ R. Proof Since 0 ∈ Hφ we have 0 = Uφ ψ0 for some ψ0 ∈ L 2 (Rn ); the condition 0 ∈ L 1vs (R2n ) implies that ψ0 ∈ Mv1s (Rn ). Let ψ be the unique solution of the Cauchy problem i h¯
∂ψ = H ψ, ψ(·, 0) = ψ0 ; ∂t
that solution is ψ = St ψ0 in view of Proposition 29, hence ψ(·, t) ∈ Mv1s (Rn ) for every t ∈ R. We claim that the (unique) solution of (64) with 0 ∈ Hφ ∩ L 1vs (R2n ) is = Uφ ψ; the proposition will follows in view of the definition of Mv1s (Rn ). Set Uφ = Uφ H (x, −i h¯ ∂x ) in view of the second equality (53) in = Uφ ψ. Since H Proposition 14, we have i h¯
∂ ∂ Uφ ψ = H . = Uφ (i h¯ ψ) = H ∂t ∂t
Now (·, 0) = 0 hence = .
7 Generalization: application to deformation quantization γ ,µ 7.1 The operators A γ ,µ correLet us show how to generalize the constructions above to the operators A sponding to the more general quantization rule (1). γ ,µ γ ,µ are obtained from j X j and Y We begin by noting that the operators X j and Y j γ ,µ by conjugation with the metaplectic rescalings S defined in Lemma 11: γ ,µ γ ,µ = j ( Xj = S γ ,µ S γ ,µ )−1 , Y S γ ,µ Y S γ ,µ )−1 . X j ( j
(65)
Spectral and regularity properties of a pseudo-differential calculus
29
(the proof is purely computational and is therefore omitted). These formulae suggest and (γ , µ) ∈ R2 such that γ µ = 0 the following definition: for any LW operator A we set γ ,µ = S γ ,µ A( A S γ ,µ )−1 .
(66)
We have: γ ,µ : S(R2n ) −→ S (R2n ) is the Proposition 31 (i) The contravariant symbol of A function a γ ,µ (x, y; px , p y ) = a( γ2 x −
1 µ
p y , µ2 y +
1 γ
px )
(67)
where a is the contravariant symbol of A. (ii) We have γ ,µ = A
1 2π h¯
n
γ ,µ (z)dz aσγ ,µ (z)T
(68)
R2n γ ,µ
(Bochner integral) where aσ by
γ ,µ (z) is the unitary operator defined = S γ ,µ aσ and T
γ ,µ (z 0 )(z) = e− T
iγ µ 2h¯ σ (z,z 0 )
(z − z 0 ).
(69)
Proof Formula (67) follows from the symplectic covariance (30) of Weyl calculus taking (41) into account. Formula (68) follows, by a change of variables in definition (32) of A. The following intertwining result is a straightforward consequence of Proposition 14: γ ,µ
S γ ,µ Uφ is an isometry of L 2 (Rn ) onto the Corollary 32 (i) The mapping Uφ = γ ,µ γ ,µ closed subspace Hφ = S Hφ of L 2 (R2n ); explicitly γ ,µ
Uφ ψ(z) =
π γ µh¯ 2
n/2
W (ψ, φ)( 21 γ x, 21 µy).
(70)
γ ,µ satisfies the intertwining formula (ii) The operator A γ ,µ U γ ,µ = U γ ,µ A A φ φ γ ,µ
γ ,µ
with Uφ
= S γ ,µ Uφ .
(71) γ ,µ
Proof (i) Uφ is the compose of two isometries hence an isometry. Hφ is closed because Hφ is, and S γ ,µ is an isomorphism L 2 (Rn ) −→ L 2 (Rn ). (ii) Formula (71) γ ,µ and U γ ,µ and the second intertwinimmediately follows from the definitions of A φ ing formula (53).
