Distributional approach to point interactions in one-dimensional quantum mechanics

Abstract

We consider the one-dimensional quantum mechanical problem of defining interactions concentrated at a single point in the framework of the theory of distributions. The often ill-defined product which describes the interaction term in the Schrödinger and Dirac equations is replaced by a well-defined distribution satisfying some simple mathematical conditions and, in addition, the physical requirement of probability current conservation is imposed. A four-parameter family of interactions thus emerges as the most general point interaction both in the non-relativistic and in the relativistic theories (in agreement with results obtained by self-adjoint extensions). Since the interaction is given explicitly, the distributional method allows one to carry out symmetry investigations in a simple way, and it proves to be useful to clarify some ambiguities related to the so-called ${\delta }^{\prime }$ interaction.

Open Access link: http://journal.frontiersin.org/Journal/10.3389/fphy.2014.00023/abstract

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• Two-point one-dimensional $\delta$-${\delta }^{\prime }$ interactions: non-abelian addition law and decoupling limit
• Distribution theory for Schrödinger's integral equation
• Point interactions with bias potentials
• The Solution of the Relativistic Schrodinger Equation for the ${\delta }^{\prime }$-Function Potential in 1-dimension Using Cutoff Regularization
• The bound-state solutions of the one-dimensional hydrogen atom
• On negative eigenvalues of 1D Schrödinger operators with ${\delta }^{\prime }$-like potentials
• Comparison between two models of absorption of matter waves by a thin time-dependent barrier
• Regularization of a strong-weak duality between pointlike interactions in one dimension
• Spontaneous emission and atomic line shift in causal perturbation theory
• Bound States and Resonance Analysis of One-Dimensional Relativistic Parity-Symmetric Two Point Interactions
• Black holes and quantum theory on manifolds with singular potentials