Generalized Bargmann functions, their growth and von Neumann lattices

0.15 MB·English

by  A. Vourdas

Generalized Bargmann functions, their growth and von Neumann
lattices
A. Vourdas 1,3, K. A. Penson 2, G. H. E. Duchamp 3, and A. I. Solomon 2,4
1 Department of Computing, University of Bradford, Bradford BD7 1DP, UK
2 Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee,
Universit´e Pierre and Marie Curie, CNRS UMR 7600, Tour 13 5-i`eme et.,
2
B.C. 121, 4 place Jussieu, F 75252 Paris Cedex 05, France
1 3 Universit´e Paris 13, LIPN, Institut Galil´ee, CNRS UMR 7030,
0
99 Av.J.B. Clement, F 93430 Villetaneuse, France
2
4 The Open University, Physics and Astronomy Department, Milton Keynes MK7 6AA, UK

n
a GeneralizedBargmannrepresentationswhicharebasedongeneralizedcoherentstatesareconsid-
J ered. The growth of the corresponding analytic functions in the complex plane is studied. Results

7 abouttheovercompletenessorundercompletenessofdiscretesetsofthesegeneralizedcoherentstates
2 are given. Several examples are discussed in detail.

]
h
I. INTRODUCTION
p
-
t
n TheBargmannrepresentation[1]representsquantumstatesbyanalyticfunctionsinthecomplex
a plane. This allows the powerful theory of analytic functions to be used in a quantum mechanical
u
context. An example of a result which can be proved only by the use of the theory of analytic
q
functions is related to the overcompleteness or undercompleteness of discrete sets of coherent
[
states, e.g. the von Neumann lattice of coherent states [2–15]. This is based on deep theorems

2 relating the growth of analytic functions and the density of their zeros[16–18]. The study of the
v Bargmann representation determines the nature of admissible functions belonging to the Hilbert
9
space. Furthermore the study of the paths of the zeros of the Bargmann functions under time
8
evolution has led to important physical insight for many systems [19–27].
8
Many generalizations of coherent states [28–30] have been considered in the literature. In the
0
. presentpaperweconsiderthegeneralizationsstudiedin[31–37]whichhaveledtomanyinteresting
1
sets of generalized coherent states. We use them to define generalized Bargmann representations,
1
which represent the various quantum states by analytic functions in the complex plane. The
1
1 requirementofconvergenceforthescalarproductintheserepresentationsdeterminesthemaximum
: growth of the generalized Bargmann functions and defines the corresponding Bargmann spaces.
v
Theorems that relate the growthof analytic functions in the complex plane to the density of their
i
X zeros lead to results about the overcompleteness or undercompleteness of discrete sets of these
r generalizedcoherentstates. Thegeneraltheoryisappliedtofourexamples: thestandardcoherent
a statesandthree examplesofgeneralizedcoherentstates. Thereforewestudyexplicitly three novel
types of generalized Bargmann representations.
In sectionII we briefly review knownresults on the growthof analytic functions in the complex
plane, and the relation to the density of their zeros. In section III we define generalized coherent
states. InsectionIVweintroducegeneralizedBargmannfunctionsandstudytheirgrowth. Wealso
show that a discrete set ofthese generalizedcoherentstates is overcomplete[resp. undercomplete]
if its density is greater [resp. smaller] than a critical density. In section V we discuss explicitly
2
four examples. We conclude in section VI with a discussion of our results.
II. GROWTH OF ANALYTIC FUNCTIONS IN THE COMPLEX PLANE AND THE
DENSITY OF THEIR ZEROS
Analyticfunctionsarecharacterizedbytheirgrowthandthedensityoftheirzerosandwebriefly
summarize these concepts [16–18]. Let M(R) be the maximum modulus of an analytic function
f(z)for|z|=R. Itsgrowthisdescribedbytheorderrandthetypes,whicharedefinedasfollows:
lnlnM(R) lnM(R)
r = lim sup ; s= lim sup . (1)
R→∞ lnR R→∞ Rr
These definitions imply that M(R) ∼ exp(sRr) as R goes to infinity (here the ∼ indicates that
M(R) is log-asymptotic to exp(sRr)).
Definition II.1.
(1) B(r,s) is the set of analytic functions in the complex plane with order smaller than r, and
also functions with order r and type smaller or equal to s.
(2) B (r,s) is the set of analytic functions in the complex plane with order smaller than r, and
1
also functions with order r and type smaller than s.
B(r,s)−B (r,s) is the set of analytic functions with order r and type s.
