November  2017, 16(6): 2299-2319. doi: 10.3934/cpaa.2017113

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  July 2017

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.

Citation: Jessica Hyde, Daniel Kelleher, Jesse Moeller, Luke Rogers, Luis Seda. Magnetic Laplacians of locally exact forms on the Sierpinski Gasket. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2299-2319. doi: 10.3934/cpaa.2017113
References:
[1]

Skye AaronZach ConnRobert S. Strichartz and Hui Yu, Hodge-de Rham theory on fractal graphs and fractals, Commun. Pure Appl. Anal., 13 (2014), 903-928. doi: 10.3934/cpaa.2014.13.903. Google Scholar

[2]

Eric Akkermans, Statistical mechanics and quantum fields on fractals, 601 (2013), 1-21 doi: 10.1090/conm/601/11962. Google Scholar

[3]

Eric Akkermans, Olivier Benichou, Gerald V Dunne, Alexander Teplyaev and Raphael Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Physical Review E 86 (2012).Google Scholar

[4]

Eric Akkermans, Gerald V Dunne and Alexander Teplyaev, Physical consequences of complex dimensions of fractals, Europhysics Letters, 88 (2009), 40007.Google Scholar

[5]

Eric Akkermans, Gerald V Dunne and Alexander Teplyaev, Thermodynamics of photons on fractals, Physical review letters, 105 (2010), 230407.Google Scholar

[6]

Shlomo Alexander, Some properties of the spectrum of the Sierpiński gasket in a magnetic field, Phys. Rev. B, 29 (1984), 5504-5508. Google Scholar

[7]

Shlomo Alexander and Raymond Orbach, Density of states on fractals: fractons, Journal de Physique Lettres, 43 (1982), 625-631. Google Scholar

[8]

Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785. Google Scholar

[9]

J. Bellissard, Renormalization group analysis and quasicrystals, in Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992, pp. 118-148. Google Scholar

[10]

Nicolas Bouleau and Francis Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co. , Berlin, 1991. doi: 10.1515/9783110858389. Google Scholar

[11]

Fabio CiprianiDaniele GuidoTommaso Isola and Jean-Luc Sauvageot, Integrals and potentials of differential 1-forms on the Sierpinski gasket, Adv. Math., 239 (2013), 128-163. doi: 10.1016/j.aim.2013.02.014. Google Scholar

[12]

Fabio Cipriani and Jean-Luc Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Anal., 201 (2003), 78-120. doi: 10.1016/S0022-1236(03)00085-5. Google Scholar

[13]

Fabio Cipriani and Jean-Luc Sauvageot, Fredholm modules on P.C.F. self-similar fractals and their conformal geometry, Comm. Math. Phys., 286 (2009), 541-558. doi: 10.1007/s00220-008-0673-4. Google Scholar

[14]

Kyallee DalrympleRobert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284. doi: 10.1007/BF01261610. Google Scholar

[15]

Jessica L. DeGradoLuke G. Rogers and Robert S. Strichartz, Gradients of Laplacian eigenfunctions on the Sierpinski gasket, Proc. Amer. Math. Soc., 137 (2009), 531-540. doi: 10.1090/S0002-9939-08-09711-6. Google Scholar

[16]

Eytan DomanyShlomo AlexanderDavid Bensimon and Leo P. Kadanoff, Solutions to the Schrödinger equation on some fractal lattices, Phys. Rev. B, 28 (1983), 3110-3123. Google Scholar

[17]

Gerald V. Dunne, Heat kernels and zeta functions on fractals J. Phys. A 45 (2012), 374016, 22. doi: 10.1088/1751-8113/45/37/374016. Google Scholar

[18]

Patrick J. Fitzsimmons, Even and odd continuous additive functionals, in Dirichlet Forms and Stochastic Processes (Beijing, 1993), de Gruyter, Berlin, 1995, pp. 139-154. Google Scholar

[19]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. Google Scholar

[20]

