Dirac and magnetic Schrödinger operators on fractals

Abstract

In this paper we define (local) Dirac operators and magnetic Schrödinger 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 Röckner, 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.

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Cited by in corpus (19)

• Complex Quantum Networks: a Topical Review
• Local Dirichlet forms, Hodge theory, and the Navier-Stokes equations on topologically one-dimensional fractals
• Frustration index and Cheeger inequalities for discrete and continuous magnetic Laplacians
• The mass of simple and higher-order networks
• Dirac gauge theory for topological spinors in 3+1 dimensional networks
• Spectral dimension and Bohr's formula for Schrodinger operators on unbounded fractal spaces
• Power dissipation in fractal AC circuits
• Fractal snowflake domain diffusion with boundary and interior drifts
• Finite energy coordinates and vector analysis on fractals
• A Feynman-Kac-Itô Formula for magnetic Schrödinger operators on graphs
• Backward problems for stochastic differential equations on the Sierpinski gasket
• Closability, regularity, and approximation by graphs for separable bilinear forms
• Parabolic type equations associated with the Dirichlet form on the Sierpinski gasket
• Canonical diffusions on the pattern spaces of aperiodic Delone sets
• From non-symmetric particle systems to non-linear PDEs on fractals
• Spectral decimation of the magnetic Laplacian on the Sierpinski gasket: Solving the Hofstadter-Sierpinski butterfly
• Gaps in the spectrum of the Laplacian on $3N$-Gaskets
• Sobolev inequalities on product Sierpinski spaces
• Dual graphs and modified Barlow--Bass resistance estimates for repeated barycentric subdivisions