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.

Derivations and Dirichlet forms on fractals

“…For earlier approaches to vector analysis on fractals, see [42,46,53,63,65]. For some related results obtained independently from and at the same time as our work see [35], and for further developments see [36][37][38].…”

Backward problems for stochastic differential equations on the Sierpinski gasket

“…Lemma 4. 16. Let h i be the harmonic functions with boundary values h i | V 0 = 1 {p i } , i = 1, 2, 3 respectively.…”

Densely defined non‐closable curl on carpet‐like metric measure spaces

“…Our paper is a part of a broader program that aims to connect research on derivatives on fractals ( [8,9,[13][14][15][25][26][27][28]31,33,35,[37][38][39][40][41]46,51,54] and references therein) and on more general regular Dirichlet spaces [29,30,34] with classical and geometric analysis on metric measure spaces ( [6,10,12,[21][22][23][24][42][43][44][45]50] and references therein). In our previous article [36] we showed that on certain topologically one-dimensional spaces with a strongly local regular Dirichlet form one can prove a natural version of the Hodge theorem for 1-forms defined in 2 -sense: the set of harmonic 1-forms is dense in the orthogonal complement of the exact 1-forms.…”

“…In particular the reader can consult the papers [7,10,[42][43][44][45] and references therein. Our current paper is a step in a long-term program (see [29][30][31][32][33][34][35][36][37][38]) to develop parts of differential geometry and their applications to mathematical physics (see [3][4][5]) for spaces that carry diffusion processes but no other smooth structure. Our approach is somewhat complementary to the celebrated works [12,21,22] because, although our spaces are metrizable, we do not use any particular metric in an essential way, and we do not use functional inequalities.…”