30
M. de Gosson, F. Luef
7.2 The Moyal product and deformation quantization The Moyal product plays a central role in deformation quantization of Flato and Sternheimer [1,2]. Let H be a Hamiltonian function and assume that ∈ S(R2n ); the Moyal product [1,2] H h¯ is defined by (H h¯ )(z) =
1 4π h¯
2n
i
e 2h¯ σ (u,v) H (z + 21 u)(z − 21 v)dudv;
(72)
R2n ×R2n
when h¯ = 1/2π it reduces to the twisted product # familiar from standard Weyl calculus: H 1/2π = H #. We claim that 2,1 = H (x − 1 i h¯ ∂ p , p + 1 i h¯ ∂x ). H h¯ = H 2 2
(73)
The proof is similar to that of Theorem 6. Let us view −→ H h¯ as a Weyl operator, denoted by H h¯ . Using formula (72) the distributional kernel of H h¯ is given by K H h¯ (z, y) =
1 2π h¯
2n
i
e h¯ σ (u,z−y) H (z − 21 u)du
(74)
R2n
hence, using (22) and the Fourier inversion formula, the contravariant symbol of H h¯ is given by H(z, ζ ) =
i
e− h¯ ζ ·η K H h¯ (z + 21 η, z − 21 η)dη.
R2n
Using (74) and performing the change of variables u = 2z + η − z we get K H h¯ (z +
1 2 η, z
−
1 2 η)dη
=
1 2π h¯
2n
e
2i h¯
σ (z,η)
i
e h¯ σ (η,z ) H ( 21 z )dz ;
R2n
setting H ( 21 z ) = H1/2 (z ) the integral is (2π h¯ )n times the symplectic Fourier transform Fσh¯ H1/2 (−η) = (H1/2 )σ (−η) so that H( 21 z, ζ )
= =
1 2π h¯
n
i
i
e− h¯ ζ ·η e h¯ σ (z,η) (H1/2 )σ (−η)dη
R2n
1 2π h¯
n
R2n
i
e− h¯ σ (z+J ζ,η) (H1/2 )σ (η)dη
Spectral and regularity properties of a pseudo-differential calculus
31
Since the second equality is the inverse symplectic Fourier transform of (H1/2 )σ calculated at z + J ζ we finally get H(z, ζ ) = H (x + 21 ζ p , p − 21 ζx )
(75)
with ζ = (ζx , ζ p ). An immediate consequence of these results is: Proposition 33 The isometries Uφ2,1 defined by the formula Uφ2,1 ψ(z) = (π h¯ )n/2 W (ψ, φ)(z)
(76)
satisfy the intertwining relation . (H h¯ )Uφ2,1 = Uφ2,1 H Proof Formula (76) is just (70) with γ = 2, µ = 1.
8 Concluding remarks We note that Iftimie et al. [44,51,52,71] have also worked on generalizations of the magnetic operators; also see Karasev and Osborn [46]. It would certainly be interesting to compare our results to those obtained by these authors. Due to limitations of length and time we have only been able to give a few applications of the theory of modulation spaces to the Landau–Weyl calculus. Modulation spaces and related topics shave turned out to be the proper setting for the discussion of pseudodifferential operators in the last decade, see for instance the papers [32,37] and the references therein; also see [5,28,47–49,64–66]. For related topics such as the spaces q (L p ) see for instance the work of Birman and Solomjak [3], Christ and Kiselev [6] or Simon [57]. Recently modulation spaces have also found various applications in the study of Schrödinger operators [9,10]. In a forthcoming work we will extend the methods described in this paper to problems coming from “noncommutative mechanics”, which is a generalization of standard quantum mechanics. References 1. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. I. Deformation of symplectic structures. Ann. Phys. 111, 6–110 (1978) 2. Bayen, F., Flato, M., Fronsdal, C., Lichnerowicz, A., Sternheimer, D.: Deformation theory and quantization. II. Phys. Appl. 110, 111–151 (1978) 3. Birman, M.Sh., Solomjak, M.Z.: Spectral theory of selfadjoint operators in Hilbert space, mathematics and its applications (Soviet Series). Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. Reidel, Dordrecht (1987) 4. Boggiatto, P., Buzano, E., Rodino, L.: Global hypoellipticity and spectral theory. In: Mathematical Research, vol. 92. Akademie Verlag, Berlin (1996)
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