1
We next consider the sequence ζ ,...,ζ ,... where 0 < |ζ | ≤ |ζ | ≤ .... and the limit is infinite
1 N 1 2
(as N goes to infinity). We denote by n(R) the number of terms of this sequence within the circle
|z|<R. The density of this sequence is described by the numbers
lnn(R) n(R) n(R)
t= lim sup ; δ = lim sup ; δ = lim inf . (2)
R→∞ lnR R→∞ Rt R→∞ Rt
In the cases considered in this paper the lim n(R) exists and therefore δ = δ. Below we will
R→∞ Rt
use the simpler notation δ =δ =δ. These definitions imply that asymptotically n(R)∼δRt as R
goes to infinity.
Definition II.2. Wesaythatthedensity(t,δ)ofasequenceissmallerthan(t ,δ )andwedenote
1 1
this by (t,δ)≺(t ,δ ) if t<t and also if t=t and δ <δ (lexicographic order).
1 1 1 1 1
Remark II.3. Thedensitydependsonlyontheabsolutevaluesofζ ,i.e. thesequencesζ ,...,ζ ,...
n 1 N
and ζ exp(iθ ),...,ζ exp(iθ ),..., where θ are arbitrary real numbers, have the same density.
1 1 N N N
Also if we add or subtract a finite number of complex numbers in a sequence, its density remains
the same.
An example of a sequence which has density (t,δ) is
1 N
ζ =exp ln +iθ (3)
N N
(cid:20)t (cid:18) δ (cid:19) (cid:21)
3
where θ are arbitrary real numbers.
N
Therelationshipbetweenthegrowthofananalyticfunctioninthecomplexplaneandthedensity
of its zeros is described through the following inequalities [16–18]:
t≤r; sr ≥δ. (4)
III. GENERALIZED COHERENT STATES
Let H be the Hilbert space corresponding to a Hamiltonian operator h and |ni its number
eigenstates. Following [31–37], we consider the generalized coherent states
∞ zn ∞ |z|2n
|z;ρi=[N (|z|2)]−1/2 |ni; N (|z|2)= . (5)
ρ ρ(n)1/2 ρ ρ(n)
nX=0 nX=0
Here ρ(n) is a positive function of n with ρ(0) = 1. The series in the normalization constant
N (|z|2) converges within some disc |z| < R ≤ ∞. In this paper we only consider cases
ρ
where ρ(n) is increasing fast enough as a function of n, so that this series converges
in the whole complex plane.
The overlapof two generalized coherent states is
∞ (ζz)n
hz′;ρ|z;ρi=[N (|z′|2)]−1/2[N (|z|2)]−1/2K (z′∗,z); K (ζ,z)= . (6)
ρ ρ ρ ρ
ρ(n)
nX=0
Here K (ζ,z) is the reproducing kernel. Clearly K (z∗,z)=N (|z|2).
ρ ρ ρ
Thechoiceρ(n)=n!isanexample,andleadstothestandardcoherentstates. Forρ(n)=n!the
overlap hz′;ρ|z;ρi is everywhere different from zero. For other ρ(n) the ovelap might have zeros.
The resolution of the identity in terms of the generalized coherent states is a weak operator
equality given by
d2zW (|z|2)|z;ρihz;ρ|=1, (7)
ρ
ZC
f
where W (x)>0 is a function such that
ρ
f ∞ πW (x)
dxxnW (x)=ρ(n); W (x)= ρ . (8)
ρ ρ
Z N (x)
0 fρ
We stress that for arbitrary ρ(n) the W (x) does not always exist.
ρ
Thepropertyoftemporalstabilitystatesthatifweactwiththeevolutionoperatorcorresponding
f
to a certain Hamiltonian h on the generalized coherent states, we get other coherent states. Our
generalized coherent states have this property:
exp(iτh)|z;ρi=|zexp(iτω);ρi. (9)
For a study of states having the above property, see[38].
4
IV. GENERALIZED BARGMANN FUNCTIONS
Let |fi be an arbitrary state in the Hilbert space H:
∞ ∞
|fi= f |ni; |f |2 =1. (10)
n n
nX=0 nX=0
We represent this state by the following analytic function in the complex plane:
∞ f zn
F(z;ρ)=[N (|z|2)]1/2hz∗;ρ|fi= n . (11)
ρ [ρ(n)]1/2
nX=0
For ρ(n) =n! this is the standard Bargmann function. But other choices of ρ(n) lead to general-
izations of the Bargmann function.
