The Solution of the Relativistic Schrodinger Equation for the ${\delta }^{\prime }$-Function Potential in 1-dimension Using Cutoff Regularization

Abstract

We study the relativistic version of Schrödinger equation for a point particle in 1-d with potential of the first derivative of the delta function. The momentum cutoff regularization is used to study the bound state and scattering states. The initial calculations show that the reciprocal of the bare coupling constant is ultra-violet divergent, and the resultant expression cannot be renormalized in the usual sense. Therefore a general procedure has been developed to derive different physical properties of the system. The procedure is used first on the non-relativistic case for the purpose of clarification and comparisons. The results from the relativistic case show that this system behaves exactly like the delta function potential, which means it also shares the same features with quantum field theories, like being asymptotically free, and in the massless limit, it undergoes dimensional transmutation and it possesses an infrared conformal fixed point.

32 pages, 5 figures

References in corpus (3)

• Distributional approach to point interactions in one-dimensional quantum mechanics
• Asymptotic Freedom, Dimensional Transmutation, and an Infra-red Conformal Fixed Point for the $\delta$-Function Potential in 1-dimensional Relativistic Quantum Mechanics
• A Solution of the Relativistic Schrödinger Equation for the $\delta$-Function Potential in 1-dimensiona with Cutoff Regularization

Cited by in corpus (5)

• One-Dimensional Semirelativistic Hamiltonian with Multiple Dirac Delta Potentials
• Analysis of a one-dimensional Hamiltonian with a singular double well consisting of two nonlocal ${\delta }^{\prime }$ interactions
• Quantum anomaly and thermodynamics of one-dimensional fermions with antisymmetric two-body interactions
• Contact interactions, self-adjoint extensions, and low-energy scattering
• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions