Modulation space estimates for Schrödinger type equations with time-dependent potentials

Wei Wei

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 539-566
  • ISSN: 0011-4642

Abstract

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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 ( - Δ ) κ / 2 with 1 ≤ κ ≤ 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.

How to cite

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Wei, Wei. "Modulation space estimates for Schrödinger type equations with time-dependent potentials." Czechoslovak Mathematical Journal 64.2 (2014): 539-566. <http://eudml.org/doc/262029>.

@article{Wei2014,
abstract = {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 $1\le \kappa \le 2$. 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.},
author = {Wei, Wei},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {2},
pages = {539-566},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Modulation space estimates for Schrödinger type equations with time-dependent potentials},
url = {http://eudml.org/doc/262029},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Wei, Wei
TI - Modulation space estimates for Schrödinger type equations with time-dependent potentials
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 539
EP - 566
AB - 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 $1\le \kappa \le 2$. 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.
LA - eng
KW - 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
UR - http://eudml.org/doc/262029
ER -

References

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  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. (1981) MR0650272
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  7. Kato, K., Kobayashi, M., Ito, S., 10.2748/tmj/1341249372, Tohoku Math. J. (2) 64 (2012), 223-231. (2012) Zbl1246.35171MR2948820DOI10.2748/tmj/1341249372
  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. (2011) Zbl1256.35104MR2953118
  9. Kato, K., Kobayashi, M., Ito, S., 10.1016/j.jfa.2013.08.017, J. Funct. Anal. 266 (2014), 733-753. (2014) Zbl1294.35010MR3132728DOI10.1016/j.jfa.2013.08.017
  10. Kitada, H., Scattering theory for the fractional power of negative Laplacian, J. Abstr. Differ. Equ. Appl. 1 (2010), 1-26. (2010) Zbl1208.35097MR2747652
  11. Wang, B., Hudzik, H., 10.1016/j.jde.2006.09.004, J. Differ. Equations 232 (2007), 36-73. (2007) Zbl1121.35132MR2281189DOI10.1016/j.jde.2006.09.004

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