Analysis & PDE

  • Anal. PDE
  • Volume 10, Number 6 (2017), 1285-1315.

Local energy decay and smoothing effect for the damped Schrödinger equation

Moez Khenissi and Julien Royer

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Abstract

We prove the local energy decay and the global smoothing effect for the damped Schrödinger equation on d . The self-adjoint part is a Laplacian associated to a long-range perturbation of the flat metric. The proofs are based on uniform resolvent estimates obtained by the dissipative Mourre method. All the results depend on the strength of the dissipation that we consider.

Article information

Source
Anal. PDE, Volume 10, Number 6 (2017), 1285-1315.

Dates
Received: 3 June 2015
Revised: 14 March 2017
Accepted: 24 April 2017
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1510843524

Digital Object Identifier
doi:10.2140/apde.2017.10.1285

Mathematical Reviews number (MathSciNet)
MR3678489

Zentralblatt MATH identifier
1372.35262

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35Q41: Time-dependent Schrödinger equations, Dirac equations 35B65: Smoothness and regularity of solutions 47A55: Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15] 47B44: Accretive operators, dissipative operators, etc.

Keywords
local energy decay smoothing effect damped Schrödinger equation resolvent estimates

Citation

Khenissi, Moez; Royer, Julien. Local energy decay and smoothing effect for the damped Schrödinger equation. Anal. PDE 10 (2017), no. 6, 1285--1315. doi:10.2140/apde.2017.10.1285. https://projecteuclid.org/euclid.apde/1510843524


