GABOR FRAMES, DISPLACED STATES AND THE LANDAU LEVELS: A TOUR IN POLYANALYTIC FOCK SPACES LUÍS DANIEL ABREU Abstract. We will observe that several results concerning Gabor frames, Euclidean Landau levels and displaced Fock states, can be translated to the setting of polyanalytic Fock spaces. This will allow several observations. For instance, in the higher Landau level eigenspaces and in displaced Fock states there is an analogue of Perelomov results on completeness of systems of coherent states, which is equivalent to the completeness of Gabor systems with Hermite functions. Another observation is that the parameter m in the weight 2 function e mjzj of polyanalytic Fock spaces corresponds to the strengh of the magnetic …eld of the Schrödinger operator which leads to the Landau Laplacian. We will motivate our presentation with an analogy to a very simple multiplexing problem from signal analysis.
1. Introduction The solutions of the Cauchy-Riemann equation 1 d F (z) = dz 2
@ @ +i @x @
F (x + i ) = 0,
known as analytic functions, provide one of the most well studied and useful objects in Mathematics. The analiticity restriction is so important that non-analytic functions are often seen as “bad objects” and therefore not worthy of further study. However, there are intermediate classes of non-analytic functions which posses signi…cant structure. They are called polyanalytic functions. A function F (z); de…ned on a subset of C, is said to be polyanalytic of order n 1 if it satis…es the generalized Cauchy-Riemann equations (1.1)
d dz
n
F (z) =
1 2n
@ @ +i @x @
n
F (x + i ) = 0.
This is equivalent to saying that F (z) is a polynomial of order n functions f'k (z)gnk=01 as coe¢ cients: (1.2)
F (z) =
n 1 X
1 in z with analytic
z k 'k (z)
k=0
Date: October 31, 2011. Partially supported by the ESF activity “Harmonic and Complex Analysis and its Applications”, by FCT project PTDC/MAT/114394/2009, and Austrian Science Foundation (FWF) project “Frames and Harmonic Analysis”. 1
2
LUÍS DANIEL ABREU
To gain insight from a simple example, we consider F (z) = 1 jzj2 = 1 zz and observe that 2 d d F (z) = z; while F (z) = 0. dz dz As a result, F (z) is not analytic in z, but is polyanalytic with n = 2. This simple example highlights one of the reasons why the properties of polyanalytic functions can be rather intricated when compared with those of analytic functions: they can vanish on closed curves without vanishing identically, while analytic functions can not even vanish on a accumulation set of the complex plane! Still, many properties of analytic functions have found an extension to polyanalytic functions, often in a nontrivial form. The function theoretical aspects of polyanalytic functions have been investigated thoroughly, notably by the Russian school led by Balk [4]. More recently the subject gained a renewed interest within operator theory [5], [18]. Our investigations in the topic were originally motivated by applications in signal analysis, in particular by the results of Gröchenig and Lyubarskii on Gabor frames with Hermite functions [10],[11] but soon it was clear that Hilbert spaces of polyanalytic functions lie at the heart of several interesting mathematical topics and that they provide an explicit representation of the eigenspaces of the Landau levels. We have organized the paper as follows. We start with a section on the Hilbert space theory of polyanalytic Fock spaces. The third section explains the connections to the theory of Gabor frames with Hermite functions. We quote some applications in Quantum Physics in section 4, namely the interpretation of the so-called true polyanalytic Fock spaces as the eigenspaces of the Euclidean Landau Hamiltonian with a constant magnetic …eld. In section 5 we take a close look to the reproducing kernels and some asymptotic results recently obtained in the study of random matrices. 2. Fock spaces of polyanalytic functions 2.1. The orthogonal decomposition. Write L2 (C) to denote the Hilbert space of all measurable functions equiped with the norm Z 2 2 kF kL2 (C) = jF (z)j2 e jzj d (z), C
where d (z) stands for area measure on C. If we require the elements of the space to be analytic, we are lead to the Fock space F2 (C): Polyanalytic Fock spaces Fn2 (C) arise in an analogous manner, by requiring its elements to be polyanalytic of order n 1. They seem to have been …rst considered by Balk [4, pag. 170] and, more recently, by Vasilevski [18], who obtained the following decompositions in terms of spaces F2n (C) which he called true poly-Fock spaces: (2.1)
Fn2 (C) = F21 (C)
:::
F2n (C):
A TOUR IN POLYANALYTIC FOCK SPACES
L2 (C) =
1 M n=1
3
F2n (C):
F2n (C)
We will use the following de…nition of which is equivalent to the one given by Vasilevski: a function F belongs to the true polyanalytic Fock space F2n+1 (C) if kF kL2 (C) < 1 and there exists an entire function H such that 1 nh i n 2 2 d jzj2 (2.2) F (z) = e e jzj H(z) : n! dz With this de…nition it is easy to verify that the spaces F2n (C) are orthogonal using Green´s formula.
