Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
Schrödinger type equation; short-time Fourier transform; modulation space; classical Hamiltonian; complex interpolation
Schrödinger type equation; short-time Fourier transform; modulation space; classical Hamiltonian; complex interpolation
Summary:
We give a new representation of solutions to a class of time-dependent Schrödinger type equations via the short-time Fourier transform and the method of characteristics. Moreover, we also establish some novel estimates for oscillatory integrals which are associated with the fractional power of negative Laplacian $(-\Delta )^{\kappa /2}$ with $1\leq \kappa \leq 2$. Consequently the classical Hamiltonian corresponding to the previous Schrödinger type equations is studied. As applications, a series of new boundedness results for the corresponding propagator are obtained in the framework of modulation spaces. The main results of the present article include the case of wave equations.
References:
[1] Bényi, Á., Gröchenig, K., Okoudjou, K., Rogers, L. G.: Unimodular Fourier multipliers for modulation spaces. J. Funct. Anal. 246 (2007), 366-384. DOI 10.1016/j.jfa.2006.12.019 | MR 2321047 | Zbl 1120.42010
[2] Córdoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Commun. Partial Differ. Equations 3 (1978), 979-1005. DOI 10.1080/03605307808820083 | MR 0507783 | Zbl 0389.35046
[3] Feichtinger, H. G.: Modulation spaces on locally compact abelian groups. Wavelets and Their Applications, Proc. Internat. Conf., Chennai, India M. Krishna, R. Radha, S. Thangavelu Allied Publishers, New Delhi (2003), 99-140 Updated version of a technical report, University of Vienna, 1983.
[4] Feichtinger, H. G.: Banach spaces of distributions of Wiener's type and interpolation. Functional Analysis and Approximation, Proc. Conf., Oberwolfach 1980 Internat. Ser. Numer. Math. 60 Birkhäuser, Basel (1981), 153-165. MR 0650272
[5] Gröchenig, K.: Foundations of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis Birkhäuser, Boston (2001). MR 1843717 | Zbl 0966.42020
[6] Hörmander, L.: Estimates for translation invariant operators in $L^{p}$-spaces. Acta Math. 104 (1960), 93-140. DOI 10.1007/BF02547187 | MR 0121655 | Zbl 0093.11402
[7] Kato, K., Kobayashi, M., Ito, S.: Representation of Schrödinger operator of a free particle via short-time Fourier transform and its applications. Tohoku Math. J. (2) 64 (2012), 223-231. DOI 10.2748/tmj/1341249372 | MR 2948820 | Zbl 1246.35171
[8] Kato, K., Kobayashi, M., Ito, S.: Remark on wave front sets of solutions to Schrödinger equation of a free particle and a harmonic oscillator. SUT J. Math. 47 (2011), 175-183. MR 2953118 | Zbl 1256.35104
[9] Kato, K., Kobayashi, M., Ito, S.: Estimates on modulation spaces for Schrödinger evolution operators with quadratic and sub-quadratic potentials. J. Funct. Anal. 266 (2014), 733-753. DOI 10.1016/j.jfa.2013.08.017 | MR 3132728 | Zbl 1294.35010
[10] Kitada, H.: Scattering theory for the fractional power of negative Laplacian. J. Abstr. Differ. Equ. Appl. 1 (2010), 1-26. MR 2747652 | Zbl 1208.35097
[11] Wang, B., Hudzik, H.: The global Cauchy problem for the NLS and NLKG with small rough data. J. Differ. Equations 232 (2007), 36-73. DOI 10.1016/j.jde.2006.09.004 | MR 2281189 | Zbl 1121.35132
Search
Partner of
