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Abstract theory of pointwise decay with applications to wave and Schrödinger equations. (English) Zbl 1367.47029

The model example of the pointwise decay for the Schrödinger operator \(H=-\Delta +V(x)\) with a short-range potential, \(x\in \mathbb R^3\), is the following Kato-Jensen estimate: \[ \| (1+|x|^2)^{-\sigma}e^{-itH}P_c(H)(1+|x|^2)^{-\sigma}\| =O(t^{-3/2}) \] where \(\sigma\) is large enough, \(P_c(H)\) is the projection on the continuous spectral part of \(H\); see A. Jensen and T. Kato [Duke Math. J. 46, 583–611 (1979; Zbl 0448.35080)].
In the paper under review, the authors develop an abstract theory to prove pointwise decay estimates in weighted spaces, assuming a commutation identity for the Hamiltonian. As specific nontrivial examples, they consider an operator with electric field, the fractional Laplacian, the Dirac operator, the wave and Klein-Gordon equations.

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B47 Commutators, derivations, elementary operators, etc.
37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems
35Q41 Time-dependent Schrödinger equations and Dirac equations
47D08 Schrödinger and Feynman-Kac semigroups
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 0448.35080
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References:

[1] Amrein W., Boutetde Monvel A., Georgescu V.: C0-Groups, Commutator Methods and Spectral Theory of N-Body Hamiltonians. Birkhäuser, Basel (1996) · Zbl 0962.47500
[2] Baumgärtel H., Wollenberg M.: Mathematical Scattering Theory. Springer, New York (1983) · Zbl 0536.47007
[3] Blue P., Soffer A.: Semilinear wave equations on the Schwarzschild manifold I: Local decay estimates. Adv. Differ. Equ. 8(5), 595-614 (2003) · Zbl 1044.58033
[4] Blue P., Soffer A.: Phase space analysis on some black hole manifolds. J. Funct. Anal. 256(1), 1-90 (2009) · Zbl 1158.83007
[5] Bony J.-F., Häfner D.: The semilinear wave equation on asymptotically Euclidean manifolds. Commun. Partial Differ. Equ. 35(1), 23-67 (2010) · Zbl 1191.35181
[6] Boussaid N., Golénia S.: Limiting absorption principle for some long range perturbations of Dirac systems at threshold energies. Commun. Math. Phys. 299(3), 677-708 (2010) · Zbl 1205.81084
[7] Boutet de Monvel, A., Georgescu, V.: Boundary values of the resolvent of a self-adjoint operator: higher order estimates. In: Boutet de Monvel, A., Marchenko, V. (eds.) Algebraic and Geometric Methods in Mathematical Physics. Proceedings of the Kaciveli Summer School, Crimea, 1993, pp. 9-52. Kluwer, New York (1996) · Zbl 0917.47022
[8] Boutetde Monvel A., Georgescu V., Sahbani J.: Boundary values of resolvent families and propagation properties. C. R. Acad. Sci. Paris Sér. I 322, 289-294 (1996) · Zbl 0844.47002
[9] Cycon H.L., Froese R., Kirsch W., Simon B.: Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry, 2nd edn. Springer, New York (2008) · Zbl 0619.47005
[10] Dafermos, M., Rodnianski, I.: The black hole stability problem for linear scalar perturbations. In: Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, Singapore, pp. 132-189 (2011). arXiv:1010.5137 · Zbl 1136.35428
[11] Donninger R., Schlag W., Soffer A.: A proof of Price’s law on Schwarzschild blackhole manifolds for all angular momenta. Adv. Math. 226, 484-540 (2011) · Zbl 1205.83041
[12] Erdogan B., Goldberg M., Green W.: Dispersive estimates for four dimensional Schödinger and wave equations with obstructions at zero energy. Commun. PDE 39(10), 1936-1964 (2014) · Zbl 1325.35017