2.2. The multiplexing problem. A classical problem in the Theory of Signals is the one of Multiplexing, that is, transmitting several signals over a single channel in such a way that it is possible to recover the original signal at the receiver. The orthogonal decomposition (2.1) can be used for this purpose, once we construct a map B n sending an arbitrary f 2 L2 (R) to the space F2n (C). Then we can, at least theoretically, proceed as follows.
(1) Given n signals f1 ; : : : ; fn , with …nite energy (fk 2 L2 (R) for every k), process each individual signal by evaluating B k fk . This encodes each signal into one of the n orthogonal spaces F 1 (C); : : : ; F n (C). (2) Construct a new signal F = Bf = B 1 f1 + ::: + B n fn as a superposition of the n processed signals. (3) Sample, transmit, or process F: (4) Let P k denote the orthogonal projection from Fn2 (C) onto F k (C), then P k (F ) = B k fk by virtue of (2.1). (5) Finally, after inverting each of the transforms B k , we recover each component fk in its original form.
The combination of n independent signals into a single signal Bn f and the subsequent processing provides our multiplexing device. With two signals this can be outlined in the following scheme. f1 !
Bf1
f2 ! B 2 f2
& %
1
Bf1 + B 2 f2 = Bf
P % P2 &
Bf1
! f1
B 2 f2 ! f2
We will use the above scheme as a source of ideas for our results. With some poetic license, we may consider that we apply signal analysis to mathematics.
4
LUÍS DANIEL ABREU
2.3. The polyanalytic Bargmann transform. The construction of the map B k above can be done as follows. To map the …rst signal f1 2 L2 (R) to the space F21 (C) = F2 (C) R 2 2 we can of course use the good old Bargmann transform Bf (z) = R f (t)e2 tz z 2 t dt. The remaining signals are mapped using 1 2
k
B
k+1
f (z) =
e
k!
k
d dz
jzj2
h
jzj2
e
i Bf (z)
It can be proved that B k : L2 (R) ! F2k (C) is a Hilbert space isomorphism, by observing that the Hermite functions are mapped to ithe orthogonal basis fek;m : m 0g of F2k (C), h where ek;m (z) =
n
n!
1 2
e
jzj2
d k dz
jzj2 m
e
z
.
2.4. A polyanalytic Weierstrass function. In order to transmit the signal, we use the following analogue of the Whittaker-Shannon-Kotel´nikov sampling theorem. Let be the Weierstrass sigma function corresponding to de…ned by (z) = z
Y
z
1
e
z
+
z2 2 2
;
2 nf0g
To simplify our notations we will write the results in terms of the square lattice, = (Z+iZ) consisting of the points = l + i m, k; m 2 Z; but most of what we will say is also true for general lattices. To write down our explicit sampling formulas, the following polyanalytic extension of the Weierstrass sigma function is required: " # 1 n n+1 n 2 2 2( d (z)) S n+1 (z) = e jzj e jzj . n! dz n!z Clearly, S 1 (z) = (z)=z. Let (z) be the Weierstrass sigma function associated to the ad1 joint lattice = (Z + iZ) of and consider the corresponding “polyanalytic Weierstrass function”S n0 (z). With this terminology we have: 1 Theorem If 2 < n+1 , then every F 2 Fn+1 (C) can be written as: 2 F (z) =
X
F ( )e
j j2
z
Sn+1 0 (z
2 (Z+iZ)
P where Sn 0 (z) = nk=1 S k 0 (z): Combining this with the decomposition (2.1) gives: 1 Corollary If 2 < n+1 , every F 2 F2n+1 (C) can be written as: (2.3)
F (z) =
X
2 (Z+iZ)
F ( )e
z
j j2
S n+1 0 (z);
);
A TOUR IN POLYANALYTIC FOCK SPACES
5
3. The Gabor connection 3.1. The Gabor transform. The study of polyanalytic Fock spaces can be signi…cantly enriched via a connection to Gabor analysis. Recall that the Gabor transform of a function or distribution f with respect to a window function g is de…ned to be Z (3.1) Vg f (x; ) = f (t)g(t x)e 2 i t dt: R
Given a point
= ( 1;
2)
in phase-space R2 , the corresponding time-frequency shift is f (t) = e2
i
2t
f (t
1 ), 1
t 2 R.
