From resolvent estimates to unique continuation for the Schrödinger equation
HTML articles powered by AMS MathViewer
- by Ihyeok Seo PDF
- Trans. Amer. Math. Soc. 368 (2016), 8755-8784 Request permission
Abstract:
In this paper we develop an abstract method to handle the problem of unique continuation for the Schrödinger equation $(i\partial _t+\Delta )u=V(x)u$. In general the problem is to find a class of potentials $V$ which allows the unique continuation. The key point of our work is to make a direct link between the problem and the weighted $L^2$ resolvent estimates $\|(-\Delta -z)^{-1}f\|_{L^2(|V|)}\leq C\|f\|_{L^2(|V|^{-1})}$. We carry it out in an abstract way, and thereby we do not need to deal with each of the potential classes. To do so, we will make use of the limiting absorption principle and Kato $H$-smoothing theorem in spectral theory, and employ some tools from harmonic analysis. Once the resolvent estimate is set up for a potential class, from our abstract theory the unique continuation would follow from the same potential class. Also, it turns out that there can be no dented surface on the boundary of the maximal open zero set of the solution $u$. In this regard, another main issue for us is to know which class of potentials allows the resolvent estimate. We establish such a new class which contains previously known ones, and we will also apply it to the problem of well-posedness for the equation.References
- Shmuel Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 2, 151–218. MR 397194
- Juan A. Barceló, Jonathan M. Bennett, Anthony Carbery, Alberto Ruiz, and M. Cruz Vilela, A note on weighted estimates for the Schrödinger operator, Rev. Mat. Complut. 21 (2008), no. 2, 481–488. MR 2441963, DOI 10.5209/rev_{r}ema.2008.v21.n2.16405
- J. A. Barceló, J. M. Bennett, A. Ruiz, and M. C. Vilela, Local smoothing for Kato potentials in three dimensions, Math. Nachr. 282 (2009), no. 10, 1391–1405. MR 2571701, DOI 10.1002/mana.200610808
- J. A. Barcelo, A. Ruiz, and L. Vega, Weighted estimates for the Helmholtz equation and some applications, J. Funct. Anal. 150 (1997), no. 2, 356–382. MR 1479544, DOI 10.1006/jfan.1997.3131
- A. P. Calderón, Notes on singular integrals, MIT Press, Cambridge, Mass., 1961.
- T. Carleman, Sur un problème d’unicité pur les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat., Astr. Fys. 26 (1939), no. 17, 9 (French). MR 0000334
- K. M. Case, Singular potentials, Phys. Rev. (2) 80 (1950), 797–806. MR 40532, DOI 10.1103/PhysRev.80.797
- Sagun Chanillo and Eric Sawyer, Unique continuation for $\Delta +v$ and the C. Fefferman-Phong class, Trans. Amer. Math. Soc. 318 (1990), no. 1, 275–300. MR 958886, DOI 10.1090/S0002-9947-1990-0958886-6
- R. R. Coifman and R. Rochberg, Another characterization of BMO, Proc. Amer. Math. Soc. 79 (1980), no. 2, 249–254. MR 565349, DOI 10.1090/S0002-9939-1980-0565349-8
- Elena Cordero and Fabio Nicola, Sharpness of some properties of Wiener amalgam and modulation spaces, Bull. Aust. Math. Soc. 80 (2009), no. 1, 105–116. MR 2520527, DOI 10.1017/S0004972709000070
- Piero D’Ancona and Luca Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations 33 (2008), no. 4-6, 1082–1112. MR 2424390, DOI 10.1080/03605300701743749
- Donatella Danielli, A Fefferman-Phong type inequality and applications to quasilinear subelliptic equations, Potential Anal. 11 (1999), no. 4, 387–413. MR 1719837, DOI 10.1023/A:1008674906902
- Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
- Damiano Foschi, Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ. 2 (2005), no. 1, 1–24. MR 2134950, DOI 10.1142/S0219891605000361
- I. M. Gel’fand and G. E. Shilov, Generalized functions. Vol. I: Properties and operations, Academic Press, New York-London, 1964. Translated by Eugene Saletan. MR 0166596
- Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316, DOI 10.1007/978-0-387-09434-2
- Alexandru D. Ionescu and Carlos E. Kenig, $L^p$ Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations, Acta Math. 193 (2004), no. 2, 193–239. MR 2134866, DOI 10.1007/BF02392564
- Tosio Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann. 162 (1965/66), 258–279. MR 190801, DOI 10.1007/BF01360915
- Tosio Kato and Kenji Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys. 1 (1989), no. 4, 481–496. MR 1061120, DOI 10.1142/S0129055X89000171
