# American Institute of Mathematical Sciences

May  2022, 21(5): 1691-1714. doi: 10.3934/cpaa.2022042

## Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs

 1 Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg, 199034, Russia 2 HSE University, 3A Kantemirovskaya ulitsa, St. Petersburg, 194100, Russia 3 Northern (Arctic) Federal University, Severnaya Dvina Emb. 17, Arkhangelsk, 163002, Russia

*Corresponding author

Received  June 2021 Revised  October 2021 Published  May 2022 Early access  February 2022

Fund Project: Our study was supported by RFBR grant 19-01-00094

We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.

Citation: Evgeny Korotyaev, Natalia Saburova. Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1691-1714. doi: 10.3934/cpaa.2022042
##### References:
 [1] G. Ahumada, Fonctions periodiques et formule des traces de Selberg sur les arbres, C. R. Acad. Sci. Paris, 305 (1987), 709-712. [2] R. Brooks, The Spectral Geometry of $k$-Regular Graphs, J. Anal. Math., 57 (1991), 120-151.  doi: 10.1007/BF03041067. [3] G. Chinta, J. Jorgenson and A. Karlsson, Heat kernels on regular graphs and generalized Ihara zeta function formulas, Monatsh. Math., 178 (2015), 171-190.  doi: 10.1007/s00605-014-0685-4. [4] D. Chelkak and E. Korotyaev, Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line, Int. Math. Res. Not., 2006 (2006), 1-41.  doi: 10.1155/IMRN/2006/60314. [5] P. Deift and B. Simon, Almost periodic Schrödinger operators Ⅲ. The absolutely continuous spectrum in one dimension, Commun. Math. Phys., 90 (1983), 389-411. [6] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral gaps and discrete magnetic Laplacians, Linear Algebra Appl., 547 (2018), 183-216.  doi: 10.1016/j.laa.2018.02.006. [7] J. S. Fabila-Carrasco, F. Lledó and O. Post, Spectral preorder and perturbations of discrete weighted graphs, Math. Ann., (2020), 49 pp. doi: 10.1007/s00208-020-02091-5. [8] E. Korotyaev, Estimates of periodic potentials in terms of gap lengths, Commun. Math. Phys., 197 (1998), 521-526.  doi: 10.1007/s002200050462. [9] E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions, Int. Math. Res. Not., 2003 (2003), 2019-2031.  doi: 10.1155/S1073792803209107. [10] E. Korotyaev, Estimates for the Hill operator. I, J. Differ. Equ., 162 (2000), 1-26.  doi: 10.1006/jdeq.1999.3684. [11] E. Korotyaev, Effective masses for zigzag nanotubes in magnetic fields, Lett. Math. Phys., 83 (2008), 83-95.  doi: 10.1007/s11005-007-0212-9. [12] E. Korotyaev and I. Krasovsky, Spectral estimates for periodic Jacobi matrices, Commun. Math. Phys., 234 (2003), 517-532.  doi: 10.1007/s00220-002-0768-2. [13] E. Korotyaev and A. Kutsenko, Inverse problem for the discrete periodic Schrödinger operator, Zapiski Nauchnyh Seminarov POMI, 315 (2004), 96-101.  doi: 10.1007/s10958-006-0104-z. [14] E. Korotyaev and N. Saburova, Schrödinger operators on periodic discrete graphs, J. Math. Anal. Appl., 420 (2014), 576-611.  doi: 10.1016/j.jmaa.2014.05.088. [15] E. Korotyaev and N. Saburova, Spectral band localization for Schrödinger operators on periodic graphs, Proc. Amer. Math. Soc., 143 (2015), 3951-3967.  doi: 10.1090/S0002-9939-2015-12586-5. [16] E. Korotyaev and N. Saburova, Effective masses for Laplacians on periodic graphs, J. Math. Anal. Appl., 436 (2016), 104-130.  doi: 10.1016/j.jmaa.2015.11.051. [17] E. Korotyaev and N. Saburova, Magnetic Schrödinger operators on periodic discrete graphs, J. Funct. Anal., 272 (2017), 1625-1660.  doi: 10.1016/j.jfa.2016.12.015. [18] E. Korotyaev and N. Saburova, Spectral estimates for Schrödinger operators on periodic discrete graphs, St. Petersburg Math. J., 30 (2018), 667-698.  doi: 10.1090/spmj/1565. [19] E. Korotyaev and N. Saburova, Invariants for Laplacians on periodic graphs, Math. Ann., 377 (2020), 723-758.  doi: 10.1007/s00208-019-01842-3. [20] E. Korotyaev and N. Saburova, Trace formulas for Schrödinger operators on periodic graphs, J. Math. Anal. Appl., 508 (2022), 33 pp. doi: 10.1016/j.jmaa.2021.125888. [21] Y. Last, On the measure of gaps and spectra for discrete 1D Schrödinger operators, Commun. Math. Phys., 149 (1992), 347-360. [22] F. Lledó and O. Post, Eigenvalue bracketing for discrete and metric graphs, J. Math. Anal. Appl., 348 (2008), 806-833.  doi: 10.1016/j.jmaa.2008.07.029. [23] P. Mnëv, Discrete path integral approach to the Selberg trace formula for regular graphs, Commun. Math. Phys., 274 (2007), 233-241.  doi: 10.1007/s00220-007-0257-8. [24] B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc., 21 (1989), 209-234.  doi: 10.1112/blms/21.3.209. [25] K. S. Novoselov, A.K. Geim and et al., Electric field effect in atomically thin carbon films, Science, 306 (2004), 666-669. [26] P. W. Sy and T. Sunada, Discrete Schrödinger operator on a graph, Nagoya Math. J., 125 (1992), 141-150.  doi: 10.1017/S0027763000003949. [27] A. Terras and D. Wallace, Selberg's trace formula on the $k$-regular tree and applications, Int. J.Math. Math. Sci., 2003 (2003), 501-526.  doi: 10.1155/S016117120311126X. [28] M. Toda, Theory of Nonlinear Lattices, 2$^{nd}$ edition, Springer, Berlin, 1989. doi: 10.1007/978-3-642-83219-2. [29] P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81.  doi: 10.1007/BF01418827.

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##### References:
 [1] G. Ahumada, Fonctions periodiques et formule des traces de Selberg sur les arbres, C. R. Acad. Sci. Paris, 305 (1987), 709-712. [2] R. Brooks, The Spectral Geometry of $k$-Regular Graphs, J. Anal. Math., 57 (1991), 120-151.  doi: 10.1007/BF03041067. [3] G. Chinta, J. Jorgenson and A. Karlsson, Heat kernels on regular graphs and generalized Ihara zeta function formulas, Monatsh. Math., 178 (2015), 171-190.  doi: 10.1007/s00605-014-0685-4. [4] D. Chelkak and E. Korotyaev, Spectral estimates for Schrödinger operators with periodic matrix potentials on the real line, Int. Math. Res. Not., 2006 (2006), 1-41.  doi: 10.1155/IMRN/2006/60314. [5] P. Deift and B. Simon, Almost periodic Schrödinger operators Ⅲ. The absolutely continuous spectrum in one dimension, Commun. Math. Phys., 90 (1983), 389-411. [6] J.S. Fabila-Carrasco, F. Lledó and O. Post, Spectral gaps and discrete magnetic Laplacians, Linear Algebra Appl., 547 (2018), 183-216.  doi: 10.1016/j.laa.2018.02.006. [7] J. S. Fabila-Carrasco, F. Lledó and O. Post, Spectral preorder and perturbations of discrete weighted graphs, Math. Ann., (2020), 49 pp. doi: 10.1007/s00208-020-02091-5. [8] E. Korotyaev, Estimates of periodic potentials in terms of gap lengths, Commun. Math. Phys., 197 (1998), 521-526.  doi: 10.1007/s002200050462. [9] E. Korotyaev, Characterization of the spectrum of Schrödinger operators with periodic distributions, Int. Math. Res. Not., 2003 (2003), 2019-2031.  doi: 10.1155/S1073792803209107. [10] E. Korotyaev, Estimates for the Hill operator. I, J. Differ. Equ., 162 (2000), 1-26.  doi: 10.1006/jdeq.1999.3684. [11] E. Korotyaev, Effective masses for zigzag nanotubes in magnetic fields, Lett. Math. Phys., 83 (2008), 83-95.  doi: 10.1007/s11005-007-0212-9. [12] E. Korotyaev and I. Krasovsky, Spectral estimates for periodic Jacobi matrices, Commun. Math. Phys., 234 (2003), 517-532.  doi: 10.1007/s00220-002-0768-2. [13] E. Korotyaev and A. Kutsenko, Inverse problem for the discrete periodic Schrödinger operator, Zapiski Nauchnyh Seminarov POMI, 315 (2004), 96-101.  doi: 10.1007/s10958-006-0104-z. [14] E. Korotyaev and N. Saburova, Schrödinger operators on periodic discrete graphs, J. Math. Anal. Appl., 420 (2014), 576-611.  doi: 10.1016/j.jmaa.2014.05.088. [15] E. Korotyaev and N. Saburova, Spectral band localization for Schrödinger operators on periodic graphs, Proc. Amer. Math. Soc., 143 (2015), 3951-3967.  doi: 10.1090/S0002-9939-2015-12586-5. [16] E. Korotyaev and N. Saburova, Effective masses for Laplacians on periodic graphs, J. Math. Anal. Appl., 436 (2016), 104-130.  doi: 10.1016/j.jmaa.2015.11.051. [17] E. Korotyaev and N. Saburova, Magnetic Schrödinger operators on periodic discrete graphs, J. Funct. Anal., 272 (2017), 1625-1660.  