If the resolution of the identity in Eq.(7) exists, then it leads to the following expression for
the scalar product of two states |fi and |gi represented by the functions F(z;ρ) and G(z;ρ),
correspondingly:
d2z ∞
hg|fi=(G,F)= W (|z|2)[G(z;ρ)]∗F(z;ρ)= g∗f . (12)
ZC π ρ nX=0 n n
It is known that a pointwise bound for F(z;ρ) is |F(z;ρ)|2 ≤K (z∗,z)(F,F) [39].
ρ
As an example, we consider the generalized coherent state |ζ;ρi which is represented by the
generalized Bargmann function
F (z;ρ)=[N (|ζ|2)]−1/2K (ζ,z). (13)
coh ρ ρ
Definition IV.1. TheBargmannspaceB(W )consistsofanalyticfunctionsinthecomplexplane
ρ
F(z;ρ), such that (F,F) is finite, with the scalar product given by Eq.(12).
Let S be a set of quantum states in the space H. S is called a total set in H, if there exists no
state |si in H which is orthogonalto all states in S.
S is called undercomplete in H, if there exists a state |si in H which is orthogonalto all states
in S.
A total set S in H is called overcomplete in H, if there exist at least one state |ui in S such
that the S−{|ui} is also a total set.
A total set S in H is called complete if every proper subset is not a total set. Below we will
mainly use the terms overcomplete, complete and undercomplete.
Proposition IV.2.
(1) If
W (|z|2)∼exp −2b(ρ)|z|a(ρ) (14)
ρ
h i
5
as |z|→∞, then the set of functions in the generalized Bargmann space B(W ) satisfies the
ρ
B [a(ρ),b(ρ)]⊂B(W )⊂B[a(ρ),b(ρ)]. (15)
1 ρ
Functions with order of growth a(ρ) and type b(ρ), might or might not belong to B(W ).
ρ
(2) Let{z }beasequenceofcomplexnumberswithdensity(t,δ). Thesetofgeneralizedcoherent
N
states |z ;ρi is overcomplete [resp. undercomplete] when (t,δ) ≻ [a(ρ),b(ρ)a(ρ)] [resp. t <
N
a(ρ)].
Proof.
(1) The integral in Eq.(12) diverges for functions with order of growth greater than a(ρ), or for
functionswithorderequaltoa(ρ)andtypegreaterthanb(ρ). Thereforethefunctions inthe
generalized Bargmann space B(W ), belong to the set B[a(ρ),b(ρ)].
ρ
Alsoif the functions haveorderless thana(ρ)or orderequalto a(ρ)andtype less thanb(ρ),
the integralin Eq.(12) converges. Therefore functions in the set B [a(ρ),b(ρ)] belong to the
1
generalized Bargmann space B(W ).
ρ
We next show with examples,that functions inB[a(ρ),b(ρ)]−B [a(ρ),b(ρ)]might ormight
1
not belong to B(W ). We consider the special case W (|z|2) = exp(−2λ|z|2), i.e., a(ρ) = 2
ρ ρ
and b(ρ)=λ>0. We also consider the functions
exp(λz2)−1
F (z;ρ)= ; F (z;ρ)=znexp(λz2). (16)
1 2
z
They both have growthwith order 2 and type λ and belong to the space B(2,λ)−B (2,λ).
1
The integral of Eq.(12) converges with the first function and diverges with the second func-
tion. Thereforethefirstfunctionbelongsto B(W ),andthe secondfunctiondoesnotbelong
ρ
to it.
(2) It follows from Eq.(11) that if F(ζ;ρ)=0 then the corresponding state |fi is orthogonal to
the generalized coherent state |ζ∗;ρi.
We consider a set of generalized coherent states |z ;ρi with density (t,δ)≻[a(ρ),b(ρ)a(ρ)].
N
If this is not a total set then there exists a function F(z;ρ) in the set B[a(ρ),b(ρ)], which is
equal to zero for all z . But this is not possible because it violates the relations in Eq.(4).
N
Therefore the set of states |z ;ρi is a total set in the space H. The same result is also
N
true if we subtract a finite number of terms from the sequence {z }. Therefore the set of
N
generalized coherent states |z ;ρi is overcomplete.