Masatoshi Fukushima, Yoichi Oshima and Masayoshi Takeda, Dirichlet Forms and Symmetric Markov Processes extended ed. , De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co. , Berlin, 2011. Google Scholar

[21]

J. M. GhezW. WangR. RammalB. Pannetier and J. Bellissard, Band spectrum for an electron on a Sierpinsky gasket in a magnetic field, Phys. Rev. B, 64 (1987), 1291-1294. Google Scholar

[22]

Batu GüneysuMatthias Keller and Marcel Schmidt, A Feynman-Kac-Itȏ formula for magnetic Schrödinger operators on graphs, Probab. Theory Related Fields, 165 (2016), 365-399. doi: 10.1007/s00440-015-0633-9. Google Scholar

[23]

Michael Hinz, Magnetic energies and Fey-Kac-Itȏ formulas for symmetric Markov processes, Stoch. Anal. Appl., 33 (2015), 1020-1049. doi: 10.1080/07362994.2015.1077715. Google Scholar

[24]

Michael HinzMichael Röckner and Alexander Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406. doi: 10.1016/j.spa.2013.06.009. Google Scholar

[25]

Michael Hinz and Luke G. Rogers, Magnetic fields on resistance spaces, J. Fractal Geom., 3 (2016), 75-93. doi: 10.4171/JFG/30. Google Scholar

[26]

Michael Hin and Alexander Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854. doi: 10.1016/j.jfa.2013.07.021. Google Scholar

[27]

Marius IonescuLuke G. Rogers and Alexander Teplyaev, Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169. doi: 10.1016/j.jfa.2012.05.021. Google Scholar

[28]

Jun Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[29]

Jun Kigami, Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets, J. Funct. Anal., 156 (1998), 170-198. doi: 10.1006/jfan.1998.3243. Google Scholar

[30]

Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar

[31]

Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125. Google Scholar

[32]

Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. , 83 (1990), no. 420, ⅳ+128. doi: 10.1090/memo/0420. Google Scholar

[33]

Leonid Malozemov and Alexander Teplyaev, Self-similarity, operators and dynamics, Math. Phys. Anal. Geom., 6 (2003), 201-218. doi: 10.1023/A:1024931603110. Google Scholar

[34]

Shintaro Nakao, Stochastic calculus for continuous additive functionals of zero energy, Z. Wahrsch. Verw. Gebiete, 68 (1985), 557-578. doi: 10.1007/BF00535345. Google Scholar

[35]

R. Rammal, Harmonic analysis in fractal spaces: random walk statistics and spectrum of the Schrödinger equation, Phys. Rep., 103 (1984), 151-159, Common trends in particle and condensed matter physics (Les Houches, 1983). doi: 10.1016/0370-1573(84)90075-9. Google Scholar

[36]

R. Rammal, Spectrum of harmonic excitations on fractals, J. Physique, 45 (1984), 191-206. Google Scholar

[37]

R. Rammal and G. Toulouse, Random walks on fractal structure and percolation cluster, J. Physique Letters, 44 (1983), L13-L22. Google Scholar

[38]

Tadashi Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math., 13 (1996), 1-23. doi: 10.1007/BF03167295. Google Scholar

[39]

Barry Simon, Almost periodic Schrödinger operators: a review, Adv. in Appl. Math., 3 (1982), 463-490. doi: 10.1016/S0196-8858(82)80018-3. Google Scholar

[40]

Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043 (electronic). doi: 10.1090/S0002-9947-03-03171-4. Google Scholar

[41]

Robert S. Strichartz, Differential Equations on Fractals: A Tutorial Princeton University Press, Princeton, NJ, 2006. Google Scholar

[42]

Robert S. Strichartz and Alexander Teplyaev, Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297. doi: 10.1007/s11854-012-0007-5. Google Scholar

show all references

References:
[1]

Skye AaronZach ConnRobert S. Strichartz and Hui Yu, Hodge-de Rham theory on fractal graphs and fractals, Commun. Pure Appl. Anal., 13 (2014), 903-928. doi: 10.3934/cpaa.2014.13.903. Google Scholar

[2]

