Smoothing effects for Schrödinger equations with electro-magnetic potentials and applications to the Maxwell-Schrödinger equations. (English) Zbl 1251.35095

Summary: We consider Schrödinger equations in \(\mathbf R^{1+2}\) with electro-magnetic potentials. The potentials belong to \(H^{1}\), and typically they are time-independent or determined as solutions to inhomogeneous wave equations. We prove Kato type smoothing estimates for solutions. We also apply this result to the Maxwell-Schrödinger equations in the Lorentz gauge and prove unique solvability of this system in the energy space.


35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35Q61 Maxwell equations
Full Text: DOI


[1] Brenner, P., On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186, 383-391, (1984) · Zbl 0524.35084
[2] Constantin, P.; Saut, J.-C., Local smoothing properties of dispersive equations, J. amer. math. soc., 1, 413-439, (1988) · Zbl 0667.35061
[3] DʼAncona, P.; Fanelli, L., Strichartz and smoothing estimates of dispersive equations with magnetic potentials, Comm. partial differential equations, 33, 1082-1112, (2008) · Zbl 1160.35363
[4] Erdoğan, M.; Goldberg, M.; Schlag, W., Strichartz and smoothing estimates for Schrödinger operators with large magnetic potentials in \(\boldsymbol{R}^3\), J. eur. math. soc. (JEMS), 10, 507-531, (2008) · Zbl 1152.35021
[5] Erdoğan, M.; Goldberg, M.; Schlag, W., Strichartz and smoothing estimates for Schrödinger operators with almost critical magnetic potentials in three and higher dimensions, Forum math., 21, 687-722, (2009) · Zbl 1181.35208
[6] Georgiev, V.; Stefanov, A.; Tarulli, M., Smoothing-Strichartz estimates for the Schrödinger equation with small magnetic potential, Discrete contin. dyn. syst., 17, 771-786, (2007) · Zbl 1125.35077
[7] Ginibre, J.; Velo, G., The Cauchy problem for coupled Yang-Mills and scalar fields in the temporal gauge, Comm. math. phys., 82, 1-28, (1981/1982) · Zbl 0486.35048
[8] Ginibre, J.; Velo, G., Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. inst. H. Poincaré phys. théor., 43, 399-442, (1985) · Zbl 0595.35089
[9] Ginibre, J.; Velo, G., Propriétés de lissage et existence de solutions pour lʼéquation de Benjamin-Ono généralisée, C. R. acad. sci. Paris Sér. I math., 308, 309-314, (1989) · Zbl 0694.35156
[10] Ginibre, J.; Velo, G., Commutator expansions and smoothing properties of generalized Benjamin-Ono equations, Ann. inst. H. Poincaré phys. théor., 51, 221-229, (1989) · Zbl 0705.35126
[11] Ginibre, J.; Velo, G., Generalized Strichartz inequalities for the wave equation, J. funct. anal., 133, 50-68, (1995) · Zbl 0849.35064
[12] Guo, Y.; Nakamitsu, K.; Strauss, W., Global finite-energy solutions of the Maxwell-Schrödinger system, Comm. math. phys., 170, 181-196, (1995) · Zbl 0830.35131
[13] Kato, J., Existence and uniqueness of the solution to the modified Schrödinger map, Math. res. lett., 12, 171-186, (2005) · Zbl 1082.35140
[14] Kato, T., Linear evolution equations of “hyperbolic” type, J. fac. sci. univ. Tokyo sect. I, 17, 241-258, (1970) · Zbl 0222.47011
[15] Kato, T., Linear evolution equations of “hyperbolic” type. II, J. math. soc. Japan, 25, 648-666, (1973) · Zbl 0262.34048
[16] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, (), 93-128
[17] Kato, T.; Yajima, K., Some examples of smooth operators and the associated smoothing effect, Rev. math. phys., 1, 481-496, (1989) · Zbl 0833.47005
[18] Kenig, C.E.; Koenig, K.D., On the local well-posedness of the Benjamin-Ono and modified Benjamin-Ono equations, Math. res. lett., 10, 879-895, (2003) · Zbl 1044.35072
[19] Kenig, C.E.; Ponce, G.; Vega, L., Oscillatory integrals and regularity of dispersive equations, Indiana univ. math. J., 40, 33-69, (1991) · Zbl 0738.35022
[20] Kenig, C.E.; Ponce, G.; Vega, L., Small solutions to nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 10, 255-288, (1993) · Zbl 0786.35121
[21] Koch, H.; Tzvetkov, N., On the local well-posedness of the Benjamin-Ono equation in \(H^s(\mathbb{R})\), Int. math. res. not., 26, 1449-1464, (2003) · Zbl 1039.35106
[22] Nakamitsu, K.; Tsutsumi, M., The Cauchy problem for the coupled Maxwell-Schrödinger equations, J. math. phys., 27, 211-216, (1986) · Zbl 0606.35015
[23] Nakamura, M.; Wada, T., Local well-posedness for the Maxwell-Schrödinger equation, Math. ann., 332, 565-604, (2005) · Zbl 1075.35065
[24] Nakamura, M.; Wada, T., Global existence and uniqueness of solutions to the Maxwell-Schrödinger equations, Comm. math. phys., 276, 315-339, (2007) · Zbl 1134.81020
[25] Sjölin, P., Regularity of solutions to the Schrödinger equation, Duke math. J., 55, 699-715, (1987) · Zbl 0631.42010
[26] Strichartz, R., Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke math. J., 44, 705-714, (1977) · Zbl 0372.35001
[27] Tanabe, H., Equations of evolution, Monogr. stud. math., vol. 6, (1979), Pitman Boston, MA
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.