  We considered a charged quantum mechanical particle with spin ${1\over 2}$
and gyromagnetic ratio $g\ne 2$ in the field af a magnetic string. Whereas the
interaction of the charge with the string is the well kown Aharonov-Bohm effect
and the contribution of magnetic moment associated with the spin in the case
$g=2$ is known to yield an additional scattering and zero modes (one for each
flux quantum), an anomaly of the magnetic moment (i.e. $g>2$) leads to bound
states. We considered two methods for treating the case $g>2$. \\ The first is
the method of self adjoint extension of the corresponding Hamilton operator. It
yields one bound state as well as additional scattering. In the second we
consider three exactly solvable models for finite flux tubes and take the limit
of shrinking its radius to zero. For finite radius, there are $N+1$ bound
states ($N$ is the number of flux quanta in the tube).\\ For $R\to 0$ the bound
state energies tend to infinity so that this limit is not physical unless $g\to
2$ along with $R\to 0$. Thereby only for fluxes less than unity the results of
the method of self adjoint extension are reproduced whereas for larger fluxes
$N$ bound states exist and we conclude that this method is not applicable.\\ We
discuss the physically interesting case of small but finite radius whereby the
natural scale is given by the anomaly of the magnetic moment of the electron
$a_e=(g-2)/2\approx 10^{-3}$.
