We study some classes of symmetric operators for the discrete series
representations of the quantum algebra U_q(su_{1,1}), which may serve as
Hamiltonians of various physical systems. The problem of diagonalization of
these operators (eigenfunctions, spectra, overlap coefficients, etc.) is solved
by expressing their overlap coefficients in terms of the known families of
q-orthogonal polynomials. We consider both bounded and unbounded operators. In
the latter case they are not selfadjoint and have deficiency indices (1,1),
which means that they have infinitely many selfadjoint extensions. We find
possible sets of point spectrum (which depends on the representation space
under consideration) for one of such symmetric operators by using the
orthogonality relations for q-Laguerre polynomials. In another case we are led
to new orthogonality relations for {}_3\phi_1-hypergeometric polynomials. Many
new realizations for the discrete series representations are constructed, which
follows from the diagonalization of the operators considered. In particular, a
new system of orthogonal functions on a discrete set is shown to emerge.