Monday, July 16, 2012

1207.3077 (Michael Hinz et al.)

Dirac and magnetic Schrödinger operators on fractals    [PDF]

Michael Hinz, Alexander Teplyaev
In this paper we define 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 finitely ramified fractals and some other spaces carrying a regular resistance form, such as the classical two-dimensional Sierpinski carpet.
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