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Semiclassical Cauchy estimates and applications


Author: Long Jin
Journal: Trans. Amer. Math. Soc. 369 (2017), 975-995
MSC (2010): Primary 35J10; Secondary 32D15, 58J50
DOI: https://doi.org/10.1090/tran/6715
Published electronically: March 18, 2016
MathSciNet review: 3572261
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Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we consider the solutions to semiclassical analytic Schrödinger equations and prove a semiclassical version of the Cauchy estimate. As an application, we study the asymptotics for the Hausdorff measures of the nodal sets of the solutions on a compact analytic manifold as $ h\to 0$.


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  • [1] Laurent Bakri and Jean-Baptiste Casteras, Quantitative uniqueness for Schrödinger operator with regular potentials, Math. Methods Appl. Sci. 37 (2014), no. 13, 1992-2008. MR 3245115, https://doi.org/10.1002/mma.2951
  • [2] W. E. Bies and E. J. Heller, Nodal structure of chaotic eigenfunctions, J. Phys. A 35 (2002), no. 27, 5673-5685. MR 1917256 (2003h:81066), https://doi.org/10.1088/0305-4470/35/27/309
  • [3] Tobias H. Colding and William P. Minicozzi II, Lower bounds for nodal sets of eigenfunctions, Comm. Math. Phys. 306 (2011), no. 3, 777-784. MR 2825508, https://doi.org/10.1007/s00220-011-1225-x
  • [4] Jean-Marc Delort, F.B.I. transformation, Lecture Notes in Mathematics, vol. 1522, Springer-Verlag, Berlin, 1992. Second microlocalization and semilinear caustics. MR 1186645 (93i:35010)
  • [5] Harold Donnelly and Charles Fefferman, Nodal sets of eigenfunctions on Riemannian manifolds, Invent. Math. 93 (1988), no. 1, 161-183. MR 943927 (89m:58207), https://doi.org/10.1007/BF01393691
  • [6] Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366 (92k:22017)
  • [7] Nicola Garofalo and Fang-Hua Lin, Unique continuation for elliptic operators: a geometric-variational approach, Comm. Pure Appl. Math. 40 (1987), no. 3, 347-366. MR 882069 (88j:35046), https://doi.org/10.1002/cpa.3160400305
  • [8] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998 edition. MR 1814364 (2001k:35004)
  • [9] Q. Han and F.-H. Lin, Nodal sets of Solutions of Elliptic Differential Equations, Book in preparation, 2007, available at http://www.nd.edu/$ \sim $qhan.
  • [10] Robert Hardt and Leon Simon, Nodal sets for solutions of elliptic equations, J. Differential Geom. 30 (1989), no. 2, 505-522. MR 1010169 (90m:58031)
  • [11] Hamid Hezari and Christopher D. Sogge, A natural lower bound for the size of nodal sets, Anal. PDE 5 (2012), no. 5, 1133-1137. MR 3022851, https://doi.org/10.2140/apde.2012.5.1133
  • [12] Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035 (85g:35002a)
  • [13] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639 (91a:32001)
  • [14] B. Hanin, S. Zelditch, and P. Zhou, Nodal sets of random eigenfunctions for the isotropic harmonic oscillator, arXiv:1310.4532
  • [15] André Martinez, An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, New York, 2002. MR 1872698 (2003b:35010)
  • [16] J. Sjöstrand, Singularités analytiques microlocales. Astérisque, 95 (1982), 1-166.
  • [17] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets, Math. Res. Lett. 18 (2011), no. 1, 25-37. MR 2770580 (2012c:58055), https://doi.org/10.4310/MRL.2011.v18.n1.a3
  • [18] Christopher D. Sogge and Steve Zelditch, Lower bounds on the Hausdorff measure of nodal sets II, Math. Res. Lett. 19 (2012), no. 6, 1361-1364. MR 3091613, https://doi.org/10.4310/MRL.2012.v19.n6.a14
  • [19] Shing-Tung Yau, Open problems in geometry, Differential geometry: partial differential equations on manifolds (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 1-28. MR 1216573 (94k:53001)
  • [20] Maciej Zworski, Semiclassical analysis, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR 2952218

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Additional Information

Long Jin
Affiliation: Department of Mathematics, Evans Hall, University of California, Berkeley, California 94720
Email: jinlong@math.berkeley.edu

DOI: https://doi.org/10.1090/tran/6715
Received by editor(s): March 26, 2013
Received by editor(s) in revised form: February 2, 2015
Published electronically: March 18, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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