## Semiclassical Cauchy estimates and applications

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## 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 $h\to 0$.## References

- 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**, DOI 10.1002/mma.2951 - W. E. Bies and E. J. Heller,
*Nodal structure of chaotic eigenfunctions*, J. Phys. A**35**(2002), no. 27, 5673–5685. MR**1917256**, DOI 10.1088/0305-4470/35/27/309 - 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**, DOI 10.1007/s00220-011-1225-x - Jean-Marc Delort,
*F.B.I. transformation*, Lecture Notes in Mathematics, vol. 1522, Springer-Verlag, Berlin, 1992. Second microlocalization and semilinear caustics. MR**1186645**, DOI 10.1007/BFb0095604 - Harold Donnelly and Charles Fefferman,
*Nodal sets of eigenfunctions on Riemannian manifolds*, Invent. Math.**93**(1988), no. 1, 161–183. MR**943927**, DOI 10.1007/BF01393691 - Gerald B. Folland,
*Harmonic analysis in phase space*, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR**983366**, DOI 10.1515/9781400882427 - 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**, DOI 10.1002/cpa.3160400305 - 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** - 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. - Robert Hardt and Leon Simon,
*Nodal sets for solutions of elliptic equations*, J. Differential Geom.**30**(1989), no. 2, 505–522. MR**1010169** - 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**, DOI 10.2140/apde.2012.5.1133 - 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**, DOI 10.1007/978-3-642-96750-4 - 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** - B. Hanin, S. Zelditch, and P. Zhou,
*Nodal sets of random eigenfunctions for the isotropic harmonic oscillator,*arXiv:1310.4532 - André Martinez,
*An introduction to semiclassical and microlocal analysis*, Universitext, Springer-Verlag, New York, 2002. MR**1872698**, DOI 10.1007/978-1-4757-4495-8 - J. Sjöstrand,
*Singularités analytiques microlocales*. Astérisque,**95**(1982), 1-166. - 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**, DOI 10.4310/MRL.2011.v18.n1.a3 - 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**, DOI 10.4310/MRL.2012.v19.n6.a14 - 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** - Maciej Zworski,
*Semiclassical analysis*, Graduate Studies in Mathematics, vol. 138, American Mathematical Society, Providence, RI, 2012. MR**2952218**, DOI 10.1090/gsm/138

## Additional Information

**Long Jin**- Affiliation: Department of Mathematics, Evans Hall, University of California, Berkeley, California 94720
- MR Author ID: 1076590
- Email: jinlong@math.berkeley.edu
- Received by editor(s): March 26, 2013
- Received by editor(s) in revised form: February 2, 2015
- Published electronically: March 18, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 975-995 - MSC (2010): Primary 35J10; Secondary 32D15, 58J50
- DOI: https://doi.org/10.1090/tran/6715
- MathSciNet review: 3572261