Scattering by a contact potential in three and lower dimensions

Abstract

We consider the scattering of nonrelativistic particles in three dimensions by a contact potential $Ω\hbar^2δ(r)/ 2μr^α$ which is defined as the $a\to 0$ limit of $Ω\hbar^2δ(r-a)/2μr^α$. It is surprising that it gives a nonvanishing cross section when $α=1$ and $Ω=-1$. When the contact potential is approached by a spherical square well potential instead of the above spherical shell one, one obtains basically the same result except that the parameter $Ω$ that gives a nonvanishing cross section is different. Similar problems in two and one dimensions are studied and results of the same nature are obtained.