N
Wenextconsiderasetofgeneralizedcoherentstates|z ;ρiwithdensityofthecorresponding
N
sequence t < a(ρ). In this case we can construct a state which is orthogonal to all |z ;ρi
N
[13, 15]. We use the Hadamard theorem[16–18] and consider the analytic function
F(z,ρ)=P(z)exp[Q (z)], (17)
q
where
∞
P(z)=zm E(z ,p), (18)
N
NY=1
6
z
E(A ,0)=1− , (19)
N
z
N
z z z2 zp
E(A ,p)=(1− )exp + +...+ . (20)
N z (cid:20)z 2z2 pzp (cid:21)
N N N N
Here E(A ,p) are the Weierstrass factors, Q (z) is a polynomial of degree q, and p is an
N q
integer. The maximum of (p,q) is called the genus of F(z,ρ) and does not exceed its order.
m is a non-negative integer and it is the multiplicity of the zero at the origin. The z are
N
clearly zeros of the F(z,ρ) in Eq.(17).
It remains to show that for some Q (z) this function belongs to the generalized Bargmann
q
spaceB(W ). WetakeQ (z)=0andthentheorderofthegrowthofF(z,ρ)=P(z)isequal
ρ q
tot(theorem7inp.16in[17]). Thereforeift<a(ρ)thisfunctionisindeedinthegeneralized
BargmannspaceB(W ),andthecorrespondingstateisorthogonaltoallgeneralizedcoherent
ρ
states |z ;ρi. Consequently the set of these coherent states is undercomplete.
N
Remark IV.3. In the ‘boundary case’ where the density of the sequence {z } has t = a(ρ) and
n
δ ≤b(ρ)a(ρ), we can not state general results. We mention some known results for special cases.
For example, when t =a(ρ) is non-integral, we consider the function F(z;ρ)=P(z), as we did
above. This function hasgrowthwith ordert andits type s satisfiess≤Cδ where C is a constant
thatdependsontheordert=a(ρ)(p.32in[18]). Inthiscaseforsequences{z }withδ <bC−1 the
n
function F(z;ρ) = P(z) has growth less than (a(ρ),b(ρ)) and it belongs to the Bargmann space.
Therefore a set of generalized coherent states |z ;ρi with density of the corresponding sequence
N
(t,δ) where t is non-integral, is undercomplete if δ is smaller than a critical value. This result is
not valid for integral t(p.32 in [18]).
Another example, is the set of standard coherent states which on a von Neumann lattice with
A = π is known to be overcomplete by one state [29]. If we subtract a finite number of coherent
states we get an undercomplete set of standard coherent states which is described by the same
density. This example shows two sequences with the same density (in the ‘boundary case’), with
corresponding coherent states forming an overcomplete and an undercomplete set.
V. EXAMPLES
A. ρ (n)=n! : standard coherent states
0
For the standard coherent states ρ (n)=n! we have
0
N (|z|2)=exp |z|2 ; W (|z|2)=exp(−|z|2). (21)
ρ0 ρ0
(cid:0) (cid:1)
Therefore a(ρ ) = 2 and b(ρ ) = 1/2. In this case B (2,1/2) ⊂ B(W ) ⊂ B(2,1/2). The set of
0 0 1 ρ
coherentstates |z ;ρ i is overcompleteor undercomplete in the cases that the density (t,δ) of the
n 0
sequence {z } is (t,δ)≻(2,1) or t<2, correspondingly.
n
7
AnexampleistherectangularvonNeumannlatticez =A(N+iM)whereN,M areintegers
NM
and A is the area of each cell. In this case n(R) = πR2/A and the density is described by t = 2
and δ = π/A. Our results show that the set of coherent states {|z ;ρ i} is overcomplete for
NM 0
A<π. For this particular example, it is also known[29] that it is undercomplete for A>π.
We have explained in remark II.3 that instead of the lattice z = A(N +iM) we can also
NM
use the complex numbers z =A(N +iM)exp(iθ ) with arbitraryphases θ . The angular
NM NM NM
distribution of the zeros is totally irrelevant.
Another example is the sequence
1
ζ =exp ln(2N)+iθ (22)
N N
(cid:20)t (cid:21)
whereθ arearbitraryrealnumbers. We haveseeninEq.(3)thatitsdensityis (t,1/2). Therefore
N
the setofthecorrespondingcoherentstatesis overcompletefort>2andundercompletefort<2.
Also the sequence
1 N
ζ =exp ln +iθ (23)
N N
(cid:20)2 (cid:18) δ (cid:19) (cid:21)
where θ are arbitrary real numbers, has density (2,δ). Therefore the set of the corresponding
N
coherent states is overcomplete for δ >1.
B. ρ (n)=(n!)2
1
We consider the case that ρ (n)=(n!)2. It was proved in [37] that
1
N (|z|2)=I (2|z|); W (|z|2)=2K (2|z|) (24)
ρ1 0 ρ1 0
where I and K are modified Bessel functions of first and second kind, respectively. But as
0 0
|z|→∞
π 1/2
2K (2|z|)∼ exp(−2|z|) (25)
0
(cid:18)|z|(cid:19)
and therefore a(ρ )=1 and b(ρ )=1. In this case B (1,1)⊂B(W )⊂B(1,1).