2
If we choose the Gaussian function h0 (t) = 2 4 e t as a window in (3.1), then a simple calculation shows that the Bargmann transform is related to these special Gabor transforms as follows: (3.2)
Bf (z) = e
i x +
jzj2 2
Vh0 f (x;
). 2
n
2
e 2 t With a bit more e¤ort, we can choose the nth Hermite function hn (t) = cn e t dtd as a special window in (3.1), and …nd a similar relation between Gabor transforms with Hermite functions and true polyanalytic Bargmann transforms:n (3.3)
B n+1 f (z) = e
i x + 2 jzj2
Vhn f (x;
):
This formula connects polyanalytic Fock spaces with Gabor analysis. Using this connection it was possible to prove results that seemed hopeless using only complex variables. For instance, it was possible to prove that the sampling and interpolation lattices of Fn2 (C) can be characterized by their density. Previously, this result was known only for n = 1, the case where the functions are analytic. 3.2. Gabor expansions with Hermite functions. To give a more concrete idea of what we are talking about, let us see what Theorem 1 tells about Gabor expansions, more precisely 1 , then every about the required size of the square lattice. From Corollary 1, if 2 < n+1 F 2 F2 (C) can be written in the form (2.3). Now, applying the inverse Bargmann transform and doing some calculations involving the interwining property between the time-frequency shifts and the Fock shifts (see [2] for the details), one can see that this expansion is exactly 1 , every equivalent to the Gabor expansion of an L2 (R) function. More precisely, if 2 < n+1 f 2 L2 (R) admits the following representation as a Gabor series X (3.4) f (t) = ck;l e2 i lt hn (t k). l;k2Z
This sort of expansions are have been used before for practical purposes, for instance, in image analysis [9]. Their mathematical study ([10], [11], [1], [2], [12]) exposed a aesthetic
6
LUÍS DANIEL ABREU
blend of ideas from signal and complex analysis, leading to one of those scarse examples in mathematics. Stable Gabor expansions of the form (3.4) can be obtained from frame theory. For a countable subset 2 R2 , one says that the Gabor system G (hn ; ) = f hn : 2 g is a Gabor frame or Weyl-Heisenberg frame in L2 (R), whenever there exist constants A; B > 0 such that, for all f 2 L2 (R), (3.5)
A kf k2L2 (R)
X 2
2
hf;
hn iL2 (R)
B kf k2L2 (R) :
1 The …rst proof of the su¢ ciency of the condition 2 < n+1 for the expansion (3.4) is due to Gröchenig and Lyubarskii [10]. In the same paper, the authors provide some evidence to support the conjecture that the condition may even be sharp (it is known from a general result of Ramanathan and Steger [14] that 2 < 1 is necessary), a statement which would 1 be surprising, since 2 < n+1 is exactly the sampling rate necessary and su¢ cient for the expansion of n functions using the superframe (the superframe [11] is a vectorial version of frame which has be seen to be equivalent to sampling in the polyanalytic space [1]). The following problem seems to be quite hard. Problem. [10] Find the exact range of such that G (hn ; (Z + iZ)) is a frame. Recently, Lyubarskii and Nes [12] found that 2 = 53 > 12 is a su¢ cient condition for the case n = 1. They also proved that, if 2 = 1 1j , no odd function in the Feichtinger algebra [7] generates a Gabor frame. In [12], supported by their results and by some numerical evidence, the authors formulated a conjecture. Conjecture. [12] If 2 < 1 and 2 = 1 1j , then G (h1 ; (Z + iZ)) is a frame.
