Shnol’s theorem and the spectrum of long range operators

Han, Rui

doi:10.1090/proc/14388

Shnol’s theorem and the spectrum of long range operators
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Proc. Amer. Math. Soc. 147 (2019), 2887-2897 Request permission

Abstract:

We extend some basic results known for finite range operators to long range operators with off-diagonal decay. Namely, we prove an analogue of Shnol’s theorem. We also establish the connection between the almost sure spectrum of long range random operators and the spectra of deterministic periodic operators.

References

Michael Aizenman and Stanislav Molchanov, Localization at large disorder and at extreme energies: an elementary derivation, Comm. Math. Phys. 157 (1993), no. 2, 245–278. MR 1244867
Slim Ayadi, Fabian Schwarzenberger, and Ivan Veselić, Uniform existence of the integrated density of states for randomly weighted Hamiltonians on long-range percolation graphs, Math. Phys. Anal. Geom. 16 (2013), no. 4, 309–330. MR 3133851, DOI 10.1007/s11040-013-9133-2
Ju. M. Berezans′kiĭ, Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, American Mathematical Society, Providence, R.I., 1968. Translated from the Russian by R. Bolstein, J. M. Danskin, J. Rovnyak and L. Shulman. MR 0222718
Anne Boutet de Monvel, Daniel Lenz, and Peter Stollmann, Sch’nol’s theorem for strongly local forms, Israel J. Math. 173 (2009), 189–211. MR 2570665, DOI 10.1007/s11856-009-0088-8
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
David Damanik, Jake Fillman, Milivoje Lukic, and William Yessen, Characterizations of uniform hyperbolicity and spectra of CMV matrices, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 4, 1009–1023. MR 3543643, DOI 10.3934/dcdss.2016039
Rupert L. Frank, Daniel Lenz, and Daniel Wingert, Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory, J. Funct. Anal. 266 (2014), no. 8, 4765–4808. MR 3177322, DOI 10.1016/j.jfa.2014.02.008
V. Z. Grinshpun, The point spectrum of an infinite-order random operator that acts in $l^2(\textbf {Z}^d)$, Dokl. Akad. Nauk Ukrainy 8 (1992), 18–21, 169 (Russian, with English and Russian summaries). MR 1192887
Alex Haro and Joaquim Puig, A Thouless formula and Aubry duality for long-range Schrödinger skew-products, Nonlinearity 26 (2013), no. 5, 1163–1187. MR 3043377, DOI 10.1088/0951-7715/26/5/1163
Vojkan Jakšić and Stanislav Molchanov, On the surface spectrum in dimension two, Helv. Phys. Acta 71 (1998), no. 6, 629–657. MR 1669046
Vojkan Jakšić and Stanislav Molchanov, Localization for one-dimensional long range random Hamiltonians, Rev. Math. Phys. 11 (1999), no. 1, 103–135. MR 1668075, DOI 10.1142/S0129055X99000052
V. Jakšić, S. Molchanov, and L. Pastur, On the propagation properties of surface waves, Wave propagation in complex media (Minneapolis, MN, 1994) IMA Vol. Math. Appl., vol. 96, Springer, New York, 1998, pp. 143–154. MR 1489748, DOI 10.1007/978-1-4612-1678-0_{7}
Russell A. Johnson, Exponential dichotomy, rotation number, and linear differential operators with bounded coefficients, J. Differential Equations 61 (1986), no. 1, 54–78. MR 818861, DOI 10.1016/0022-0396(86)90125-7
Werner Kirsch, An invitation to random Schrödinger operators, Random Schrödinger operators, Panor. Synthèses, vol. 25, Soc. Math. France, Paris, 2008, pp. 1–119 (English, with English and French summaries). With an appendix by Frédéric Klopp. MR 2509110
Werner Kirsch and Fabio Martinelli, On the spectrum of Schrödinger operators with a random potential, Comm. Math. Phys. 85 (1982), no. 3, 329–350. MR 678150
W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, J. Reine Angew. Math. 334 (1982), 141–156. MR 667454, DOI 10.1515/crll.1982.334.141
Abel Klein, Andrew Koines, and Maximilian Seifert, Generalized eigenfunctions for waves in inhomogeneous media, J. Funct. Anal. 190 (2002), no. 1, 255–291. Special issue dedicated to the memory of I. E. Segal. MR 1895534, DOI 10.1006/jfan.2001.3887
Peter Kuchment, Quantum graphs. II. Some spectral properties of quantum and combinatorial graphs, J. Phys. A 38 (2005), no. 22, 4887–4900. MR 2148631, DOI 10.1088/0305-4470/38/22/013
Daniel Lenz and Alexander Teplyaev, Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces, Trans. Amer. Math. Soc. 368 (2016), no. 7, 4933–4956. MR 3456166, DOI 10.1090/tran/6639
C. A. Marx, Dominated splittings and the spectrum of quasi-periodic Jacobi operators, Nonlinearity 27 (2014), no. 12, 3059–3072. MR 3291142, DOI 10.1088/0951-7715/27/12/3059
È. È. Šnol′, On the behavior of the eigenfunctions of Schrödinger’s equation, Mat. Sb. (N.S.) 42 (84) (1957), 273-286; erratum 46 (88) (1957), 259 (Russian). MR 0125315
Christoph Schumacher, Fabian Schwarzenberger, and Ivan Veselić, A Glivenko-Cantelli theorem for almost additive functions on lattices, Stochastic Process. Appl. 127 (2017), no. 1, 179–208. MR 3575539, DOI 10.1016/j.spa.2016.06.005
Barry Simon, Spectrum and continuum eigenfunctions of Schrödinger operators, J. Functional Analysis 42 (1981), no. 3, 347–355. MR 626449, DOI 10.1016/0022-1236(81)90094-X
Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
Barry Simon and Thomas Spencer, Trace class perturbations and the absence of absolutely continuous spectra, Comm. Math. Phys. 125 (1989), no. 1, 113–125. MR 1017742