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Resonances for open quantum maps and a fractal uncertainty principle. (English) Zbl 1372.81101

Open quantum maps are useful models in the study of scattering phenomena and in particular scattering resonances. This paper investigates eigenvalues for a family of open quantum maps known as quantum open Baker’s maps. The corresponding trapped orbits form Cantor sets. The combinatorial and number theoretic properties of these sets make it possible to prove results on spectral gaps that lie well beyond what is known for other models. A fractal Weyl upper bound also is given and numerical results are provided.

MSC:

81S22 Open systems, reduced dynamics, master equations, decoherence
28A80 Fractals
81U05 \(2\)-body potential quantum scattering theory
35P05 General topics in linear spectral theory for PDEs

References:

[1] Balázs N.L., Voros A.: The quantized baker’s transformation. Ann. Phys. 190, 1-31 (1989) · Zbl 0664.58045 · doi:10.1016/0003-4916(89)90259-5
[2] Barkhofen S., Weich T., Potzuweit A., Stöckmann H.-J., Kuhl U., Zworski M.: Experimental observation of the spectral gap in microwave n-disk systems. Phys. Rev. Lett. 110, 164102 (2013) · doi:10.1103/PhysRevLett.110.164102
[3] Borthwick D.: Distribution of resonances for hyperbolic surfaces. Exp. Math. 23, 25-45 (2014) · Zbl 1321.58021 · doi:10.1080/10586458.2013.857282
[4] Borthwick D., Weich T.: Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions. J. Spectr. Th. 6, 267-329 (2016) · Zbl 1366.37053 · doi:10.4171/JST/125
[5] Bourgain J.: Bounded orthogonal systems and the \[{\Lambda(p)}\] Λ(p)-set problem. Acta Math. 162, 227-245 (1989) · Zbl 0674.43004 · doi:10.1007/BF02392838
[6] Bourgain, J., Dyatlov, S.: Spectral gaps without the pressure condition, preprint, arXiv:1612.09040 · Zbl 1421.11071
[7] Brun T.A., Schack R.: Realizing the quantum baker’s map on a NMR quantum computer. Phys. Rev. A 59, 2649 (1999) · doi:10.1103/PhysRevA.59.2649
[8] Carlo G.G., Benito R.M., Borondo F.: Theory of short periodic orbits for partially open quantum maps. Phys. Rev. E 94, 012222 (2016) · doi:10.1103/PhysRevE.94.012222
[9] Chen, X., Seeger, A.: Convolution powers of Salem measures with applications, preprint, arXiv:1509.00460 · Zbl 1375.42002
[10] Datchev K., Dyatlov S.: Fractal Weyl laws for asymptotically hyperbolic manifolds. Geom. Funct. Anal. 23, 1145-1206 (2013) · Zbl 1297.58006 · doi:10.1007/s00039-013-0225-8
[11] Degli Esposti M., Nonnenmacher S., Winn B.: Quantum variance and ergodicity for the baker’s map. Commun. Math. Phys. 263, 325-352 (2006) · Zbl 1107.81025 · doi:10.1007/s00220-005-1397-3
[12] Dolgopyat D.: On decay of correlations in Anosov flows. Ann. Math. 147(2), 357-390 (1998) · Zbl 0911.58029 · doi:10.2307/121012
[13] Dorin D., Chun-Kit L.: Some reductions of the spectral set conjecture to integers. Math. Proc. Camb. Philos. Soc. 156, 123-135 (2014) · Zbl 1294.52010 · doi:10.1017/S0305004113000558
[14] Dyatlov S.: Resonance projectors and asymptotics for r-normally hyperbolic trapped sets. J. Am. Math. Soc. 28, 311-381 (2015) · Zbl 1338.35316 · doi:10.1090/S0894-0347-2014-00822-5
[15] Dyatlov, S.: Improved fractal Weyl bounds for hyperbolic manifolds, with an appendix with David Borthwick and Tobias Weich. J. Europ. Math. Soc. arXiv:1512.00836 · Zbl 1420.35035
[16] Dyatlov, S., Jin, L.: Dolgopyat’s method and the fractal uncertainty principle, preprint, arXiv:1702.03619 · Zbl 1390.28016
[17] Dyatlov S., Zahl J.: Spectral gaps, additive energy, and a fractal uncertainty principle. Geom. Funct. Anal. 26, 1011-1094 (2016) · Zbl 1384.58019 · doi:10.1007/s00039-016-0378-3
[18] Ermann L., Frahm K.M., Shepelyansky D.L.: Google matrix analysis of directed networks. Rev. Mod. Phys. 87, 1261 (2015) · doi:10.1103/RevModPhys.87.1261
[19] Faure F., Tsujii M.: Band structure of the Ruelle spectrum of contact Anosov flows. C R Math. Acad. Sci. Paris 351, 385-391 (2013) · Zbl 1346.37030 · doi:10.1016/j.crma.2013.04.022
[20] Faure, F., Tsujii, M.: The semiclassical zeta function for geodesic flows on negatively curved manifolds, preprint. Invent. Math. arXiv:1311.4932 · Zbl 1379.37054
[21] Faure, F., Tsujii, M.: Prequantum transfer operator for Anosov diffeomorphism, Astérisque 375(2015) · Zbl 1417.37012
[22] Fuglede B.: Commuting self-adjoint partial differential operators and a group theoretic problem. J. Funct. Anal. 16, 101-121 (1974) · Zbl 0279.47014 · doi:10.1016/0022-1236(74)90072-X
[23] Gaspard P., Rice S.: Scattering from a classically chaotic repeller. J. Chem. Phys. 90, 2225-2241 (1989) · doi:10.1063/1.456017
[24] Guillopé L., Lin K.K., Zworski M.: The Selberg zeta function for convex co-compact Schottky groups. Commun. Math. Phys. 245:1, 149-176 (2004) · Zbl 1075.11059 · doi:10.1007/s00220-003-1007-1
[25] Hannay J.H., Keating J.P., de Almeida A.M.O.: Optical realization of the baker’s transformation. Nonlinearity 7, 1327-1342 (1994) · Zbl 0803.58025 · doi:10.1088/0951-7715/7/5/003
[26] Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, (1934) · Zbl 0010.10703
[27] Horn, R., Johnson, C.: Topics in Matrix Analysis. Cambridge University Press, (1991) · Zbl 0729.15001
[28] Ikawa M.: Decay of solutions of the wave equation in the exterior of several convex bodies. Ann. Inst. Fourier 38, 113-146 (1988) · Zbl 0636.35045 · doi:10.5802/aif.1137
[29] Jakobson D., Naud F.: On the critical line of convex co-compact hyperbolic surfaces. Geom. Funct. Anal. 22, 352-368 (2012) · Zbl 1284.30035 · doi:10.1007/s00039-012-0154-y
[30] Keating J.P., Nonnenmacher S., Novaes M., Sieber M.: On the resonance eigenstates of an open quantum baker map. Nonlinearity 21, 2591-2624 (2008) · Zbl 1158.35319 · doi:10.1088/0951-7715/21/11/007
[31] Keating J.P., Novaes M., Prado S.D., Sieber M.: Semiclassical structure of chaotic resonance eigenfunctions. Phys. Rev. Lett. 97, 150406 (2006) · doi:10.1103/PhysRevLett.97.150406
[32] Łaba I.: The spectral set conjecture and multiplicative properties of roots of polynomials. J. Lond. Math. Soc. 65, 661-671 (2002) · Zbl 1059.42005 · doi:10.1112/S0024610702003149
[33] Łaba, I., Wang, H.: Decoupling and near-optimal restriction estimates for Cantor sets, preprint, arXiv:1607.08302 · Zbl 1442.42031
[34] Lu W., Sridhar S., Zworski M.: Fractal Weyl laws for chaotic open systems. Phys. Rev. Lett. 91, 154101 (2003) · doi:10.1103/PhysRevLett.91.154101
[35] Malikiosis, R.-D., Kolountzakis, M.: Fuglede’s conjecture on cyclic groups of order pnq, preprint, arXiv:1612.01328 · Zbl 1404.42023
[36] Naud F.: Expanding maps on Cantor sets and analytic continuation of zeta functions. Ann. de l’ENS 38(4), 116-153 (2005) · Zbl 1110.37021
[37] Naud F.: Density and location of resonances for convex co-compact hyperbolic surfaces. Invent. Math. 195, 723-750 (2014) · Zbl 1291.58016 · doi:10.1007/s00222-013-0463-2
[38] Nonnenmacher S.: Spectral problems in open quantum chaos. Nonlinearity 24, R123 (2011) · Zbl 1229.35223 · doi:10.1088/0951-7715/24/12/R02
[39] Nonnenmacher S., Rubin M.: Resonant eigenstates in quantum chaotic scattering. Nonlinearity 20, 1387-1420 (2007) · Zbl 1138.81021 · doi:10.1088/0951-7715/20/6/004
[40] Nonnenmacher S., Sjöstrand J., Zworski M.: From open quantum systems to open quantum maps. Commun. Math. Phys. 304, 1, 1-48 (2011) · Zbl 1223.81127 · doi:10.1007/s00220-011-1214-0
[41] Nonnenmacher S., Sjöstrand J., Zworski M.: Fractal Weyl law for open quantum chaotic maps. Ann. Math. 179(2), 179-251 (2014) · Zbl 1293.81022 · doi:10.4007/annals.2014.179.1.3
[42] Nonnenmacher S., Zworski M.: Fractal Weyl laws in discrete models of chaotic scattering. J. Phys. A 38, 10683-10702 (2005) · Zbl 1082.81079 · doi:10.1088/0305-4470/38/49/014
[43] Nonnenmacher S., Zworski M.: Distribution of resonances for open quantum maps. Commun. Math. Phys. 269, 311-365 (2007) · Zbl 1114.81043 · doi:10.1007/s00220-006-0131-0
[44] Nonnenmacher S., Zworski M.: Quantum decay rates in chaotic scattering. Acta Math. 203, 149-233 (2009) · Zbl 1226.35061 · doi:10.1007/s11511-009-0041-z
[45] Novaes M.: Resonances in open quantum maps. J. Phys. A 46, 143001 (2013) · Zbl 1267.81170 · doi:10.1088/1751-8113/46/14/143001
[46] Novaes M., Pedrosa J.M., Wisniacki D., Carlo G.G., Keating J.P.: Quantum chaotic resonances from short periodic orbits. Phys. Rev. E 80, 035202 (2009) · doi:10.1103/PhysRevE.80.035202
[47] Patterson S.J.: On a lattice-point problem in hyperbolic space and related questions in spectral theory. Ark. Mat. 26, 167-172 (1988) · Zbl 0645.10040 · doi:10.1007/BF02386116
[48] Petkov V., Stoyanov L.: Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function. Anal. PDE 3, 427-489 (2010) · Zbl 1251.37031 · doi:10.2140/apde.2010.3.427
[49] Potzuweit A., Weich T., Barkhofen S., Kuhl U., Stöckmann H.-J., Zworski M.: Weyl asymptotics: from closed to open systems. Phys. Rev. E 86, 066205 (2012) · doi:10.1103/PhysRevE.86.066205
[50] Saraceno M.: Classical structures in the quantized baker transformation. Ann. Phys. 199, 37-60 (1990) · Zbl 0724.58059 · doi:10.1016/0003-4916(90)90367-W
[51] Saraceno M., Voros A.: Towards a semiclassical theory of the quantum baker’s map. Phys. D Nonlinear Phenom. 79, 206-268 (1994) · Zbl 0888.58036 · doi:10.1016/S0167-2789(05)80007-7
[52] Shmerkin, P., Suomala, V.: A class of random Cantor measures, with applications, preprint, arXiv:1603.08156 · Zbl 1317.28004
[53] Sjöstrand J.: Geometric bounds on the density of resonances for semiclassical problems. Duke Math. J. 60:1, 1-57 (1990) · Zbl 0702.35188 · doi:10.1215/S0012-7094-90-06001-6
[54] Sjöstrand J., Zworski M.: Fractal upper bounds on the density of semiclassical resonances. Duke Math. J. 137, 381-459 (2007) · Zbl 1201.35189 · doi:10.1215/S0012-7094-07-13731-1
[55] Stoyanov L.: Spectra of Ruelle transfer operators for axiom A flows. Nonlinearity 24, 1089-1120 (2011) · Zbl 1230.37040 · doi:10.1088/0951-7715/24/4/005
[56] Stoyanov L.: Non-integrability of open billiard flows and Dolgopyat-type estimates. Erg. Theory Dyn. Syst. 32, 295-313 (2012) · Zbl 1323.37017 · doi:10.1017/S0143385710000933
[57] Tao, T., Vu, V.: Additive Combinatorics, Cambridge Studies in Advanced Mathematics 105. Cambridge University Press, (2006)
[58] Titchmarsh, E.C.: The Theory of Functions, Second Edition. Oxford University Press, (1939) · Zbl 0702.35188
[59] Wiener N., Wintner A.: Fourier-Stieltjes transforms and singular infinite convolutions. Am. J. Math. 60, 513-522 (1938) · JFM 64.0223.02 · doi:10.2307/2371591
[60] Zelditch, S.: Recent developments in mathematical quantum chaos, Curr. Dev. Math. 115-204 (2009) · Zbl 1223.37113
[61] Zworski M.: Dimension of the limit set and the density of resonances for convex co-compact hyperbolic surfaces. Invent. Math. 136, 353-409 (1999) · Zbl 1016.58014 · doi:10.1007/s002220050313
[62] Zworski, M.: Semiclassical analysis, Graduate Studies in Mathematics, vol. 138. AMS, Providence (2012) · Zbl 1252.58001
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