3.3. Sampling and Interpolation in Fn2 (C). We say that a set is a set of sampling for Fn2 (C) if there exist constants A; B > 0 such that, for all F 2 Fn2 (C), X 2 A kF k2Fn+1 (C) jF ( )j2 e j j B kF k2Fn+1 (C) : 2
2
2
A set is a set of interpolation for Fn2 (C) if for every sequence fai( ) g 2 l2 , we can …nd a 2 function F 2 Fn2 (C) such that ei 1 2 2 j j F ( ) = ai( ) , for every 2 . The sampling and interpolation lattices of Fn2 (C) can be characterized by their density. For the square lattice the results are as follows. 1 . Theorem. The lattice (Z + iZ) is a set of sampling for Fn2 (C) if and only if 2 < n+1 1 n 2 and it is a set of interpolation for F2 (C) if and only if > n+1 . So far, nobody has been able to …nd a proof of these results using only complex variables. The proof in [1] is based on the observation that, using the polyanalytic Bargmann transform, the sampling problem can be transformed in a problem about Gabor superframes with
A TOUR IN POLYANALYTIC FOCK SPACES
7
Hermite functions. Then, a remarkable structure result of Gabor analysis, the so called RonShen duality [16] transforms the problem in a problem about Riesz sequences, which can be further transformed in a problem about multiple interpolation in the Fock space which has been solved in [6]. The dual of this argument proves the second theorem. The characterization of the lattices yielding Gabor superframes with Hermite functions had been previously obtained by Gröchenig and Lyubarskii in [11], using the Wexler-Rax orthogonality relations. 4. The Quantum connection 4.1. The Landau levels. The motion of a charged particle in a constant uniform magnetic …eld in R2 is described by the Schrödinger operator HB =
1 (@x + iBy)2 + (@y 4
iBx)2
1 2
acting on L2 (R2 ). Here B > 0 is the strength of the magnetic …eld. Writing fz = e B2 jzj2 HB e
B jzj2 2
we obtain the following Laplacian on C (4.1)
fz =
d d d + Bz . dz dz dz
This Laplacian is positive and selfadjoint operator in the Hilbert space L2 (C) and the set fn; n 2 Z+ g can be shown to be the pure point spectrum of fz in L2 (C). The eigenspaces of fz ; are known as the Landau levels. In [3] the authors consider A2n;B (C) = fF 2 L2 (C) : ] z;B F = nf g,
and obtain an orthogonal basis for the spaces A2n;B . When B = we can use the results in [3] (comparing either the orthogonal basis or the reproducing kernels of both spaces) to see that (4.2)
A2m; (C) = F2n (C).
We now can apply the results about sampling in the true polyanalytic space to the subsystems of the states in each Landau Level. Using the true polyanalytic transform, the results about Gabor frames with Hermite function translate to sampling in true polyanalytic Fock: The lattice (Z + iZ) is a set of sampling for F2n (C) if and only G (hn ; (Z + iZ)) is a 1 Gabor frame. Thus, we conclude that, in particular, if 2 < n+1 , the subsystems of states constituted by the lattice (Z + iZ) are complete in the Landau levels. Now, take B = 1 and observe that d d fz = +z . dz dz
8
LUÍS DANIEL ABREU
This suggests us to consider the operators d d + z; a = , dz dz which are formally adjoint to each other and satisfy the commutation relations for the quantum mechanic creation and annihilation operators. Vasilevski [18, Theorem 2.9] proved that the operators s (k 1)! + l k jF2k (C) : F2k (C) ! F2l (C) a (l 1)! s (k 1)! l k a jF2k (C) : F2l (C) ! F2k (C) (l 1)! a+ =
are Hilbert spaces isomorphisms (and one is the inverse of the other). Given our identi…cation (4.2) we conclude that the operators a+ and a are, respectively, the raising and lowering operators between two di¤erent Landau levels. 4.2. Displaced Fock states. In [17], Wünsche derives the following representation for the displaced Fock states jz; n >:
( 1)n (4.3) jz; n >= p n! h i 2 2 d d Observing that ejzj dz e jzj F (z) = dz F (z)
d +z dz
jzj2
jz > :
zF (z), one realizes that (4.3) is essentially
the map T : F2 (C) ! F2n+1 (C) such that T : F (z) ! e
n
d dz
n
h
e
jzj2
i F (z) .
Thus, the displaced Fock states are also true polyanalytic Fock spaces. We can now use 1 then the subsystem of these coherGröchenig and Lyubarskii result to show that if 2 < n+1 ent states constituted by the square lattice on the plane is overcomplete. From Ramathan and Steeger general result [14], we know that if 2 > 1 they are not. This can be seen as analogues of Perelomov completeness result [13] in the setting of displaced Fock states. 5. Reproducing kernels The reproducing kernels of the polyanalytic Fock spaces have been computed using several di¤erent methods: invariance properties of the Landau laplacian fz [3], composition of unitary operators [18], Gabor transforms with Hermite functions [2], and the expansion in the kernel basis functions [8]. Nice formulas are obtained using the Laguerre polynomials Lk (x) =
k X i=0
( 1)i
k+ xi . k i i!
