Generalized Levinson theorem for singular potentials in two dimensions

The Levinson theorem for two-dimensional scattering is generalized for potentials with inverse square singularities. By this theorem, the number of bound states $N_m^{\mathrm{b}}$ in a given $m\mathrm{th}$ partial wave is related to the phase shift ${\delta }_m\mathrm{}\left(k\right)\mathrm{}$ and the singularity strength of the potential. When the effective potential has an inverse square singularity at the origin of the form ${\nu }^2/{\rho }^2$ and inverse square tail at infinity such as ${\mu }^2/{\rho }^2,$ Levinson’s relation gives ${\delta }_m\mathrm{}\left(0\right)\mathrm{}-{\delta }_m\left(\infty \right)=\pi [N_m^{\mathrm{b}}+(|\nu |-|\mu |)/2\mathrm{}].$