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References

  • L. Aloui, “Smoothing effect for regularized Schrödinger equation on bounded domains”, Asymptot. Anal. 59:3-4 (2008), 179–193.
  • L. Aloui, “Smoothing effect for regularized Schrödinger equation on compact manifolds”, Collect. Math. 59:1 (2008), 53–62.
  • L. Aloui and M. Khenissi, “Stabilisation pour l'équation des ondes dans un domaine extérieur”, Rev. Mat. Iberoamericana 18:1 (2002), 1–16.
  • L. Aloui and M. Khenissi, “Stabilization of Schrödinger equation in exterior domains”, ESAIM Control Optim. Calc. Var. 13:3 (2007), 570–579.
  • L. Aloui and M. Khenissi, “Boundary stabilization of the wave and Schrödinger equations in exterior domains”, Discrete Contin. Dyn. Syst. 27:3 (2010), 919–934.
  • L. Aloui, M. Khenissi, and G. Vodev, “Smoothing effect for the regularized Schrödinger equation with non-controlled orbits”, Comm. Partial Differential Equations 38:2 (2013), 265–275.
  • L. Aloui, M. Khenissi, and L. Robbiano, “The Kato smoothing effect for regularized Schrödinger equations in exterior domains”, Ann. Inst. H. Poincaré Anal. Non Linéaire (online publication January 2017).
  • W. O. Amrein, A. Boutet de Monvel, and V. Georgescu, $C_0$-groups, commutator methods and spectral theory of $N$-body Hamiltonians, Progress in Mathematics 135, Birkhäuser, Basel, 1996.
  • C. Bardos, G. Lebeau, and J. Rauch, “Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary”, SIAM J. Control Optim. 30:5 (1992), 1024–1065.
  • M. Ben-Artzi and S. Klainerman, “Decay and regularity for the Schrödinger equation”, J. Anal. Math. 58 (1992), 25–37.
  • J.-F. Bony and D. Häfner, “Local energy decay for several evolution equations on asymptotically Euclidean manifolds”, Ann. Sci. Éc. Norm. Supér. $(4)$ 45:2 (2012), 311–335.
  • C. A. Bortot and M. M. Cavalcanti, “Asymptotic stability for the damped Schrödinger equation on noncompact Riemannian manifolds and exterior domains”, Comm. Partial Differential Equations 39:9 (2014), 1791–1820.
  • J.-M. Bouclet, “Low frequency estimates and local energy decay for asymptotically Euclidean Laplacians”, Comm. Partial Differential Equations 36:7 (2011), 1239–1286.
  • J.-M. Bouclet and J. Royer, “Local energy decay for the damped wave equation”, J. Funct. Anal. 266:7 (2014), 4538–4615.
  • J.-M. Bouclet and J. Royer, “Sharp low frequency resolvent estimates on asymptotically conical manifolds”, Comm. Math. Phys. 335:2 (2015), 809–850.
  • N. Burq, “Décroissance de l'énergie locale de l'équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel”, Acta Math. 180:1 (1998), 1–29.
  • N. Burq, “Smoothing effect for Schrödinger boundary value problems”, Duke Math. J. 123:2 (2004), 403–427.
  • N. Burq, P. Gérard, and N. Tzvetkov, “On nonlinear Schrödinger equations in exterior domains”, Ann. Inst. H. Poincaré Anal. Non Linéaire 21:3 (2004), 295–318.
  • P. Constantin and J.-C. Saut, “Local smoothing properties of dispersive equations”, J. Amer. Math. Soc. 1:2 (1988), 413–439.
  • P. D'Ancona and R. Racke, “Evolution equations on non-flat waveguides”, Arch. Ration. Mech. Anal. 206:1 (2012), 81–110.
  • E. B. Davies, “The functional calculus”, J. London Math. Soc. $(2)$ 52:1 (1995), 166–176.
  • M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series 268, Cambridge University Press, 1999.
  • S.-i. Doi, “Smoothing effects of Schrödinger evolution groups on Riemannian manifolds”, Duke Math. J. 82:3 (1996), 679–706.
  • S.-i. Doi, “Smoothing effects for Schrödinger evolution equation and global behavior of geodesic flow”, Math. Ann. 318:2 (2000), 355–389.
  • M. B. Erdo\u gan, M. Goldberg, and W. Schlag, “Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions”, Forum Math. 21:4 (2009), 687–722.
  • C. Guillarmou, A. Hassell, and A. Sikora, “Resolvent at low energy, III: The spectral measure”, Trans. Amer. Math. Soc. 365:11 (2013), 6103–6148.
  • T. Kato, “Wave operators and similarity for some non-selfadjoint operators”, Math. Ann. 162 (1966), 258–279.
  • H. Koch and D. Tataru, “Carleman estimates and absence of embedded eigenvalues”, Comm. Math. Phys. 267:2 (2006), 419–449.
  • P. D. Lax, C. S. Morawetz, and R. S. Phillips, “Exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle”, Comm. Pure Appl. Math. 16 (1963), 477–486.
  • C. S. Morawetz, J. V. Ralston, and W. A. Strauss, “Decay of solutions of the wave equation outside nontrapping obstacles”, Comm. Pure Appl. Math. 30:4 (1977), 447–508.
  • E. Mourre, “Absence of singular continuous spectrum for certain selfadjoint operators”, Comm. Math. Phys. 78:3 (1981), 391–408.
  • J. V. Ralston, “Solutions of the wave equation with localized energy”, Comm. Pure Appl. Math. 22 (1969), 807–823.
  • J. Rauch, “Local decay of scattering solutions to Schrödinger's equation”, Comm. Math. Phys. 61:2 (1978), 149–168.
  • J. Rauch and M. Taylor, “Exponential decay of solutions to hyperbolic equations in bounded domains”, Indiana Univ. Math. J. 24 (1974), 79–86.
  • M. Reed and B. Simon, Methods of modern mathematical physics, IV: Analysis of operators, Academic Press, New York, 1978.
  • J. Royer, “Limiting absorption principle for the dissipative Helmholtz equation”, Comm. Partial Differential Equations 35:8 (2010), 1458–1489.
  • J. Royer, “Exponential decay for the Schrödinger equation on a dissipative waveguide”, Ann. Henri Poincaré 16:8 (2015), 1807–1836.
  • J. Royer, “Mourre's commutators method for a dissipative form perturbation”, J. Operator Theory 76:2 (2016), 351–385.
  • P. Sjölin, “Regularity of solutions to the Schrödinger equation”, Duke Math. J. 55:3 (1987), 699–715.
  • B. Sz.-Nagy, C. Foias, H. Bercovici, and L. Kérchy, Harmonic analysis of operators on Hilbert space, 2nd ed., Springer, 2010.
  • D. Tataru, “Local decay of waves on asymptotically flat stationary space-times”, Amer. J. Math. 135:2 (2013), 361–401.
  • L. Thomann, “A remark on the Schrödinger smoothing effect”, Asymptot. Anal. 69:1-2 (2010), 117–123.
  • Y. Tsutsumi, “Local energy decay of solutions to the free Schrödinger equation in exterior domains”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31:1 (1984), 97–108.
  • M. Zworski, Semiclassical analysis, Graduate Studies in Mathematics 138, American Mathematical Society, Providence, RI, 2012.