[13] Fernández C., Richard S., Tiedrade Aldecoa R.: Commutator methods for unitary operators. J. Spectr. Theory 3, 271-292 (2013) · Zbl 1290.47036
[14] Gérard C.: A proof of the abstract limiting absorption principle by energy estimates. J. Funct. Anal. 254, 2070-2704 (2008) · Zbl 1141.47017
[15] Golénia S.: Positive commutators, Fermi golden rule and the spectrum of the zero temperature Pauli-Fierz hamiltonians. J. Funct. Anal. 256(8), 2587-2620 (2009) · Zbl 1166.81017
[16] Georgescu V., Gérard C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208(2), 275-281 (1999) · Zbl 0961.81009
[17] Golénia S., Jecko T.: A new look at Mourre’s commutator theory. Complex Anal. Oper. Theory 1(3), 399-422 (2007) · Zbl 1167.47010
[18] Golénia S., Jecko T.: Rescaled Mourre’s commutator theory, application to Schrödinger operators with oscillating potential. J. Oper. Theory 70(1), 109-144 (2013) · Zbl 1289.47022
[19] Georgescu V., Golénia S.: Isometries, Fock spaces and spectral analysis of Schrödinger operators on trees. J. Funct. Anal. 227, 389-429 (2005) · Zbl 1106.47006
[20] Golénia S., Moroianu S.: Spectral analysis of magnetic Laplacians on conformally cusp manifolds. Ann. Henri Poincaré 9(1), 131-179 (2008) · Zbl 1134.58009
[21] Golénia S., Moroianu S.: The spectrum of Schrödinger operators and Hodge Laplacians on conformally cusp manifolds. Trans. AMS 364(1), 1-29 (2012) · Zbl 1239.58021
[22] Hunziker W., Sigal I.M., Soffer A.: Minimal escape velocities. Commun. Partial Differ. Equ. 24, 2279-2295 (1999) · Zbl 0944.35014
[23] Jensen A., Kato T.: Spectral properties of Schrödinger operators and time-decay of the wave functions. Duke Math. J. 46(3), 583-611 (1979) · Zbl 0448.35080
[24] Jensen A., Mourre E., Perry P.: Multiple commutator estimates and resolvent smoothness in scattering theory. Ann. Inst. Henri Poincaré 41, 207-225 (1984) · Zbl 0561.47007
[25] Jensen A., Nenciu G.: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 13(6), 717-754 (2001) · Zbl 1029.81067
[26] Jensen, A., Nenciu, G.: Erratum: A unified approach to resolvent expansions at thresholds. Rev. Math. Phys. 16(5), 675-677 (2004) (Rev. Math. Phys. 13(6), 717-754, 2001) · Zbl 1055.81624
[27] Journé J.-L., Soffer A., Sogge C.D.: Decay estimates for Schrödinger operators. Commun. Pure Appl. Math. 44, 573-604 (1991) · Zbl 0743.35008
[28] Komech A., Kopylova E.: Dispersion Decay and Scattering Theory. Wiley, Hoboken (2012) · Zbl 1317.35162
[29] Leinfelder H., Simader C.: Schrödinger operators with singular magnetic vector potentials. Math. Z. 176(1), 1-19 (1981) · Zbl 0468.35038
[30] Mourre E.: Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys. 78(3), 519-567 (1981) · Zbl 0489.47010
[31] Mǎntoiu M., Richard S.: Absence of singular spectrum for Schrödinger operators with anisotropic potentials and magnetic fields. J. Math. Phys. 41, 2732-2740 (2000) · Zbl 0987.81025
[32] Mǎntoiu M., Tiedra de Aldecoa R.: Spectral analysis for convolution operators on locally compact groups. J. Funct. Anal. 253(2), 675-691 (2007) · Zbl 1132.43002
[33] Mǎntoiu M., Richard S., Tiedra de Aldecoa R.: Spectral analysis for adjacency operators on graphs. Ann. Henri Poincaré 8, 1401-1423 (2007) · Zbl 1137.81013