Eric Akkermans, Statistical mechanics and quantum fields on fractals, 601 (2013), 1-21 doi: 10.1090/conm/601/11962. Google Scholar

[3]

Eric Akkermans, Olivier Benichou, Gerald V Dunne, Alexander Teplyaev and Raphael Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Physical Review E 86 (2012).Google Scholar

[4]

Eric Akkermans, Gerald V Dunne and Alexander Teplyaev, Physical consequences of complex dimensions of fractals, Europhysics Letters, 88 (2009), 40007.Google Scholar

[5]

Eric Akkermans, Gerald V Dunne and Alexander Teplyaev, Thermodynamics of photons on fractals, Physical review letters, 105 (2010), 230407.Google Scholar

[6]

Shlomo Alexander, Some properties of the spectrum of the Sierpiński gasket in a magnetic field, Phys. Rev. B, 29 (1984), 5504-5508. Google Scholar

[7]

Shlomo Alexander and Raymond Orbach, Density of states on fractals: fractons, Journal de Physique Lettres, 43 (1982), 625-631. Google Scholar

[8]

Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-623. doi: 10.1007/BF00318785. Google Scholar

[9]

J. Bellissard, Renormalization group analysis and quasicrystals, in Ideas and Methods in Quantum and Statistical Physics (Oslo, 1988), Cambridge Univ. Press, Cambridge, 1992, pp. 118-148. Google Scholar

[10]

Nicolas Bouleau and Francis Hirsch, Dirichlet Forms and Analysis on Wiener Space, De Gruyter Studies in Mathematics, vol. 14, Walter de Gruyter & Co. , Berlin, 1991. doi: 10.1515/9783110858389. Google Scholar

[11]

Fabio CiprianiDaniele GuidoTommaso Isola and Jean-Luc Sauvageot, Integrals and potentials of differential 1-forms on the Sierpinski gasket, Adv. Math., 239 (2013), 128-163. doi: 10.1016/j.aim.2013.02.014. Google Scholar

[12]

Fabio Cipriani and Jean-Luc Sauvageot, Derivations as square roots of Dirichlet forms, J. Funct. Anal., 201 (2003), 78-120. doi: 10.1016/S0022-1236(03)00085-5. Google Scholar

[13]

Fabio Cipriani and Jean-Luc Sauvageot, Fredholm modules on P.C.F. self-similar fractals and their conformal geometry, Comm. Math. Phys., 286 (2009), 541-558. doi: 10.1007/s00220-008-0673-4. Google Scholar

[14]

Kyallee DalrympleRobert S. Strichartz and Jade P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284. doi: 10.1007/BF01261610. Google Scholar

[15]

Jessica L. DeGradoLuke G. Rogers and Robert S. Strichartz, Gradients of Laplacian eigenfunctions on the Sierpinski gasket, Proc. Amer. Math. Soc., 137 (2009), 531-540. doi: 10.1090/S0002-9939-08-09711-6. Google Scholar

[16]

Eytan DomanyShlomo AlexanderDavid Bensimon and Leo P. Kadanoff, Solutions to the Schrödinger equation on some fractal lattices, Phys. Rev. B, 28 (1983), 3110-3123. Google Scholar

[17]

Gerald V. Dunne, Heat kernels and zeta functions on fractals J. Phys. A 45 (2012), 374016, 22. doi: 10.1088/1751-8113/45/37/374016. Google Scholar

[18]

Patrick J. Fitzsimmons, Even and odd continuous additive functionals, in Dirichlet Forms and Stochastic Processes (Beijing, 1993), de Gruyter, Berlin, 1995, pp. 139-154. Google Scholar

[19]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35. doi: 10.1007/BF00249784. Google Scholar

[20]

Masatoshi Fukushima, Yoichi Oshima and Masayoshi Takeda, Dirichlet Forms and Symmetric Markov Processes extended ed. , De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co. , Berlin, 2011. Google Scholar

[21]

