January  2020, 19(1): 609-640. doi: 10.3934/cpaa.2020029

On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations

Dipartimento di Matematica, Sapienza Università di Roma, Piero D'Ancona, Piazzale A. Moro 2, 00185 Roma, Italy

Received  November 2018 Revised  February 2019 Published  July 2019

We prove a sharp resolvent estimate in scale invariant norms of Amgon–Hörmander type for a magnetic Schrödinger operator on
$ \mathbb{R}^{n} $
,
$ n\ge3 $
$ \begin{equation*} L = -(\partial+iA)^{2}+V \end{equation*} $
with large potentials
$ A, V $
of almost critical decay and regularity.
The estimate is applied to prove sharp smoothing and Strichartz estimates for the Schrödinger, wave and Klein–Gordon flows associated to
$ L $
.
Citation: P. D'Ancona. On large potential perturbations of the Schrödinger, wave and Klein–Gordon equations. Communications on Pure and Applied Analysis, 2020, 19 (1) : 609-640. doi: 10.3934/cpaa.2020029
References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151–218.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30 (1976), 1–38. doi: 10.1007/BF02786703.

[3]

G. Artbazar and K. Yajima, The Lp-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo, 7 (2000), 221-240. 

[4]

P. Auscher and J. M. Martell, Weighted norm inequalities for fractional operators, Indiana Univ. Math. J., 57 (2008), 1845-1869.  doi: 10.1512/iumj.2008.57.3236.

[5]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.

[6]

P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.  doi: 10.1016/0022-0396(85)90083-X.

[7]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.  doi: 10.1512/iumj.2004.53.2541.

[8]

F. Cacciafesta and P. D'Ancona, Weighted Lp estimates for powers of selfadjoint operators, Adv. Math., 229 (2002), 501–530. doi: 10.1016/j.aim.2011.09.007.

[9]

F. CacciafestaP. D'Ancona and R. Lucà, Helmholtz and dispersive equations with variable coefficients on exterior domains, SIAM J. Appl. Math., 48 (2016), 1798-1832.  doi: 10.1137/15M103769X.

[10]

S. Cuccagna, On the wave equation with a potential, Comm. Partial Differential Equations, 25 (2000), 1549-1565.  doi: 10.1080/03605300008821559.

[11]

P. D'Ancona, Kato smoothing and strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1–16. doi: 10.1007/s00220-014-2169-8.

[12]

P. D'Ancona and L. Fanelli, Lp-boundedness of the wave operator for the one dimensional Schrödinger operator, Comm. Math. Phys., 268 (2006), 415-438.  doi: 10.1007/s00220-006-0098-x.

[13]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.

[14]

P. D'AnconaL. FanelliL. Vega and N. Visciglia, Endpoint Strichartzz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.  doi: 10.1016/j.jfa.2010.02.007.

[15]

P. D'Ancona and M. Okamoto, On the cubic Dirac equation with potential and the Lochak–Majorana condition, J. Math. Anal. Appl., 456 (2017), 1203–1237. doi: 10.1016/j.jmaa.2017.07.055.

[16]

P. D'Ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.  doi: 10.1016/j.jfa.2005.05.013.

[17]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartzz and smoothing estimates for Schrödinger operators with large magnetic potentials in r3, Journal of the European Mathematical Society, 10 (2008), 507-531.  doi: 10.4171/JEMS/120.

[18]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Mathematicum, 21 (2009), 687-722.  doi: 10.1515/FORUM.2009.035.

[19]

L. Fanelli and L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann., 344 (2009), 249-278.  doi: 10.1007/s00208-008-0303-7.

[20]

V. Georgiev, A. Stefanov and M. Tarulli, Strichartz estimates for the Schrödinger equation with small magnetic potential, In Journées "Équations aux Dérivées Partielles", pages Exp. No. IV, 17. École Polytech., Palaiseau, 2005.

[21]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 64 (1985), 363–401.

[22]

J. Ginibre and G. Velo, Generalized Strichartzz inequalities for the wave equation, In Partial Differential Operators and Mathematical Physics (Holzhau, 1994), volume 78 of Oper. Theory Adv. Appl., pages 153–160. Birkhäuser, Basel, 1995.

[23]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, second edition, 2008.

[24]

A. Ionescu and C. Kenig, Well-posedness and local smoothing of solutions of Schrödinger equations, Mathematical Research Letters, 12 (2005), 193-205.  doi: 10.4310/MRL.2005.v12.n2.a5.

[25]

R. Johnson and C. J. Neugebauer, Change of variable results for Ap- and reverse Hölder RHr-classes, Trans. Amer. Math. Soc., 328 (1991), 639-666.  doi: 10.2307/2001798.

