Hamiltonians and Green's functions which interpolate between two and three dimensions

TL;DR

This paper introduces Hamiltonians combining 2D and 3D kinetic terms to model systems that transition between two- and three-dimensional behaviors, predicting new correlation and transmission properties.

Contribution

It presents a novel Hamiltonian framework that interpolates between 2D and 3D physics, providing insights into correlation functions, transmission probabilities, and potentials across dimensions.

Findings

Correlation functions interpolate between 2D and 3D laws.

Transmission probabilities depend on both transverse and longitudinal momentum.

Static potential transitions from logarithmic to Coulomb form.

Abstract

I propose to use Hamiltonians which contain two-dimensional and three-dimensional kinetic terms for the description of two-dimensional systems in physics. As a model system the evolution of three-dimensional wavefunctions in the presence of an infinitely thin layer is studied. The model predicts distance laws for correlation functions which interpolate between two-dimensional and three-dimensional behavior. It also predicts that in certain cases transmission probabilities through thin layers should depend not only on the transverse, but also on the longitudinal momentum of the infalling particles. The model also yields a static potential which interpolates between the two-dimensional logarithmic potential at small distances and the three-dimensional (1/r)-potential at large distances.