J. M. GhezW. WangR. RammalB. Pannetier and J. Bellissard, Band spectrum for an electron on a Sierpinsky gasket in a magnetic field, Phys. Rev. B, 64 (1987), 1291-1294. Google Scholar

[22]

Batu GüneysuMatthias Keller and Marcel Schmidt, A Feynman-Kac-Itȏ formula for magnetic Schrödinger operators on graphs, Probab. Theory Related Fields, 165 (2016), 365-399. doi: 10.1007/s00440-015-0633-9. Google Scholar

[23]

Michael Hinz, Magnetic energies and Fey-Kac-Itȏ formulas for symmetric Markov processes, Stoch. Anal. Appl., 33 (2015), 1020-1049. doi: 10.1080/07362994.2015.1077715. Google Scholar

[24]

Michael HinzMichael Röckner and Alexander Teplyaev, Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces, Stochastic Process. Appl., 123 (2013), 4373-4406. doi: 10.1016/j.spa.2013.06.009. Google Scholar

[25]

Michael Hinz and Luke G. Rogers, Magnetic fields on resistance spaces, J. Fractal Geom., 3 (2016), 75-93. doi: 10.4171/JFG/30. Google Scholar

[26]

Michael Hin and Alexander Teplyaev, Dirac and magnetic Schrödinger operators on fractals, J. Funct. Anal., 265 (2013), 2830-2854. doi: 10.1016/j.jfa.2013.07.021. Google Scholar

[27]

Marius IonescuLuke G. Rogers and Alexander Teplyaev, Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169. doi: 10.1016/j.jfa.2012.05.021. Google Scholar

[28]

Jun Kigami, A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290. doi: 10.1007/BF03167882. Google Scholar

[29]

Jun Kigami, Distributions of localized eigenvalues of Laplacians on post critically finite self-similar sets, J. Funct. Anal., 156 (1998), 170-198. doi: 10.1006/jfan.1998.3243. Google Scholar

[30]

Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511470943. Google Scholar

[31]

Jun Kigami and Michel L. Lapidus, Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125. Google Scholar

[32]

Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. , 83 (1990), no. 420, ⅳ+128. doi: 10.1090/memo/0420. Google Scholar

[33]

Leonid Malozemov and Alexander Teplyaev, Self-similarity, operators and dynamics, Math. Phys. Anal. Geom., 6 (2003), 201-218. doi: 10.1023/A:1024931603110. Google Scholar

[34]

Shintaro Nakao, Stochastic calculus for continuous additive functionals of zero energy, Z. Wahrsch. Verw. Gebiete, 68 (1985), 557-578. doi: 10.1007/BF00535345. Google Scholar

[35]

R. Rammal, Harmonic analysis in fractal spaces: random walk statistics and spectrum of the Schrödinger equation, Phys. Rep., 103 (1984), 151-159, Common trends in particle and condensed matter physics (Les Houches, 1983). doi: 10.1016/0370-1573(84)90075-9. Google Scholar

[36]

R. Rammal, Spectrum of harmonic excitations on fractals, J. Physique, 45 (1984), 191-206. Google Scholar

[37]

R. Rammal and G. Toulouse, Random walks on fractal structure and percolation cluster, J. Physique Letters, 44 (1983), L13-L22. Google Scholar

[38]

Tadashi Shima, On eigenvalue problems for Laplacians on p.c.f. self-similar sets, Japan J. Indust. Appl. Math., 13 (1996), 1-23. doi: 10.1007/BF03167295. Google Scholar

[39]

Barry Simon, Almost periodic Schrödinger operators: a review, Adv. in Appl. Math., 3 (1982), 463-490. doi: 10.1016/S0196-8858(82)80018-3. Google Scholar

[40]

Robert S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043 (electronic). doi: 10.1090/S0002-9947-03-03171-4. Google Scholar

[41]

Robert S. Strichartz, Differential Equations on Fractals: A Tutorial Princeton University Press, Princeton, NJ, 2006. Google Scholar

[42]

Robert S. Strichartz and Alexander Teplyaev, Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297. doi: 10.1007/s11854-012-0007-5. Google Scholar

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
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