Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems
J. R. Klauder*
Departments of Physics and Mathematics, University of Florida, Gainesville, Florida 32611
K. A Penson
†
and J.-M. Sixdeniers
‡
Laboratoire de Physique The ´orique des Liquides (CNRS-UMR 7600), Universite ´ Pierre et Marie Curie, Tour 16,
4 place Jussieu, 75252 Paris Cedex 05, France
Received 6 November 2000; published 13 June 2001
We construct a large number of coherent states satisfying the resolution of unity with a positive weight
function, obtained through analytic solutions of the Stieltjes moment problem coherent states on a plane and
the Hausdorff moment problem coherent states on a disk. These solutions are obtained through the method of
inverse Mellin transform. In addition, these coherent states induce a deformation of the metric that has been
calculated analytically.
DOI: 10.1103/PhysRevA.64.013817 PACS numbers: 42.50.Ar, 03.65.-w
I. INTRODUCTION
In this paper we shall expose a rather general method for
constructing coherent states, as defined according to a mini-
mal set of conditions, proposed by one of us 1,2. For con-
venience, we shall focus on holomorphic coherent states,
which up to normalization, are functions of a single complex
variable z. The ensemble of states | z labeled by the single
complex number z is called a set of coherent state if i | z is
normalizable, ii | z is continuous in the label z, i.e., | z
-z ' | →0 ⇒ | z -| z ' →0, and iii the states | z , z C,
form a complete in fact, an overcomplete set and that al-
lows a resolution of unity with the positive function W( | z |
2
)
completeness relation
C
d
2
z | z W | z |
2
z | =I =
n =0
| n n | , 1
where, in Eq. 1 I is the unit operator and | n is a set of
orthonormal eigenfunctions of a Hermitian operator H
ˆ
. As
already noted, without loss of generality, in Eq. 1 the states
| z are normalized to one. In Eq. 1 the integration is re-
stricted to the part of the complex plane where normalization
converges, see Eq. 3 below. The general method of con-
struction alluded to in the above consists of choosing a set of
strictly positive parameters ( n ), n =0,1, . . . , M , M ,
where (0) =1, such that the normalized state | z reads
| z =N
-1/2
| z |
2
n =0
z
n
n
| n , 2
where
N | z |
2
=
n =0
| z |
2 n
n
3
is the normalization, a convergent series in | z |
2
within the
radius of convergence | z |
2
R , 0 R , thus satisfying
condition i. While continuity in z is easily checked for the
form of Eq. 2, the condition 1 presents a severe restriction
on the choice of ( n )’s. In fact, only a relatively small num-
ber of distinct sets of ( n )’s is known, for which the func-
tion W( | z |
2
) can be extracted. As a result, the family of truly
coherent states is small in number. The standard example
leading to conventional coherent states is ( n ) =n ! 2. Re-
cently progress has been made in finding a resolution of
unity for selected choices of ( n ) 3–5. The physical moti-
vation behind the form of Eq. 2, is to propose a general
linear combination of basis states | n , whose coefficients
( n )
-1/2
are adapted to satisfy Eq. 1 and can be linked
to a specific Hamiltonian H
ˆ
H
ˆ
0
, where H
ˆ
0
is the linear
harmonic oscillator. As we will show in the following, there
exists only a very restricted set of families of ( n ) for which
the above requirements can be satisfied.
The idea of building coherent states through an appropri-
ate choice of ( n ) has been put forward in Refs. 6 and 7.
The states defined through Eq. 2 share for general ( n )
some universal features that we will enumerate now. For two
different complex numbers z and z ' the states | z and | z '
are, in general, not orthogonal and their overlap is given by
z | z ' =
N z * z '
N | z |
2
N | z ' |
2
1/2
, 4
where we have extended the definition of the normalization,
Eq. 3, to
N z * z ' =
n =0
z * z '
n
n
. 5
Whereas, through the positivity of ( n ), N( | z |
2
) is a strictly
increasing function of its argument, the overlap z | z ' is a
complex function of its arguments.
The continuity in label z follows from the continuity of
the overlap z | z ' through
*Email address: klauder@phys.ufl.edu
†
Email address: penson@lptl.jussieu.fr
‡
Email address: sixdeniers@lptl.jussieu.fr
PHYSICAL REVIEW A, VOLUME 64, 013817
1050-2947/2001/641/01381718/$20.00 ©2001 The American Physical Society 64 013817-1