Abstract
We describe smoothing effects and dispersion of singularities for the Schrödinger evolution group in the weighted Sobolev spaces. Under a fairly general assumption on the potential, it is shown that all singularities in the wavefunction vanish instantly whenever the initial state has sufficient decay. We measure the regularity gained by the wavefunction by the decay property of the initial state. No assumptions on the regularity of the initial state are imposed throughout the paper.
Similar content being viewed by others
References
Brezis, H. &Kato, T., Remarks on the Schrödinger operator with singular complex potentials. J. Math. Pures et Appl.58, 137–151 (1979).
Constantin, P. &Saut, J. C., Local smoothing properties of dispersive equations. J. Amer. Math. Soc.1, 413–439 (1988).
Cycon, H. L., Froese, R. C., Kirsch, W., &Simon, B.,Schrödinger Operators, with Appplication to Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin, Heidelberg, New York (1987).
Friedman, A.,Partial Differential Equations. Holt-Rinehart and Winston, New York (1969).
Ginibre, J., &Velo, G., Sur une équation de Schrödinger non linéaire avec interaction non locale, inNonlinear partial differential equations and their applications, Collège de France seminar Vol. II. Pitman, Boston (1981), 155–199.
Ginibre, J., A remark on some papers byN. Hayashi andT. Ozawa. J. Funct. Anal.85, 349–352 (1989).
Hayashi, N., Nakamitsu, K., &Tsutsumi, M., On solutions of the initial value problem for the nonlinear Schrödinger equations in one space dimension. Math. Z.192, 637–650 (1986).
Hayashi, N., Nakamitsu, K., &Tsutsumi, M., On solutions of the initial value problem for the nonlinear Schrödinger equations. J. Funct. Anal.71, 218–245 (1987).
Hayashi, N., &Ozawa, T., Time decay of solutions to the Cauchy problem for time-dependent Schrödinger-Hartree equations. Commun. Math. Phys.110, 467–478 (1987).
Hayashi, N., &Ozawa, T., Scattering theory in the weightedL 2(ℝn) spaces for some Schrödinger equations. Ann. Inst. Henri Poincaré48, 17–37 (1988).
Hayashi, N., &Ozawa, T., Time decay for some Schrödinger equations. Math. Z.200, 467–483 (1989).
Hayashi, N., &Ozawa, T., Smoothing effect for some Schrödinger equations. J. Funct. Anal.85, 307–348 (1989).
Herbst, I. W., Spectral theory of the operator (p 2 +m 2)1/2−Ze 2/r 2.Commun. Math. Phys.53, 285–294 (1977).
Hunziker, W., On the space-time behavior of Schrödinger wavefunctions. J. Math. Phys.7, 300–304 (1966).
Jensen, A., Commutator methods and a smoothing property of the Schrödinger evolution group. Math. Z.191, 53–59 (1986).
Lin, C. S., Interpolation inequalities with weights. Comm. PDE,11, 1515–1538 (1986).
Mourre, E., Absence of singular continuous spectrum for certain self-adjoint operators. Commun. Math. Phys.78, 391–408 (1981).
Nakamitsu, K., Smoothing effects for Schrödinger evolution groups. Tokyo Denki Univ. Kiyo10, 49–62 (1988).
Ozawa, T., Invariant subspaces for the Schrödinger evolution group. Preprint, Nagoya 1989.
Perry, P., Sigal, I. M., &Simon, B., Spectral analysis ofN-body Schrödinger operators. Ann. of Math.114, 519–567 (1981).
Radin, C., &Simon, B., Invariant domains for the time-dependent Schrödinger equation. J. Differential Eqs.29, 289–296 (1978).
Simader, C. G., Bemerkungen über Schrödinger-Operatoren mit stark singulären Potentialen. Math. Z.138, 53–70 (1974).
Simon, B.,Functional Integration and Quantum Physics. Academic Press, New York (1979).
Sjölin, P., Regularity of solutions to the Schrödinger equation. Duke Math. J.55, 699–715 (1987).
Strichartz, R. T., Multipliers on fractional Sobolev spaces. J. Math. Mech.16, 1031–1060 (1967).
Tanabe, H.,Equations of Evolution. Pitman, London (1979).
Triebel, H., Spaces of distributions with weights. Multipliers inL p-spaces with weights. Math. Nachr.78, 339–355 (1977).
Yajima, K., Existence of solutions for Schrödinger evolution equations. Commun. Math. Phys.110, 415–26 (1987).
Author information
Authors and Affiliations
Additional information
Communicated by H.Brezis
Dedicated to Professor Teruo Ikebe on his sixtieth birthday
Rights and permissions
About this article
Cite this article
Ozawa, T. Smoothing effects and dispersion of singularities for the Schrödinger evolution group. Arch. Rational Mech. Anal. 110, 165–186 (1990). https://doi.org/10.1007/BF00873497
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00873497