The general solution for the relativistic and nonrelativistic Schrödinger equation for the δ(n) -function potential in 1-dimension using cutoff regularization, and the fate of universality

Abstract

A general method has been developed to solve the Schrödinger equation for an arbitrary derivative of the δ-function potential in 1-d using cutoff regularization. The work treats both the relativistic and nonrelativistic cases. A distinction in the treatment has been made between the case when the derivative n is an even number from the one when n is an odd number. A general gap equation for each case has been derived. The case of δ(2)-function potential has been used as an example. The results from the relativistic case show that the δ(2)-function system behaves exactly like the δ-function and the δ′-function potentials, which means it also shares the same features with quantum field theories, like being asymptotically free, in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point. As a result, the evidence of universality of contact interactions has been extended further to include the δ(2)-function potential.