The General Solution for the Relativistic and Nonrelativistic Schr\"odinger Equation for the $\delta^{(n)}$-Function Potential in 1-dimension Using Cutoff Regularization, and the Fate of Universality

TL;DR

This paper presents a unified method to solve the Schrödinger equation with derivatives of delta-function potentials in 1D, revealing that the $ abla^2$-delta potential shares universal features with simpler contact interactions, both relativistically and nonrelativistically.

Contribution

A general cutoff regularization approach for solving Schrödinger equations with arbitrary derivatives of delta potentials in 1D, including relativistic and nonrelativistic cases, extending the concept of universality.

Findings

The $ abla^2$-delta potential behaves like the delta and delta-prime potentials.

It exhibits asymptotic freedom and dimensional transmutation in the massless limit.

It confirms the universality of contact interactions including higher derivatives.

Abstract

A general method has been developed to solve the Schr\"odinger equation for an arbitrary derivative of the $\delta$-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 equations for each case has been derived. The case of $\delta^{(2)}$-function potential has been used as an example. The results from the relativistic case show that the $\delta^{(2)}$-function system behaves exactly like the $\delta$-function and the $\delta'$-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 $\delta^{(2)}$-function potential.