Quantum bound states in a classically unbound system of crossed wires

We have computed the energy and the wave function for an electron caught at the intersection of two narrow channels. There are two bound energies for the case with fourfold rotational symmetry. For impenetrable walls the energies are $E_1$=0.66$E_t$ and $E_2$=3.72$E_t$, where the threshold for propagation of electrons in one channel is $E_t$=h${¯}^2$ ${\pi }^2$/2$m^{\mathrm{*}}$ $w^2$ and w is the width of the channel. The state at $E_2$ is bound only because it has odd parity and thus cannot decay into the even-parity propagating wave at the same energy. (The odd-parity propagation threshold is at 4$E_t$.) We have also computed the transmission and reflection probabilities in the propagating case for a range of energies up to slightly above the odd-parity threshold.