Dirac and magnetic Schr\"odinger operators on fractals

TL;DR

This paper introduces and rigorously defines Dirac and magnetic Schr"odinger operators on fractals, specifically on the Sierpinski gasket, establishing their self-adjointness using advanced mathematical frameworks.

Contribution

It develops a novel framework for defining these operators on fractals using Dirichlet forms and 1-forms, extending analysis on fractals.

Findings

Defined local Dirac operators on fractals.

Proved essential self-adjointness of magnetic Schr"odinger Hamiltonians.

Framework applicable to other fractals like the Sierpinski carpet.

Abstract

In this paper we define (local) Dirac operators and magnetic Schr\"odinger Hamiltonians on fractals and prove their (essential) self-adjointness. To do so we use the concept of 1-forms and derivations associated with Dirichlet forms as introduced by Cipriani and Sauvageot, and further studied by the authors jointly with R\"ockner, Ionescu and Rogers. For simplicity our definitions and results are formulated for the Sierpinski gasket with its standard self-similar energy form. We point out how they may be generalized to other spaces, such as the classical Sierpinski carpet.