  The system whose Hamiltonian is a linear combination of the generators of
SU(1,1) group with time-dependent coefficients is studied. It is shown that
there is a unitary relation between the system and a system whose Hamiltonian
is simply proportional to the generator of the compact subgroup of the SU(1,1).
The unitary relation is described by the classical solutions of a
time-dependent (harmonic) oscillator. Making use of the relation, the wave
functions satisfying the Schr\"{o}dinger equation are given for a general
unitary representation in terms of the matrix elements of a finite group
transformation (Bargmann function). The wave functions of the harmonic
oscillator with an inverse-square potential is studied in detail, and it is
shown that, through an integral, the model provides a way of deriving the
Bargmann function for the representation of positive discrete series of the
SU(1,1).
