Magnetic Laplacians of locally exact forms on the Sierpinski Gasket

Jessica Hyde; Daniel Kelleher; Jesse Moeller; Luke Rogers; Luis Seda

doi:10.3934/cpaa.2017113

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2017, Volume 16, Issue 6: 2299-2319. Doi: 10.3934/cpaa.2017113

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Magnetic Laplacians of locally exact forms on the Sierpinski Gasket

1.
Massachusetts Institute of Technology, Cambridge, MA, USA
2.
PIMS Stochastic Postdoctoral Fellow, University of Alberta, Edmonton, AB, Canada
3.
University of Nebraska, Lincoln, NE, USA
4.
University of Connecticut, Storrs, CT, USA
5.
University of Puerto Rico Mayagüez, Mayagüez, PR, USA

*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.

Received: July 2016

Revised: May 2017

Published: September 2017

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Abstract

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.

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

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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.

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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 $\mathcal M^{\beta b}$

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Figure 3. The Ladder Fractafold

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

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Figure 5. he folded Sierpinski Ladder Fractafold

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