ON THE OPERATOR EXP - Theoretical and Computational Research in the 21st Century - page 152

8.2 METHOD

8.2.1 P AND Q ARE ARBITRARY

If Eq. (1) is modified as we obtain the fol-

λλ → − , PP →− , Q →− Q ,

lowing equation:

exp( 2

λ ( PQ + ) ) = exp( -- f ( λ ) Q ) exp ( -- g ( λ ) P ) exp ( - h Q (- λ ) ),

(6)

as P and Q are arbitrary, Eq. (6) coincides with Eq. (1) if the functions
involved are odd:

f ( λ ) = − f (− λ ), g ( λ ) = − g (− λ ), h ( λ ) = − h (− λ ) (7)

The properties and

− 1

permit to de-

− 1 −−− 111

(exp A ) = exp ( − A )

( ABC ) = C B A

duce the inverse operator of Eq. (1):

exp( 2 ( - λ p Q ) ) + = ex p( - hQ ( λ ) ) exp( - g p ( λ ) ) exp ( - f ( λ ) Q ), (8)

And if in Eq. (1) only we realize the change in accordance with

λ →− λ

Eq. (7):

exp( 2 (

- λ p Q ) ) + = ex p( - hQ ( λ ) ) exp( - g p ( λ ) ) exp ( - f ( λ ) Q ),

(9)

Again, as P and Q are arbitrary, we obtain the total compatibility between
Eqs. (8) and (9) if f = h ; therefore, the factorization of Eq. (1) is equivalent
to

T ( λ ) º exp( 2 λ ( P + Q ) ) = exp( f Q ) exp( g P ) exp ( f Q ),

(10)

where f and g are odd functions of the parameter λ .

8.2.2 P AND Q VERIFY

[ ] [ ]

⎡ P , Q , P ⎤ =− 2 P and ⎡ P , Q , Q ⎤ = 2 Q

⎣ ⎦ ⎣ ⎦

Expression (3) is correct if P and Q satisfy Eq. (2); then, we shall consider
the operator: