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2011, 31(1): 109-118. doi: 10.3934/dcds.2011.31.109

Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential

1. 

Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025, United States

Received  January 2010 Revised  March 2011 Published  June 2011

We prove Strichartz estimates for the absolutely continuous evolution of a Schrödinger operator $H = (i\nabla + A)^2 + V$ in $R^n$, $n \ge 3$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div}\,A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.
Citation: Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109
References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 2 (1975), 151.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. Anal. Math., 30 (1976), 1. doi: 10.1007/BF02786703.

[3]

J.-M. Bouclet and N. Tzvetkov, On global Strichartz estimates for non-trapping metrics,, J. Funct. Anal., 254 (2008), 1661. doi: 10.1016/j.jfa.2007.11.018.

[4]

F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates for the Schrödinger equation in dimension four and five,, Asymptot. Anal., 62 (2009), 125.

[5]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687.

[6]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: 10.1090/S0894-0347-1988-0928265-0.

[7]

M. B. Erdoǧan, 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 (2009), 687. doi: 10.1515/FORUM.2009.035.

[8]

M. Goldberg and M. Visan, A Counterexample to dispersive estimates for Schrödinger operators in higher dimensions,, Comm. Math. Phys., 266 (2006), 211. doi: 10.1007/s00220-006-0013-5.

[9]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. II,'', Grundlehren der Mathematischen Wissenschaften, (1983).

[10]

A. Ionescu and W. Schlag, Agmon-Kato-Kuroda theorems for a large class of perturbations,, Duke Math. J., 131 (2006), 397. doi: 10.1215/S0012-7094-06-13131-9.

[11]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions,, Duke Math. J., 46 (1979), 583. doi: 10.1215/S0012-7094-79-04631-3.

[12]

T. Kato, Wave operators and similarity for some non-selfadjoint operators,, Math. Ann., 162 (): 258.

[13]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues,, Comm. Math. Phys., 267 (2006), 419. doi: 10.1007/s00220-006-0060-y.

[14]

J. Marzuola, J. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations,, J. Funct. Anal., 255 (2008), 1497. doi: 10.1016/j.jfa.2008.05.022.

[15]

D. Robert, Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du Laplacien,, Ann. Sci. École Norm. Sup., 25 (1992), 107.

[16]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials,, Invent. Math., 155 (2004), 451. doi: 10.1007/s00222-003-0325-4.

[17]

B. Simon, Best constants in some operator smoothness estimates,, J. Funct. Anal., 107 (1992), 66. doi: 10.1016/0022-1236(92)90100-W.

[18]

H. Smith and C. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian,, Comm. PDE, 25 (2000), 2171. doi: 10.1080/03605300008821581.

[19]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, Comm. Math. Phys., 110 (1987), 415. doi: 10.1007/BF01212420.

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory,, Ann. Sc. Norm. Super. Pisa. Cl. Sci., 2 (1975), 151.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics,, J. Anal. Math., 30 (1976), 1. doi: 10.1007/BF02786703.

[3]

J.-M. Bouclet and N. Tzvetkov, On global Strichartz estimates for non-trapping metrics,, J. Funct. Anal., 254 (2008), 1661. doi: 10.1016/j.jfa.2007.11.018.

[4]

F. Cardoso, C. Cuevas and G. Vodev, Dispersive estimates for the Schrödinger equation in dimension four and five,, Asymptot. Anal., 62 (2009), 125.

[5]

M. Christ and A. Kiselev, Maximal functions associated to filtrations,, J. Funct. Anal., 179 (2001), 409. doi: 10.1006/jfan.2000.3687.

[6]

P. Constantin and J.-C. Saut, Local smoothing properties of dispersive equations,, J. Amer. Math. Soc., 1 (1988), 413. doi: 10.1090/S0894-0347-1988-0928265-0.

[7]

M. B. Erdoǧan, 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 (2009), 687. doi: 10.1515/FORUM.2009.035.

[8]

M. Goldberg and M. Visan, A Counterexample to dispersive estimates for Schrödinger operators in higher dimensions,, Comm. Math. Phys., 266 (2006), 211. doi: 10.1007/s00220-006-0013-5.

[9]

L. Hörmander, "The Analysis of Linear Partial Differential Operators. II,'', Grundlehren der Mathematischen Wissenschaften, (1983).

[10]

A. Ionescu and W. Schlag, Agmon-Kato-Kuroda theorems for a large class of perturbations,, Duke Math. J., 131 (2006), 397. doi: 10.1215/S0012-7094-06-13131-9.

[11]

A. Jensen and T. Kato, Spectral properties of Schrödinger operators and time-decay of the wave functions,, Duke Math. J., 46 (1979), 583. doi: 10.1215/S0012-7094-79-04631-3.

[12]

T. Kato, Wave operators and similarity for some non-selfadjoint operators,, Math. Ann., 162 (): 258.

[13]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues,, Comm. Math. Phys., 267 (2006), 419. doi: 10.1007/s00220-006-0060-y.

[14]

J. Marzuola, J. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations,, J. Funct. Anal., 255 (2008), 1497. doi: 10.1016/j.jfa.2008.05.022.

[15]

D. Robert, Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du Laplacien,, Ann. Sci. École Norm. Sup., 25 (1992), 107.

[16]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials,, Invent. Math., 155 (2004), 451. doi: 10.1007/s00222-003-0325-4.

[17]

B. Simon, Best constants in some operator smoothness estimates,, J. Funct. Anal., 107 (1992), 66. doi: 10.1016/0022-1236(92)90100-W.

[18]

H. Smith and C. Sogge, Global Strichartz estimates for nontrapping perturbations of the Laplacian,, Comm. PDE, 25 (2000), 2171. doi: 10.1080/03605300008821581.

[19]

K. Yajima, Existence of solutions for Schrödinger evolution equations,, Comm. Math. Phys., 110 (1987), 415. doi: 10.1007/BF01212420.

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