In this paper we use a simple straightforward technique to investigate the emergence of a bound state in a weakly bent wire. We show that the bend behaves like an infinitely shallow potential well, and in the limit of small bending angle $(\phi \ll 1)$ and low energy the bend can be presented by a simple one-dimensional $\delta$-function potential, $V\left(x\right)=-\left(2\mathrm{}\sqrt{c_b}{\phi }^2\right)\delta \mathrm{}\left(x\right)\mathrm{}$ where $c_b\cong \mathrm{2.1.}$

• Received 31 July 2001

DOI:https://doi.org/10.1103/PhysRevB.65.233101