A TOUR IN POLYANALYTIC FOCK SPACES
9
The reproducing kernel of the space F2n (C), Kn (z; w), can be written as Kn (z; w) = L0n 1 ( jz This gives the explicit formula for the orthogonal projection P n required at the step 5. in our theoretical multiplexing device of section 2: Z n F (z) L0n 1 ( jz wj2 )e z(w z) d (z): (5.1) (P F )(w) = C
The reproducing kernel of the space Fn2 (C) is denoted by Kn (z; w): Using the formula Pn 1 +1 n L1n 1 ( jz wj2 )e zw . In [8], a variant of this k=0 Lk = Ln 1 , (2.1) gives K (z; w) = setting is used in the investigation of the polyanalytic Ginibre ensemble. The authors consider the space with reproducing kernel Knm (z; w) = mL1n 1 (m jz
wj2 )emzw
and the polynomial space P olm;n;k = spanfz j z l : 0
j
k
1; 0
l
n
1g:
Several interesting asymptotic results are obtained. For instance, denoting the reproducing kernel of P olm;n;k by Knm;k (z; w), it is proved that, if z; w 2 D, when m; k ! 1 with jm kj bounded and 1 jzwj > 0, then Knm;k (z; w) = Knm (z; w) + O(e
1 m 2 2
emjzwj ).
Remark. Comparing this set up with section 4.1, one recognizes the parameter m as the strengh of the magnetic …eld B. Therefore, the physical interpretation of the above limit m; k ! 1 consists of increasing the strength of the magnetic …eld and simultaneously the number of independent states in the system. Aknowledgement. The author wishes to thank Franz Luef and José Luis Romero for interesting discussions and comments on early versions of this paper. References [1] L. D. Abreu, Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions, Appl. Comp. Harm. Anal., 29 (2010), 287-302. [2] L. D. Abreu, K. Gröchenig, Banach Gabor frames with Hermite functions: polyanalytic functions from the Heisenberg group, Appl. Anal. DOI:10.1080/00036811.2011.584186. [3] N. Askour, A. Intissar, Z. Mouayn, Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 7, 707–712. [4] M. B. Balk, Polyanalytic Functions, Akad. Verlag, Berlin (1991). [5] H. Begehr, G. N. Hile, A hierarchy of integral operators. Rocky Mountain J. Math. 27 (1997), no. 3, 669–706. [6] S. Brekke and K. Seip. Density theorems for sampling and interpolation in the Bargmann-Fock space. III. Math. Scand., 73 (1993), 112–126. [7] H. G. Feichtinger, On a new Segal algebra. Monatsh. Math. 92 (1981), no. 4, 269–289. [8] A. Haimi, H. Hendenmalm, The polyanalytic Ginibre ensembles. preprint arXiv:1106.2975.
wj2 )e
zw
:
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LUÍS DANIEL ABREU
[9] I. Gertner, G. A. Geri, Image representation using Hermite functions, Biological Cybernetics, Vol. 71, 2 , 147-151, (1994). [10] K. Gröchenig, Y. Lyubarskii, Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007). [11] K. Gröchenig, Y. Lyubarskii, Gabor (Super)Frames with Hermite Functions, Math. Ann. , 345, no. 2, 267-286 (2009). [12] Y. Lyubarskii, P. G. Nes, Gabor frames with rational density, arXiv:1108.2684 (2011). [13] A. M. Perelomov, On the completeness of a system of coherent states, Theor. Math. Phys. 6, 156-164 (1971). [14] J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995), pp. 148–153. [15] A. K. Ramazanov, On the structure of spaces of polyanalytic functions. (Russian. Russian summary) Mat. Zametki 72 (2002), no. 5, 750–764; translation in Math. Notes 72 (2002), no. 5-6, 692–704. [16] A. Ron, Z. Shen, Weyl-Heisenberg frames and Riesz bases in L2 (Rd ), Duke Math. J. 89 (1997), 237–282. [17] A. Wünsche, Displaced Fock states and their connection to quasiprobabilities, Quantum Opt. 3 (1991) 359–383. [18] N. L. Vasilevski, Poly-Fock spaces, Di¤erential operators and related topics, Vol. I (Odessa, 1997), 371–386, Oper. Theory Adv. Appl., 117, Birkhäuser, Basel, (2000). E-mail address:
[email protected] NuHAG, Faculty of Mathematics University of Vienna, Austria. On leave from: Department of Mathematics of University of Coimbra, Portugal.