[26]

L. JournéA. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.  doi: 10.1002/cpa.3160440504.

[27]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.  doi: 10.1007/BF01360915.

[28]

T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.  doi: 10.1142/S0129055X89000171.

[29]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[30]

C. E. KenigG. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.  doi: 10.1016/S0294-1449(16)30213-X.

[31]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Comm. Math. Phys., 267 (2006), 419-449.  doi: 10.1007/s00220-006-0060-y.

[32]

S. MachiharaK. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.  doi: 10.4171/RMI/342.

[33]

J. MarzuolaJ. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal., 255 (2008), 1497-1553.  doi: 10.1016/j.jfa.2008.05.022.

[34]

H. Mizutani, Global-in-time smoothing effects for Schödinger equations with inverse–square potentials, arXiv: 1610.01745, 2016. doi: 10.1090/proc/13729.

[35]

K. Mochizuki, Uniform resolvent estimates for magnetic Schrödinger operators and smoothing effects for related evolution equations, Publ. Res. Inst. Math. Sci., 46 (2010), 741-754.  doi: 10.2977/PRIMS/24.

[36]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274.  doi: 10.2307/1996833.

[37]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.

[38]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513.  doi: 10.1007/s00222-003-0325-4.

[39]

A. Stefanov, Strichartz estimates for the magnetic Schrödinger equation, Adv. Math., 210 (2007), 246-303.  doi: 10.1016/j.aim.2006.06.006.

[40]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993.

[41]

D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math., 130 (2008), 571-634.  doi: 10.1353/ajm.0.0000.

[42]

K. Yajima, The Wk, p-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.

[43]

K. Yajima, The Wk, p-continuity of wave operators for schrödinger operators. ⅲ. even-dimensional cases m ≥ 4, J. Math. Sci. Univ. Tokyo, 2 (1995), 311-346. 

show all references

References:
[1]

S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 151–218.

[2]

S. Agmon and L. Hörmander, Asymptotic properties of solutions of differential equations with simple characteristics, J. Anal. Math., 30 (1976), 1–38. doi: 10.1007/BF02786703.

[3]

G. Artbazar and K. Yajima, The Lp-continuity of wave operators for one dimensional Schrödinger operators, J. Math. Sci. Univ. Tokyo, 7 (2000), 221-240. 

[4]

P. Auscher and J. M. Martell, Weighted norm inequalities for fractional operators, Indiana Univ. Math. J., 57 (2008), 1845-1869.  doi: 10.1512/iumj.2008.57.3236.

[5]

J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer-Verlag, Berlin, 1976.

[6]

P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equations, J. Differential Equations, 56 (1985), 310-344.  doi: 10.1016/0022-0396(85)90083-X.

[7]

N. BurqF. PlanchonJ. G. Stalker and A. S. Tahvildar-Zadeh, Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay, Indiana Univ. Math. J., 53 (2004), 1665-1680.  doi: 10.1512/iumj.2004.53.2541.

[8]

F. Cacciafesta and P. D'Ancona, Weighted Lp estimates for powers of selfadjoint operators, Adv. Math., 229 (2002), 501–530. doi: 10.1016/j.aim.2011.09.007.

[9]

F. CacciafestaP. D'Ancona and R. Lucà, Helmholtz and dispersive equations with variable coefficients on exterior domains, SIAM J. Appl. Math., 48 (2016), 1798-1832.  doi: 10.1137/15M103769X.

[10]

S. Cuccagna, On the wave equation with a potential, Comm. Partial Differential Equations, 25 (2000), 1549-1565.  doi: 10.1080/03605300008821559.

[11]

P. D'Ancona, Kato smoothing and strichartz estimates for wave equations with magnetic potentials, Comm. Math. Phys., 335 (2015), 1–16. doi: 10.1007/s00220-014-2169-8.

[12]

P. D'Ancona and L. Fanelli, Lp-boundedness of the wave operator for the one dimensional Schrödinger operator, Comm. Math. Phys., 268 (2006), 415-438.  doi: 10.1007/s00220-006-0098-x.

[13]

P. D'Ancona and L. Fanelli, Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. Partial Differential Equations, 33 (2008), 1082-1112.  doi: 10.1080/03605300701743749.

[14]

P. D'AnconaL. FanelliL. Vega and N. Visciglia, Endpoint Strichartzz estimates for the magnetic Schrödinger equation, J. Funct. Anal., 258 (2010), 3227-3240.  doi: 10.1016/j.jfa.2010.02.007.

[15]

P. D'Ancona and M. Okamoto, On the cubic Dirac equation with potential and the Lochak–Majorana condition, J. Math. Anal. Appl., 456 (2017), 1203–1237. doi: 10.1016/j.jmaa.2017.07.055.

[16]

P. D'Ancona and V. Pierfelice, On the wave equation with a large rough potential, J. Funct. Anal., 227 (2005), 30-77.  doi: 10.1016/j.jfa.2005.05.013.

[17]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartzz and smoothing estimates for Schrödinger operators with large magnetic potentials in r3, Journal of the European Mathematical Society, 10 (2008), 507-531.  doi: 10.4171/JEMS/120.

[18]

M. B. ErdoğanM. Goldberg and W. Schlag, Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum Mathematicum, 21 (2009), 687-722.  doi: 10.1515/FORUM.2009.035.

[19]

L. Fanelli and L. Vega, Magnetic virial identities, weak dispersion and Strichartz inequalities, Math. Ann., 344 (2009), 249-278.  doi: 10.1007/s00208-008-0303-7.

[20]

V. Georgiev, A. Stefanov and M. Tarulli, Strichartz estimates for the Schrödinger equation with small magnetic potential, In Journées "Équations aux Dérivées Partielles", pages Exp. No. IV, 17. École Polytech., Palaiseau, 2005.

[21]

J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9), 64 (1985), 363–401.

[22]

J. Ginibre and G. Velo, Generalized Strichartzz inequalities for the wave equation, In Partial Differential Operators and Mathematical Physics (Holzhau, 1994), volume 78 of Oper. Theory Adv. Appl., pages 153–160. Birkhäuser, Basel, 1995.

[23]

L. Grafakos, Classical Fourier Analysis, volume 249 of Graduate Texts in Mathematics, Springer, New York, second edition, 2008.

[24]

A. Ionescu and C. Kenig, Well-posedness and local smoothing of solutions of Schrödinger equations, Mathematical Research Letters, 12 (2005), 193-205.  doi: 10.4310/MRL.2005.v12.n2.a5.

[25]

R. Johnson and C. J. Neugebauer, Change of variable results for Ap- and reverse Hölder RHr-classes, Trans. Amer. Math. Soc., 328 (1991), 639-666.  doi: 10.2307/2001798.

[26]

L. JournéA. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604.  doi: 10.1002/cpa.3160440504.

[27]

T. Kato, Wave operators and similarity for some non-selfadjoint operators, Math. Ann., 162 (1965/1966), 258-279.  doi: 10.1007/BF01360915.

[28]

T. Kato and K. Yajima, Some examples of smooth operators and the associated smoothing effect, Rev. Math. Phys., 1 (1989), 481-496.  doi: 10.1142/S0129055X89000171.

[29]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 

[30]

C. E. KenigG. Ponce and L. Vega, Small solutions to nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 255-288.  doi: 10.1016/S0294-1449(16)30213-X.

[31]

H. Koch and D. Tataru, Carleman estimates and absence of embedded eigenvalues, Comm. Math. Phys., 267 (2006), 419-449.  doi: 10.1007/s00220-006-0060-y.

[32]

S. MachiharaK. Nakanishi and T. Ozawa, Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation, Rev. Mat. Iberoamericana, 19 (2003), 179-194.  doi: 10.4171/RMI/342.

[33]

J. MarzuolaJ. Metcalfe and D. Tataru, Strichartz estimates and local smoothing estimates for asymptotically flat Schrödinger equations, J. Funct. Anal., 255 (2008), 1497-1553.  doi: 10.1016/j.jfa.2008.05.022.

[34]

H. Mizutani, Global-in-time smoothing effects for Schödinger equations with inverse–square potentials, arXiv: 1610.01745, 2016. doi: 10.1090/proc/13729.

[35]

K. Mochizuki, Uniform resolvent estimates for magnetic Schrödinger operators and smoothing effects for related evolution equations, Publ. Res. Inst. Math. Sci., 46 (2010), 741-754.  doi: 10.2977/PRIMS/24.

[36]

B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261-274.  doi: 10.2307/1996833.

[37]

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.

[38]

I. Rodnianski and W. Schlag, Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. Math., 155 (2004), 451-513.  doi: 10.1007/s00222-003-0325-4.

[39]

A. Stefanov, Strichartz estimates for the magnetic Schrödinger equation, Adv. Math., 210 (2007), 246-303.  doi: 10.1016/j.aim.2006.06.006.

[40]

E. M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993.

[41]

D. Tataru, Parametrices and dispersive estimates for Schrödinger operators with variable coefficients, Amer. J. Math., 130 (2008), 571-634.  doi: 10.1353/ajm.0.0000.

[42]

K. Yajima, The Wk, p-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan, 47 (1995), 551-581.  doi: 10.2969/jmsj/04730551.

[43]

K. Yajima, The Wk, p-continuity of wave operators for schrödinger operators. ⅲ. even-dimensional cases m ≥ 4, J. Math. Sci. Univ. Tokyo, 2 (1995), 311-346. 

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