Momentum Operators in The Unit Square

TL;DR

This paper studies skew-adjoint extensions of partial derivative operators in a unit square, analyzing their spectra, unitary equivalence, and commuting properties, with implications for more complex domains including fractals.

Contribution

It provides a detailed analysis of skew-adjoint extensions and their spectra for operators on the unit square and extends results to more general domains.

Findings

Extensions may lack discrete spectrum.

Unitary equivalence classes characterized.

Results applicable to fractal domains.

Abstract

We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.