- Markus Keel and Terence Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998), no. 5, 955–980. MR 1646048, DOI 10.1353/ajm.1998.0039
- C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J. 55 (1987), no. 2, 329–347. MR 894584, DOI 10.1215/S0012-7094-87-05518-9
- Carlos E. Kenig and Christopher D. Sogge, A note on unique continuation for Schrödinger’s operator, Proc. Amer. Math. Soc. 103 (1988), no. 2, 543–546. MR 943081, DOI 10.1090/S0002-9939-1988-0943081-3
- Carlos E. Kenig and Peter A. Tomas, On conjectures of Rivière and Strichartz, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 4, 694–697. MR 532556, DOI 10.1090/S0273-0979-1979-14674-3
- R. Kerman and E. Sawyer, The trace inequality and eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4, 207–228 (English, with French summary). MR 867921, DOI 10.5802/aif.1074
- Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted $L^{p}$ spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254. MR 561835, DOI 10.1090/S0002-9947-1980-0561835-X
- R. Lascar and C. Zuily, Unicité et non unicité du problème de Cauchy pour une classe d’opérateurs différentiels à caractéristiques doubles, Duke Math. J. 49 (1982), no. 1, 137–162 (French). MR 650374, DOI 10.1215/S0012-7094-82-04910-9
- Sanghyuk Lee and Ihyeok Seo, A note on unique continuation for the Schrödinger equation, J. Math. Anal. Appl. 389 (2012), no. 1, 461–468. MR 2876512, DOI 10.1016/j.jmaa.2011.11.067
- Kiyoshi Mochizuki, Uniform resolvent estimates for magnetic Schrödinger operators and smoothing effects for related evolution equations, Publ. Res. Inst. Math. Sci. 46 (2010), no. 4, 741–754. MR 2791005, DOI 10.2977/PRIMS/24
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Michael Reed and Barry Simon, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0493420
- Michael Reed and Barry Simon, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493421
- Igor Rodnianski and Wilhelm Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math. 155 (2004), no. 3, 451–513. MR 2038194, DOI 10.1007/s00222-003-0325-4
- Alberto Ruiz and Luis Vega, On local regularity of Schrödinger equations, Internat. Math. Res. Notices 1 (1993), 13–27. MR 1201747, DOI 10.1155/S1073792893000029
- Alberto Ruiz and Luis Vega, Local regularity of solutions to wave equations with time-dependent potentials, Duke Math. J. 76 (1994), no. 3, 913–940. MR 1309336, DOI 10.1215/S0012-7094-94-07636-9
- Ihyeok Seo, Unique continuation for the Schrödinger equation with potentials in Wiener amalgam spaces, Indiana Univ. Math. J. 60 (2011), no. 4, 1203–1227. MR 2975341, DOI 10.1512/iumj.2011.60.4824
- Ihyeok Seo, Carleman estimates for the Schrödinger operator and applications to unique continuation, Commun. Pure Appl. Anal. 11 (2012), no. 3, 1013–1036. MR 2968606, DOI 10.3934/cpaa.2012.11.1013
- Ihyeok Seo, Global unique continuation from a half space for the Schrödinger equation, J. Funct. Anal. 266 (2014), no. 1, 85–98. MR 3121722, DOI 10.1016/j.jfa.2013.09.025
- Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Robert S. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), no. 3, 705–714. MR 512086
- Gerald Teschl, Mathematical methods in quantum mechanics, Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009. With applications to Schrödinger operators. MR 2499016, DOI 10.1090/gsm/099
- Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 358216, DOI 10.1090/S0002-9904-1975-13790-6
- M. C. Vilela, Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc. 359 (2007), no. 5, 2123–2136. MR 2276614, DOI 10.1090/S0002-9947-06-04099-2
- Claude Zuily, Uniqueness and nonuniqueness in the Cauchy problem, Progress in Mathematics, vol. 33, Birkhäuser Boston, Inc., Boston, MA, 1983. MR 701544, DOI 10.1007/978-1-4899-6656-8
Additional Information
- Ihyeok Seo
- Affiliation: Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
- MR Author ID: 927090
- Email: ihseo@skku.edu
- Received by editor(s): September 3, 2014
- Received by editor(s) in revised form: November 2, 2015, and November 15, 2014
- Published electronically: January 13, 2016
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 8755-8784
- MSC (2010): Primary 47A10, 35B60; Secondary 35Q40
- DOI: https://doi.org/10.1090/tran/6635
- MathSciNet review: 3551588