  The Stieltjes-Wigert polynomials, which correspond to an indeterminate moment
problem on the positive half-line, are eigenfunctions of a second order
q-difference operator. We consider the orthogonality measures for which the
difference operator is symmetric in the corresponding weighted $L^2$-spaces.
These measures are exactly the solutions to the q-Pearson equation.In the case
of discrete and absolutely continuous measures the difference operator is
essentially self-adjoint, and the corresponding spectral decomposition is given
explicitly. In particular, we find an orthogonal set of q-Bessel functions
complementing the Stieltjes-Wigert polynomials to an orthogonal basis for
$L^2(\mu)$ when $\mu$ is a discrete orthogonality measure solving the q-Pearson
equation. To obtain the spectral decomposition of the difference operator in
case of an absolutely continuous orthogonality measure we use the results from
the discrete case combined with direct integral techniques.
