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Strichartz estimates for the Schrödinger equation with time-periodic \(L^{n/2}\) potentials. (English) Zbl 1161.35004

Summary: We prove Strichartz estimates for the Schrödinger operator \(H= - \Delta +V(t,x)\) with time-periodic complex potentials \(V\) belonging to the scaling-critical space \(L_x^{n/2}L_t^{\infty}\) in dimensions \(n\geqslant 3\). This is done directly from estimates on the resolvent rather than using dispersive bounds, as the latter generally require a stronger regularity condition than what is stated above. In typical fashion, we project onto the continuous spectrum of the operator and must assume an absence of resonances. Eigenvalues are permissible at any location in the spectrum, including at threshold energies, provided that the associated eigenfunction decays sufficiently rapidly.

MSC:

35J10 Schrödinger operator, Schrödinger equation
35P25 Scattering theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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[1] Agmon, S., Spectral properties of Schrödinger operators and scattering theory, Ann. sc. norm. super. Pisa cl. sci. (4), 2, 2, 151-218, (1975) · Zbl 0315.47007
[2] Erdoğan, M.B.; Schlag, W., Dispersive estimates for Schrödinger operators in the presence of a resonance and/or an eigenvalue at zero energy in dimension three: I, Dyn. partial differ. equ., 1, 4, 359-379, (2004) · Zbl 1080.35102
[3] Galtbayar, A.; Jensen, A.; Yajima, K., Local time-decay of solutions to Schrödinger equations with time-periodic potentials, J. stat. phys., 116, 231-282, (2004) · Zbl 1138.81018
[4] Goldberg, M.; Schlag, W., A limiting absorption principle for the three-dimensional Schrödinger equation with \(L^p\) potentials, Int. math. res. not., 2004:75, 4049-4071, (2004) · Zbl 1069.35063
[5] Goldberg, M.; Visan, M., A counterexample to dispersive estimates for Schrödinger operators in higher dimensions, Comm. math. phys., 266, 211-238, (2006) · Zbl 1110.35073
[6] Ionescu, A.; Jerison, D., On the absence of positive eigenvalues of Schrödinger operators with rough potentials, Geom. funct. anal., 13, 1029-1081, (2003) · Zbl 1055.35098
[7] 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
[8] Journé, J.-L.; Soffer, A.; Sogge, C.D., Decay estimates for Schrödinger operators, Comm. pure appl. math., 44, 5, 573-604, (1991) · Zbl 0743.35008
[9] Kato, T., Wave operators and similarity for some non-selfadjoint operators, Math. ann., 162, 258-279, (1965/1966) · Zbl 0139.31203
[10] Keel, M.; Tao, T., Endpoint Strichartz inequalities, Amer. J. math., 120, 955-980, (1998) · Zbl 0922.35028
[11] Kenig, C.E.; Ruiz, A.; Sogge, C.D., Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke math. J., 55, 2, 329-347, (1987) · Zbl 0644.35012
[12] Lieb, E.; Loss, M., Analysis, Grad. stud. math., vol. 14, (2001), Amer. Math. Soc. Providence, RI
[13] Reed, M.; Simon, B., Methods of modern mathematical physics. IV. analysis of operators, (1978), Academic Press New York, London · Zbl 0401.47001
[14] Rodnianski, I.; Schlag, W., Time decay for solutions of Schrödinger equations with rough and time-dependent potentials, Invent. math., 155, 3, 451-513, (2004) · Zbl 1063.35035
[15] Ruiz, A.; Vega, L., On local regularity of Schrödinger equations, Int. math. res. not., 1, 13-27, (1993) · Zbl 0812.35016
[16] Simon, B., Best constants in some operator smoothness estimates, J. funct. anal., 107, 1, 66-71, (1992) · Zbl 0815.47003
[17] Tomas, P., Restriction theorems for the Fourier transform, Proc. sympos. pure math., 35, 111-114, (1979)
[18] Vodev, G., Dispersive estimates of solutions to the Schrödinger equation in dimensions \(n \geqslant 4\), Asymptot. anal., 49, 61-86, (2006) · Zbl 1111.35006
[19] Yajima, K., Exponential decay of quasi-stationary states of time periodic Schrödinger equations with short range potentials, Sci. papers collega arts sci. univ. Tokyo, 40, 27-36, (1990) · Zbl 0712.35017
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