\(L^{p}\) norms, nodal sets, and quantum ergodicity. (English) Zbl 1332.81067
Summary: For small range of \(p > 2\), we improve the \(L^p\) bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35P20 | Asymptotic distributions of eigenvalues in context of PDEs |
| 53C17 | Sub-Riemannian geometry |
References:
| [1] | Anantharamn, N.; Rivière, G., Dispersion and controllability for the Schrödinger equation on negatively curved manifolds, Anal. PDE, 5, 313-338 (2012) · Zbl 1267.35176 |
| [2] | Baladi, V.; Liverani, C., Exponential decay of correlations for piecewise cone hyperbolic contact flows, Comm. Math. Phys., 314, 3, 689-773 (2012) · Zbl 1386.37030 |
| [3] | Bérard, P. H., On the wave equation on a compact Riemannian manifold without conjugate points, Math. Z., 155, 3, 249-276 (1977) · Zbl 0341.35052 |
| [4] | Blair, M.; Sogge, C., On Kakeya-Nikodym averages, \(L^p\)-norms and lower bounds for nodal sets of eigenfunctions in higher dimensions, J. Eur. Math. Soc. (JEMS) (2015), in press · Zbl 1330.58023 |
| [5] | Bourgain, J., Eigenfunctions bounds for the Laplacian on the \(n\)-torus, Int. Math. Res. Not. IMRN, 3, 61-66 (1993) · Zbl 0779.58039 |
| [6] | Bourgain, J., Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces, Israel J. Math., 193, 441-458 (2013) · Zbl 1271.42039 |
| [7] | Bourgain, J.; Demeter, C., The proof of the \(l^2\) decoupling conjecture (2014) |
| [8] | Brüning, J., Über Knoten Eigenfunktionen des Laplace-Beltrami Operators, Math. Z., 158, 15-21 (1978) · Zbl 0349.58012 |
| [9] | Colding, T.; Minicozzi, W. P., Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys., 306, 3, 777-784 (2011) · Zbl 1238.58020 |
| [10] | Colin de Verdière, Y., Ergodicité et fonctions propres du Laplacien, Comm. Math. Phys., 102, 497-502 (1985) · Zbl 0592.58050 |
| [11] | Dolgopyat, D., On decay of correlations in Anosov flows, Ann. of Math. (2), 147, 2, 357-390 (1998) · Zbl 0911.58029 |
| [12] | Dong, R. T., Nodal sets of eigenfunctions on Riemann surfaces, J. Differential Geom., 36, 2, 493-506 (1992) · Zbl 0776.53024 |
| [13] | Donnelly, H.; Fefferman, C., Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math., 93, 161-183 (1988) · Zbl 0659.58047 |
| [14] | Donnelly, H.; Fefferman, C., Nodal sets for eigenfunctions of the Laplacian on surfaces, J. Amer. Math. Soc., 3, 332-353 (1990) · Zbl 0702.58077 |
| [15] | Duistermaat, J. J.; Guillemin, V., The spectrum of elliptic operators and periodic bicharacteristics, Invent. Math., 29, 39-79 (1975) · Zbl 0307.35071 |
| [16] | Dyatlov, S.; Zworski, M., Dynamical zeta function for Anosov flows via microlocal analysis (2013), preprint |
| [17] | Faure, F.; Sjöstrand, J., Upper bound on the density of Ruelle resonances for Anosov flows, Comm. Math. Phys., 308, 325-364 (2011) · Zbl 1260.37016 |
| [18] | Faure, F.; Tsujii, M., The semiclassical zeta function for geodesic flows on negatively curved manifolds (2013), preprint |
| [19] | Giulietti, P.; Liverani, C.; Pollicott, M., Anosov flows and dynamical zeta functions, Ann. of Math., 178, 2, 687-773 (2013) · Zbl 1418.37042 |
| [20] | Han, X., Small scale quantum ergodicity on negatively curved manifolds (2014) |
| [21] | Hassell, A.; Tacy, M., Improvement of eigenfunction estimates on manifolds of nonpositive curvature, Forum Math., 27, 3, 1435-1451 (2015) · Zbl 1410.35264 |
| [22] | Helffer, B.; Martinez, A.; Robert, D., Ergodicité et limite semi-classique, Comm. Math. Phys., 109, 313-326 (1987) · Zbl 0624.58039 |
| [23] | Hezari, H.; Rivière, G., Quantitative equidistribution properties of toral eigenfunctions (2015), to appear in J. Spectr. Theory · Zbl 1369.58022 |
| [24] | Hezari, H.; Sogge, C. D., A natural lower bound for the size of nodal sets, Anal. PDE, 5, 5, 1133-1137 (2012) · Zbl 1329.35224 |
| [25] | Iwaniec, H.; Sarnak, P., \(L^\infty\) norms of eigenfunctions of arithmetic surfaces, Ann. of Math., 141, 301-320 (1995) · Zbl 0833.11019 |
| [26] | Jung, J., Quantitative quantum ergodicity conjecture and the nodal domains of Maass-Hecke cusp forms (2013), preprint |
| [27] | Koch, H.; Tataru, D.; Zworski, M., Semiclassical \(L^p\) estimates, Ann. Henri Poincaré, 8, 5, 885-916 (2007) · Zbl 1133.58025 |
| [28] | Liverani, C., On contact Anosov flows, Ann. of Math., 159, 1275-1312 (2004) · Zbl 1067.37031 |
| [29] | Luo, W.; Sarnak, P., Quantum ergodicity of eigenfunctions on \(PSL_2(Z) \backslash H^2\), Publ. Math. Inst. Hautes Études Sci., 81, 207-237 (1995) · Zbl 0852.11024 |
| [30] | Marklof, J.; Rudnick, Z., Almost all eigenfunctions of a rational polygon are uniformly distributed, J. Spectr. Theory, 2, 107-113 (2012) · Zbl 1242.37027 |
| [31] | Melbourne, I.; Török, A., Convergence of moments for axiom A and non-uniformly hyperbolic flows, Ergodic Theory Dynam. Systems, 32, 3, 1091-1100 (2012) · Zbl 1263.37047 |
| [32] | Nonnenmacher, S.; Zworski, M., Decay of correlations for normally hyperbolic trapping, Invent. Math., 200, 2, 345-438 (2015) · Zbl 1377.37049 |
| [33] | Ratner, M., The central limit theorem for geodesic flows on \(n\)-dimensional manifolds, Israel J. Math., 16, 181-197 (1973) · Zbl 0283.58010 |
| [34] | Ratner, M., The rate of mixing for geodesic and horocycle flows, Ergodic Theory Dynam. Systems, 7, 267-288 (1987) · Zbl 0623.22008 |
| [35] | Rivière, G., Remarks on quantum ergodicity, J. Mod. Dyn., 7, 119-133 (2013) · Zbl 1348.58019 |
| [36] | Schubert, R., Upper bounds on the rate of quantum ergodicity, Ann. Henri Poincaré, 7, 1085-1098 (2006) · Zbl 1099.81032 |
| [37] | Shnirelman, A., Ergodic properties of eigenfunctions, Usp. Mat. Nauk, 29, 181-182 (1974) · Zbl 0324.58020 |
| [38] | Smith, H.; Zworski, M., Pointwise bounds on quasimodes of semiclassical Schrödinger operators in dimension two, Math. Res. Lett., 20, 401-408 (2013) · Zbl 1284.58009 |
| [39] | Sogge, C., Concerning the \(L^p\) norm of spectral clusters for second-order elliptic operators on compact manifolds, J. Funct. Anal., 77, 1, 123-138 (1988) · Zbl 0641.46011 |
| [40] | Sogge, D., Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, vol. 105 (1993), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0783.35001 |
| [41] | Sogge, C. D., Kakeya-Nikodym averages and \(L^p\)-norms of eigenfunctions, Tohoku Math. J. (2), 63, 4, 519-538 (2011) · Zbl 1234.35156 |
| [42] | Sogge, C. D., Localized \(L^p\)-estimates of eigenfunctions: a note on an article of Hezari and Rivière (2015), preprint · Zbl 1331.58036 |
| [43] | Sogge, C. D.; Toth, J. A.; Zelditch, S., About the blowup of quasimodes on Riemannian manifolds, J. Geom. Anal., 21, 1, 150-173 (2011) · Zbl 1214.58012 |
| [44] | Sogge, C.; Zelditch, S., On eigenfunction restriction estimates and \(L^4\)-bounds for compact surfaces with nonpositive curvature (2011) |
| [45] | Sogge, C. D.; Zelditch, S., Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett., 18, 1, 25-37 (2011) · Zbl 1242.58017 |
| [46] | Sogge, C. D.; Zelditch, S., Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett., 19, 6, 1361-1364 (2012) · Zbl 1283.58020 |
| [47] | Sogge, C. D.; Zelditch, S., Concerning the \(L^4\) norms of typical eigenfunctions on compact surfaces, (Recent Developments in Geometry and Analysis. Recent Developments in Geometry and Analysis, Adv. Lectures Math., vol. 23 (2012), Int. Press: Int. Press Somerville, MA), 407-423 · Zbl 1317.58035 |
| [48] | Tsujii, M., Quasi-compactness of transfer operators for contact Anosov flows, Nonlinearity, 23, 7, 1495-1545 (2010) · Zbl 1200.37025 |
| [49] | Tsujii, M., Contact Anosov flows and the Fourier-Bros-Iagolnitzer transform, Ergodic Theory Dynam. Systems, 32, 6, 2083-2118 (2012) · Zbl 1276.37020 |
| [50] | Yau, S.-T., Survey on partial differential equations in differential geometry, (Seminar on Differential Geometry. Seminar on Differential Geometry, Ann. of Math. Stud., vol. 102 (1982), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ), 3-71 · Zbl 0478.53001 |
| [51] | Young, M.-P., The quantum unique ergodicity conjecture for thin sets (2013), preprint |
| [52] | Zelditch, S., Uniform distribution of the eigenfunctions on compact hyperbolic surfaces, Duke Math. J., 55, 919-941 (1987) · Zbl 0643.58029 |
| [53] | Zelditch, S., On the rate of quantum ergodicity. I. Upper bounds, Comm. Math. Phys., 160, 1, 81-92 (1994) · Zbl 0788.58043 |
| [54] | Zelditch, S., Quantum ergodicity of \(C^\ast\) dynamical systems, Comm. Math. Phys., 177, 2, 507-528 (1996) · Zbl 0856.58019 |
| [55] | Zelditch, S., Eigenfunctions and nodal sets, (Geometry and Topology. Geometry and Topology, Surv. Differ. Geom., vol. 18 (2013), Int. Press: Int. Press Somerville, MA), 237-308 · Zbl 1352.53032 |
| [56] | Zworski, M., Semiclassical Analysis, Grad. Stud. Math., vol. 138 (2012), AMS · Zbl 1252.58001 |
| [57] | Zygmund, A., On Fourier coefficients and transforms of functions of two variables, Studia Math., 50, 2, 189-201 (1974) · Zbl 0278.42005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
