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Dispersion for Schrödinger operators on regular trees. (English) Zbl 1487.81096

Summary: We prove dispersive estimates for two models: the adjacency matrix on a discrete regular tree, and the Schrödinger equation on a metric regular tree with the same potential on each edge/vertex. The latter model can be thought of as an extension of the case of periodic Schrödinger operators on the real line. We establish a \(t^{-3/2}\)-decay for both models which is sharp, as we give the first-order asymptotics.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
46N50 Applications of functional analysis in quantum physics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
35P20 Asymptotic distributions of eigenvalues in context of PDEs
05C05 Trees
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References:

[1] Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. With Formulas, Graphs, and Mathematical Tables. Tenth Printing (1972) · Zbl 0543.33001
[2] Ali Mehmeti, F.; Ammari, K.; Nicaise, S., Dispersive effects for the Schrödinger equation on the tadpole graph, J. Math. Anal. Appl., 448, 262-280 (2017) · Zbl 1357.35269 · doi:10.1016/j.jmaa.2016.10.060
[3] Ali Mehmeti, F.; Ammari, K.; Nicaise, S., Dispersive effects and high frequency behaviour for the Schrödinger equation in star-shaped networks, Portugal. Math., 72, 309-355 (2015) · Zbl 1334.34064 · doi:10.4171/PM/1970
[4] Anantharaman, N.; Ingremeau, M.; Sabri, M.; Winn, B., Absolutely continuous spectrum for quantum trees, Commun. Math. Phys., 383, 537-594 (2021) · Zbl 1461.81045 · doi:10.1007/s00220-021-03994-3
[5] Anantharaman, N.; Ingremeau, M.; Sabri, M.; Winn, B., Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit, J. Funct. Anal., 280, 108988 (2021) · Zbl 1461.81046 · doi:10.1016/j.jfa.2021.108988
[6] Anantharaman, N.; Sabri, M., Poisson kernel expansions for Schrödinger operators on trees, J. Spectr. Theory, 9, 243-268 (2019) · Zbl 1410.81022 · doi:10.4171/JST/247
[7] Anantharaman, N.; Sabri, M., Recent results of quantum ergodicity on graphs and further investigation, Ann. Fac. Sci. Toulouse Math., 28, 559-592 (2019) · Zbl 1434.82037 · doi:10.5802/afst.1609
[8] Avni, N.; Breuer, J.; Simon, B., Periodic Jacobi matrices on trees, Adv. Math., 370, 107241 (2020) · Zbl 1512.47056 · doi:10.1016/j.aim.2020.107241
[9] Banica, V.; Ignat, LI, Dispersion for the Schrödinger equation on the line with multiple Dirac delta potentials and on delta trees, Anal. PDE., 7, 903-927 (2014) · Zbl 1297.35200 · doi:10.2140/apde.2014.7.903
[10] Cai, K., Dispersion for Schrödinger Operators with One-gap Periodic Potentials on \({\mathbb{R}}^1 \), Dyn. PDE, 3, 71-92 (2006) · Zbl 1238.35107
[11] Carlson, R., Hill’s equation for a homogeneous tree, Electron. J. Differ. Equ., 23, 1-30 (1997) · Zbl 0890.34066
[12] Colin de Verdière, Y., Spectres de Graphes (1998), Paris: Société Mathématique de France, Paris · Zbl 0913.05071
[13] Cuccagna, S., Stability of standing waves for NLS with perturbed Lamé potential, J. Differ. Equ., 223, 112-160 (2006) · Zbl 1115.35120 · doi:10.1016/j.jde.2005.07.017
[14] Cuccagna, S., Dispersion for Schrödinger equation with periodic potential in 1D, Commun. Part. Differ. Equ., 33, 2064-2095 (2008) · Zbl 1163.35032 · doi:10.1080/03605300802501582
[15] Firsova, NE, On the time decay of a wave packet in a one-dimensional finite band periodic lattice, J. Math. Phys., 37, 1171-1181 (1996) · Zbl 0863.35030 · doi:10.1063/1.531454
[16] Hundertmark, D., Machinek, L., Meyries, M., Schnaubelt, R.: Operator Semigroups and Dispersive Equations. In: 16th Internet Seminar on Evolution Equations. Lecture Notes (2013)
[17] Ingremeau, M.; Sabri, M.; Winn, B., Quantum ergodicity for large equilateral quantum graphs, J. Lond. Math. Soc., 101, 82-109 (2020) · Zbl 1476.58033 · doi:10.1112/jlms.12259
[18] Kawarabayashi, T.; Suzuki, M., Decay rate of the Green function in a random potential on the Bethe lattice and a criterion for localization, J. Phys. A. Math. Gen., 26, 5729-5750 (1993) · doi:10.1088/0305-4470/26/21/014
[19] Keller, M.; Lenz, D.; Warzel, S., On the spectral theory of trees with finite cone type, Israel J. Math., 194, 107-135 (2013) · Zbl 1270.47003 · doi:10.1007/s11856-012-0059-3
[20] Klein, A., Extended states in the Anderson model on the Bethe lattice, Adv. Math., 133, 163-184 (1998) · Zbl 0899.60088 · doi:10.1006/aima.1997.1688
[21] Krasikov, I., Approximations for the Bessel and Airy functions with an explicit error term, LMS J. Comput. Math., 17, 209-225 (2014) · Zbl 1294.41024 · doi:10.1112/S1461157013000351
[22] Korotyaev, E., The propagation of the waves in periodic media at large time, Asymptot. Anal., 15, 1-24 (1997) · Zbl 0951.34054
[23] Landau, LJ, Bessel Functions: Monotonicity and Bounds, J. Lond. Math. Soc., 61, 197-215 (2000) · Zbl 0948.33001 · doi:10.1112/S0024610799008352
[24] Olenko, A. Ya.: Upper bound on \(\sqrt{x}J_\nu (x)\) and its applications. Integral Transforms Spec. Funct. 17, 455-467 (2006) · Zbl 1105.33007
[25] Olver, FWJ, Error bounds for stationary phase approximations, SIAM J. Math. Anal., 5, 19-29 (1974) · Zbl 0239.65028 · doi:10.1137/0505003
[26] Parnovski, L., Bethe-Sommerfeld conjecture, Ann. Henri Poincaré, 9, 457-508 (2008) · Zbl 1201.81054 · doi:10.1007/s00023-008-0364-x
[27] Pöschel, J.; Trubowitz, E., Inverse Spectral Theory (1987), Cambridge: Academic Press, Cambridge · Zbl 0623.34001
[28] Simon, B.: Spectral analysis of rank one perturbations and applications. In: Mathematical Quantum Theory. II. Schrödinger Operators (Vancouver, BC, 1993), CRM Proceedings and Lecture Notes, 8. American Mathematical Society, Providence, RI (1995) · Zbl 0824.47019
[29] Stefanov, A.; Kevrekidis, PG, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18, 1841-1857 (2005) · Zbl 1181.35266 · doi:10.1088/0951-7715/18/4/022
[30] Stein, EM, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (1993), Princeton: Princeton University Press, Princeton · Zbl 0821.42001
[31] Tao, T.: Nonlinear dispersive equations: local and global analysis. In: CBMS Regional Conference Series in Mathematics. Number 106. AMS (2006) · Zbl 1106.35001
[32] Teschl, G., Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators (2014), Providence: American Mathematical Society, Providence · Zbl 1342.81003
[33] Veliev, OA, Perturbation theory for the periodic multidimensional Schrödinger operator and the Bethe-Sommerfeld Conjecture, Int. J. Contemp. Math. Sci., 2, 19-87 (2007) · Zbl 1118.35019 · doi:10.12988/ijcms.2007.07003
[34] Zworski, M., Semiclassical Analysis (2012), Providence: American Mathematical Society, Providence · Zbl 1252.58001 · doi:10.1090/gsm/138
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