1 1 1 ρ
The set of coherent states |z ;ρ i is overcomplete [resp, undercomplete] when the density (t,δ)
N 1
of the sequence {z } is (t,δ)≻(1,1) [resp. t<1].
N
An example is the one-dimensional lattice z = ℓNexp(iθ ), where N is an integer and θ
N N N
are arbitrary phases. In this case n(R)=2R/ℓ and the density is described by t=1 and δ =2/ℓ.
Therefore the set of coherent states {|z ;ρ i} is overcomplete for ℓ<2.
N 1
8
C. ρ (n)= (n!)3π1/2
2 2Γ(n+3/2)
We consider the case ρ (n) = (n!)3π1/2 , where Γ denotes the Gamma function, and for which
2 2Γ(n+3/2)
[37]
N (|z|2)=[I (|z|)]2+2|z|I (|z|)I (|z|); W (|z|2)=[K (|z|)]2 (26)
ρ2 0 0 1 ρ2 0
But as |z|→∞
π
[K (|z|)]2 ∼ exp(−2|z|) (27)
0
(cid:18)2|z|(cid:19)
Therefore a(ρ )=1 and b(ρ )=1. In this case B (1,1)⊂B(W )⊂B(1,1).
2 2 1 ρ
Therefore our conclusions about the overcompleteness or undercompleteness of the coherent
states |z ;ρ i are also valid for the coherent states here. In connection with this it is interesting
N 2
to compare the growth of ρ(n) in these two cases, as n → ∞. We use the formula [40]
Γ(z+a)
lim exp(−alnz)=1 (28)
|z|→∞ Γ(z)
and from this we conclude that as n → ∞
ρ (n) π1/2 n! π1/2 1
2
= ∼ exp − lnn . (29)
ρ (n) 2 Γ(n+3/2) 2 (cid:18) 2 (cid:19)
1
Thereforethe ρ (n)consideredin this subsectiongrowsmore slowlywithn thanthe ρ (n)consid-
2 1
ered in the previous subsection.
D. ρ (n)= Γ(αn+β)
3 Γ(β)
We consider the case
Γ(αn+β)
ρ (n)= ; α,β >0. (30)
3
Γ(β)
It was proved in [34] that
2(β−α)
Nρ3(|z|2)=Γ(β)Eα,β(|z|2); Wρ3(|z|2)= |zα|Γ(αβ) exp −|z|α2 (31)
(cid:16) (cid:17)
whereE (y)aregeneralizedMittag-Lefflerfunctions[34,41]. Thereforea(ρ )=α−1 andb(ρ )=
α,β 3 3
1. In this case B (α−1,1)⊂B(W )⊂B(α−1,1).
1 ρ
9
Thesetofcoherentstates|z ;ρ iisovercompleteorundercompleteinthecasesthatthedensity
N 3
(t,δ) of the sequence {z } is (t,δ) ≻ (α−1,α−1) or t < α−1, correspondingly. For example , the
N
sequence
ζ =exp[sln(N)+iθ ] (32)
N N
where θ are arbitrary real numbers has density (s−1,1) and the set of corresponding coherent
N
states is overcomplete [resp. undercomplete] when s < α or s = α > 1 [resp. s > α]. Also, the
sequence
N
ζ =exp αln +iθ (33)
N N
(cid:20) (cid:18) δ (cid:19) (cid:21)
where θ are arbitrary real numbers has density (α−1,δ). Therefore the set of corresponding
N
coherent states is overcomplete when δ >α−1.
VI. DISCUSSION
We have used the generalizedcoherentstates studied in [31–37] to define generalizedBargmann
analytic representations in the complex plane. For these states the scalar product in Eq.(12)
convergesandthisdeterminesthegrowthofthegeneralizedBargmannfunctions. Fromthegrowth
we infer the maximum density of the zeros of these functions and this in turn determines the
overcompleteness or undercompleteness of discrete sets of these generalized coherent states. In
addition to the standard coherent states, we studied three examples in detail in subsections VB,
VC and VD. Other examples in the same spirit can also be found.
Theworkprovidesadeeperinsightintotheuseofthetheoryofanalyticfunctionsinaquantum
mechanical context.
Acknowledgement: WethanktheAgenceNationaledelaRecherche(ANR)forsupportunder
the programme PHYSCOMB No ANR-08-BLAN-0243-2.
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