Generalized Coherent States and Their Applications
This monograph treats an extensively developed field in modern mathematical physics - the theory of generalized coherent states and their applications to various physical problems. Coherent states, introduced originally by Schrodinger and von Neumann, wer
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It was proven in Sect. 1.4 that the CS subsystem {IOCk>1 is complete if and only if there is no entire function t/J(oc) such that t/J(OCk) =0 and flt/J(oc)1 2 exp (-locI2)d2oc < 00. The relevance of Example i) in Sect. 1.4 was evident. Let us prove the statement of Example ii): system {lOCk>} is complete if the point set {OCk} does not contain the origin oc = 0, and
"I L.. OCk 1- 2 - e-- 00
(Al)
k
at some positive c > O. This statement is a consequence of some general theorems concerning a relation between the order of an entire function and the distribution of its zeros in the complex plane. Recall that the order A for an entire analytical function J(z) is the number A=lim In [lnM(r)] r~oo Inr '
(A2)
where M(r) is the maximum of modulus of the functionJ(z) on the circle Izl =r, lim = lim sup is the upper limit. Respectively, the number -.- In M(r) ), r--+ 00 r
(A3)
Jl= hm
is called the type of the function J(z). Let {ocd be a sequence of zeros of the entire functionJ(z). To characterize the sequence {OCk}, the so-called convergence index Al is introduced. The definition is that for arbitrary small positive numbers c and (j the series Ilockl-),1-e is k
converging, while the series I IOCkl-),1 + lJ is diverging. The series L IOCkl-),1 may be k
either converging or diverging. From the theory of entire functions [231], it is known that the order of an entire function with zeros at some points rOCk} is no less than the convergence index of the sequence {OCk}: A~AI' Condition (Al) means that AI> 2. Therefore the order A for the function J(z) with zeros at the points {OCk} is above 2: A~ AI> 2. Clearly, here
Proof of Completeness for Certain CS Subsystems
297
JIJI2 exp ( -lzI2)d2z = 00,
so the CS system {I(Xk)} is complete. Correspondingly, at At < 2, as well as at At = 2, the CS system is incomplete if the series L l(Xkl- 2 is converging. If, however, At = 2 and the series L l(Xkl 2 is diverging, then to find out whether the subsystem is complete, one must have more specific information on the distribution of zeros (Xk in the complex (X plane [231]. Here it is not difficult to construct an entire function of A= 2 with zeros at the points (Xk. According to Theorem 9.1.1 in the book by Boas [232], the type Il for such a functionJ(z) must satisfy the inequalities
1 I. N(r) 2 r .... oo
Il?:.- 1m - -
r
1 -I. N(r)
1l?:.2e 1m
T'
(A.4) (AS)
where N(r) is the number of points of the set {(Xk} within the circle Izl ~r, lim =Iim inf is the lower limit, lim = lim sup is the upper limit. Hence, the CS system is complete at At = 2 and under anyone of these conditions:
N(r) 1 I ยท ~> .l!!!. , 7-00
- . N(r) hm ~>e.
r--+oo
(A6)
,-
,-
(A 7)
Let us consider now Example iii), where the points form a regular lattice in the (X plane: (A8)
where the lattice spacings W t and W2 are linearly independent, 1m {w2wd =1=0, and m and n are arbitrary integers. Note, firstly, that for a regular lattice with cells of an area S (A8)
Now condition (A6) leads directly to the first part of the theorem in Sect. 1.4. Next con
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