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Article Contents

# Magnetic Laplacians of locally exact forms on the Sierpinski Gasket

*Authors supported in part by the National Science Foundation through grant DMS 1262929.
† Author supported in part by the National Science Foundation through grant DMS-0505622.

• We give an explicit construction of a magnetic Schrödinger operator corresponding to a field with flux through finitely many holes of the Sierpinski Gasket. The operator is shown to have discrete spectrum accumulating at ∞, and it is shown that the asymptotic distribution of eigenvalues is the same as that for the Laplacian. Most eigenfunctions may be computed using gauge transformations corresponding to the magnetic field and the remainder of the spectrum may be approximated to arbitrary precision by using a sequence of approximations by magnetic operators on finite graphs.

Mathematics Subject Classification: Primary: 28A80; Secondary: 60J35, 31E05, 47A07, 81Q10, 81Q35.

 Citation:

• Figure 1.  (a) The $1$-form $\sqrt{30}b$, with orientation clockwise around each $1$-cell, hence counterclockwise around the central hole, and (b) The harmonic function $B$ on disjoint $1$-cells.

Figure 2.  Eigenvalues less than $160$ and $0\leq \beta\leq 2$ for the (from top to bottom) 4th, 5th, and 6th level approximation to $\mathcal M^{\beta b}$

Figure 3.  The Ladder Fractafold

Figure 4.  The graphs $\Gamma_0$ (unfilled verteces and dashed edges), and $\Gamma$ (filled verteces, solid edges)

Figure 5.  he folded Sierpinski Ladder Fractafold

Figures(5)