Solution of a Singular Integral Equation from Scattering Theory

Abstract

The solution of a singular integral equation of the form $${\mathrm{\psi (\omega ,\omega }}_0\mathrm{)=-}\frac{1}{\omega }-\frac{1}{\pi }{\displaystyle {\int }_{\mu }^{\infty }}\frac{{\mathrm{\rho (\omega }}_1{\mathrm{,\omega }}_0{\mathrm{)\psi (\omega }}_1{\mathrm{,\omega }}_0{\mathrm{)\hspace{0.17em}d\omega }}_1}{{\omega }_1{\mathrm{+\omega -\omega }}_0\mathrm{-i\epsilon }}$$ is obtained. The equation first appeared in the attempt of Källén and Pauli to solve V − θ scattering in the Lee model. The methods developed by one of us to solve that problem are turned to solving the integral equation itself.