[34] Mǎntoiu, M., Richard, S., Tiedra de Aldecoa, R.: The method of the weakly conjugate operator: extensions and applications to operators on graphs and groups. In: Petroleum—Gas University of Ploiesti Bulletin, Mathematics—Informatics—Physics Series LXI, pp 1-12 (2009) · Zbl 1262.47005
[35] Richard S.: Some improvements in the method of the weakly conjugate operator. Lett. Math. Phys. 76, 27-36 (2006) · Zbl 1136.35428
[36] Rosenblum M.: Perturbation of the continuous spectrum and unitary equivalence. Pac. J. Math. 7, 997-1010 (1957) · Zbl 0081.12003
[37] Reed M., Simon B.: Methods of Modern Mathematical Physics, vol 4. Academic Press, New York · Zbl 0401.47001
[38] Schlag W., Rodnianski I.: Time decay for solutions of Schrödinger equations with rough and time-dependent potentials. Invent. Math. 155(3), 451-513 (2004) · Zbl 1063.35035
[39] Rodnianski I., Tao T.: Long time decay estimates for the Schrödinger equation on manifolds. Mathematical aspects of nonlinear dispersive equations. Ann. Math. Stud. 1, 223-253 (2007) · Zbl 1133.35022
[40] Richard S., Tiedra de Aldecoa R.: On perturbations of Dirac operators with variable magnetic field of constant direction. J. Math. Phys. 45, 4164-4173 (2004) · Zbl 1064.81028
[41] Richard S., Tiedrade Aldecoa R.: On the spectrum of magnetic Dirac operators with Coulomb-type perturbations. J. Funct. Anal. 250, 625-641 (2007) · Zbl 1152.35080
[42] Richard S., Tiedrade Aldecoa R.: A few results on Mourre theory in a two-Hilbert spaces setting. Anal. Math. Phys. 3, 183-200 (2013) · Zbl 1283.81067
[43] Richard S., Tiedrade Aldecoa R.: Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends. Rev. Math. Phys. 25, 1350003-1-1350003-40 (2013) · Zbl 1280.58021
[44] Richard, S., Tiedra de Aldecoa, R.: A new formula relating localisation operators to time operators. In: Operator Theory: Advances and Applications, vol. 224, pp. 301-338. Birkhäuser, Basel (2012) · Zbl 1294.47102
[45] Rodnianski I., Schlag W., Soffer A.: Dispersive analysis of charge transfer models. Commun. Pure Appl. Math. 58(2), 149-216 (2005) · Zbl 1130.81053
[46] Rodnianski, I., Schlag, W., Soffer, A.: Asymptotic stability of N-soliton states of NLS. arXiv:math/0309114 (2003) · Zbl 1143.35001
[47] Sahbani J.: The conjugate operator method for locally regular Hamiltonians. J. Oper. Theory 38(2), 297-322 (1997) · Zbl 0905.47003
[48] Sinha K.B.: On the absolutely and singularly continuous subspaces in scattering theory. Ann. l’I.H.P. Sect. A 26(3), 263-277 (1977)
[49] Schlag W.: Dispersive estimates for Schrödinger operators: a survey. Mathematical aspects of nonlinear dispersive equations. Ann. Math. Stud. 1, 255-285 (2007) · Zbl 1143.35001
[50] Soffer, A.: Monotonic Local Decay Estimates. arXiv:1110.6549 (2011) (revised version in preparation) · Zbl 0448.35080
[51] Sigal, I.M., Soffer, A.: Local decay and velocity bounds for time-independent and time-dependent Hamiltonians (preprint, Princeton) (1987) · Zbl 0944.35014
[52] Tataru D.: Local decay of waves on asymptotically flat stationary space-times. Am. J. Math. 135(2), 361-401 (2013) · Zbl 1266.83033
[53] Thaller B.: The Dirac Equation. Springer, Berlin (1992) · Zbl 0765.47023
[54] Tiedra de Aldecoa, R.: Commutator methods for the spectral analysis of uniquely ergodic dynamical systems. In: Ergodic Theory and Dynamical Systems, first view, pp. 1-24 (2014) · Zbl 0844.47002
[55] Yajima K.: Dispersive estimates for Schrödinger equations with threshold resonance and eigenvalue. Commun. Math. Phys. 259, 475-509 (2005) · Zbl 1079.81021
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