doi: 10.1016/j.jfa.2016.12.015. [18] E. Korotyaev and N. Saburova, Spectral estimates for Schrödinger operators on periodic discrete graphs, St. Petersburg Math. J., 30 (2018), 667-698.  doi: 10.1090/spmj/1565. [19] E. Korotyaev and N. Saburova, Invariants for Laplacians on periodic graphs, Math. Ann., 377 (2020), 723-758.  doi: 10.1007/s00208-019-01842-3. [20] E. Korotyaev and N. Saburova, Trace formulas for Schrödinger operators on periodic graphs, J. Math. Anal. Appl., 508 (2022), 33 pp. doi: 10.1016/j.jmaa.2021.125888. [21] Y. Last, On the measure of gaps and spectra for discrete 1D Schrödinger operators, Commun. Math. Phys., 149 (1992), 347-360. [22] F. Lledó and O. Post, Eigenvalue bracketing for discrete and metric graphs, J. Math. Anal. Appl., 348 (2008), 806-833.  doi: 10.1016/j.jmaa.2008.07.029. [23] P. Mnëv, Discrete path integral approach to the Selberg trace formula for regular graphs, Commun. Math. Phys., 274 (2007), 233-241.  doi: 10.1007/s00220-007-0257-8. [24] B. Mohar and W. Woess, A survey on spectra of infinite graphs, Bull. London Math. Soc., 21 (1989), 209-234.  doi: 10.1112/blms/21.3.209. [25] K. S. Novoselov, A.K. Geim and et al., Electric field effect in atomically thin carbon films, Science, 306 (2004), 666-669. [26] P. W. Sy and T. Sunada, Discrete Schrödinger operator on a graph, Nagoya Math. J., 125 (1992), 141-150.  doi: 10.1017/S0027763000003949. [27] A. Terras and D. Wallace, Selberg's trace formula on the $k$-regular tree and applications, Int. J.Math. Math. Sci., 2003 (2003), 501-526.  doi: 10.1155/S016117120311126X. [28] M. Toda, Theory of Nonlinear Lattices, 2$^{nd}$ edition, Springer, Berlin, 1989. doi: 10.1007/978-3-642-83219-2. [29] P. van Moerbeke, The spectrum of Jacobi matrices, Invent. Math., 37 (1976), 45-81.  doi: 10.1007/BF01418827.
The hexagonal lattice $\bf G$ and its fundamental graph $\bf G_*$ with edge indices; $a_1, a_2$ are the periods of $\bf G$. The fundamental cell $\Omega$ is shaded. The vertices $v_1, v_2$ of $\bf G$ from $\Omega$ are black points. Edge indices depend on the choice of the embedding of the periodic graph into $\mathbb R^2$ (i.e., the choice of $\Omega$). Cycle indices do not depend on this choice.
a) A periodic graph $\mathcal G$; $a_1, a_2$ are the periods of $\mathcal G$; the fundamental cell $\Omega$ is shaded; bold edges are bridges of $\mathcal G$; b) removal of all bridges disconnects the graph $\mathcal G$ into infinitely many connected components.
a) The Kagome lattice $\bf K$; b) the fundamental graph $\bf K_*$
a) A $\mathbb Z$-periodic graph $\mathcal G$; b) the fundamental graph $\mathcal G_*$
The spectrum of the Schrödinger operator $H = - \Delta+V$ with a $\nu$-periodic potential $V$ on $\mathbb Z$
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