The behavior of bound states in asymmetric cross, T and L shaped
configurations is considered. Because of the symmetries of the wavefunctions,
the analysis can be reduced to the case of an electron localized at the
intersection of two orthogonal crossed wires of different width. Numerical
calculations show that the fundamental mode of this system remains bound for
the widths that we have been able to study directly; moreover, the
extrapolation of the results obtained for finite widths suggests that this
state remains bound even when the width of one arm becomes infinitesimal. We
provide a qualitative argument which explains this behavior and that can be
generalized to the lowest energy states in each symmetry class. In the case of
odd-odd states of the cross we find that the lowest mode is bounded when the
width of the two arms is the same and stays bound up to a critical value of the
ratio between the widths; in the case of the even-odd states we find that the
lowest mode is unbound up to a critical value of the ratio between the widths.
Our qualitative arguments suggest that the bound state survives as the width of
the vertical arm becomes infinitesimal.Comment: 11 pages, 19 figures, 3 table