The sharp Hardy uncertainty principle for Schrödinger evolutions. (English) Zbl 1220.35008

Escauriaza, Luis; Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis

Duke Math. J. 155, No. 1, 163-187 (2010).

The authors study \(u\) in \(C([0, T],L^2(\mathbb{R}^n))\) satisfying the Schrödinger equation \[ \partial_t u= i(\Delta u+ V(x,t)u)\quad\text{in }\mathbb{R}^n\times [0,T].\tag{1} \] They give conditions in terms of the potential \(V\) and the behaviour of \(u(0)\) and \(u(T)\) at infinity which are sufficient in order that \(u\equiv 0\). For instance they prove that \(\equiv 0\), provided \(V\) is bounded, \(\lim_{R\to\infty}\| V\|_{L^1([0, T], L^\infty(\mathbb{R}^n\setminus B_R))}= 0\) and there exist positive \(\alpha\), \(\beta\) such that \[ \| e^{|x|^2/\beta^2} u(0)\|_{L^2(\mathbb{R}^n)}<\infty\quad\text{and}\quad \| e^{|x|^2/\alpha^2} u(T)\|_{L^2(\mathbb{R}^n)}< \infty \] and \(T/\alpha\beta> 1/4\). They show that this condition is sharp in the sense that there exist \(V\) and a nonzero function \(u\) in \(C^\infty([0,T],{\mathcal S}(\mathbb{R}^n))\) satisfying above conditions in the case \(T/\alpha\beta= 1/4\). As a byproduct they derive estimates for the Gaussian decay of solutions of (1).

Reviewer: Johannes F. Brasche (Clausthal)

MSC:

35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B60	Continuation and prolongation of solutions to PDEs

Keywords:

Hardy uncertainty principle; unique continuation theorem; Gaussian decay bounds

References:

[1]	A. Bonami and B. Demange, A survey on uncertainty principles related to quadratic forms , Collect. Math. Vol . Extra (2006), 1–36. · Zbl 1107.30021
[2]	A. Bonami, B. Demange, and P. Jaming, Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms , Rev. Mat. Iberoamericana 19 (2006), 23–55. · Zbl 1037.42010 · doi:10.4171/RMI/337
[3]	M. G. Cowling and J. F. Price, “Generalizations of Heisenberg’s inequality” in Harmonic Analysis (Cortona, 1982) , Lecture Notes in Math. 992 (1983), Springer, Berlin, 443–449. · Zbl 0516.43002 · doi:10.1137/0515012
[4]	-, Bandwidth versus time concentration: The Heisenberg-Pauli-Weyl inequality , SIAM J. Math. Anal. 15 (1984), 151–165. · doi:10.1137/0515012
[5]	H. Dong and W. Staubach, Unique continuation for the Schrödinger equation with gradient vector potentials , Proc. Amer. Math. Soc. 135 (2007), 2141–2149. · Zbl 1119.35083 · doi:10.1090/S0002-9939-07-08813-2
[6]	L. Escauriaza, C. E. Kenig, G. Ponce, and L. Vega, On uniqueness properties of solutions of Schrödinger equations , Comm. Partial Differential Equations 31 (2006), 1811–1823. · Zbl 1124.35068 · doi:10.1080/03605300500530446
[7]	-, Convexity properties of solutions to the free Schrödinger equation with Gaussian decay , Math. Res. Lett. 15 (2008), 957–971. · Zbl 1169.35307 · doi:10.4310/MRL.2008.v15.n5.a10
[8]	-, Hardy’s uncertainty principle, convexity and Schrödinger evolutions , J. Eur. Math. Soc. (JEMS) 10 (2008), 883–907. · Zbl 1158.35018 · doi:10.4171/JEMS/134
[9]	G. H. Hardy, A theorem concerning Fourier transforms , J. London Math. Soc. s1 -8 (1933), 227–231. · Zbl 0007.30403 · doi:10.1112/jlms/s1-8.3.227
[10]	L. HöRmander, A uniqueness theorem of Beurling for Fourier transform pairs , Ark. Mat. 29 (1991), 237–240. · Zbl 0755.42009 · doi:10.1007/BF02384339
[11]	A. D. Ionescu and C. E. Kenig, \(L^p\) Carleman inequalities and uniqueness of solutions of nonlinear Schrödinger equations , Acta Math. 193 (2004), 193–239. · Zbl 1209.35128 · doi:10.1007/BF02392564
[12]	-, Uniqueness properties of solutions of Schrödinger equations , J. Funct. Anal. 232 (2006), 90–136. · Zbl 1092.35104 · doi:10.1016/j.jfa.2005.06.005
[13]	C. E. Kenig, G. Ponce, and L. Vega, On unique continuation for nonlinear Schrödinger equations , Comm. Pure Appl. Math. 56 (2003), 1247–1262. · Zbl 1041.35072 · doi:10.1002/cpa.10094
[14]	A. Sitaram, M. Sundari, and S. Thangavelu, Uncertainty principles on certain Lie groups , Proc. Indian Acad. Sci. Math. Sci. 105 (1995), 135–151. · Zbl 0857.43011 · doi:10.1007/BF02880360
[15]	E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lecture in Analysis II , Princeton Univ. Press, Princeton, 2003. · Zbl 1020.30001