Abstract
We study the transition to the continuum of an initially bound quantum particle in ℝd, d=1,2,3, subjected, for t≥0, to a time periodic forcing of arbitrary magnitude. The analysis is carried out for compactly supported potentials, satisfying certain auxiliary conditions. It provides complete analytic information on the time Laplace transform of the wave function. From this, comprehensive time asymptotic properties (Borel summable transseries) follow.We obtain in particular a criterion for whether the wave function gets fully delocalized (complete ionization). This criterion shows that complete ionization is generic and provides a convenient test for particular cases. When satisfied it implies absence of discrete spectrum and resonances of the associated Floquet operator. As an illustration we show that the parametric harmonic perturbation of a potential chosen to be any nonzero multiple of the characteristic function of a measurable compact set has this property.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola. Norm. Sup. Pisa, Ser. IV 2:151–218 (1975).
J. Belissard, Stability and instability in quantum mechanics, in Trends and Developments in the Eighties, S. Albeverio and Ph. Blanchard, eds. (World Scientific, Singapore, 1985), pp. 1–106.
C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers(McGraw-Hill, 1978). [Springer-Verlag 1999.]
C. Cohen-Tannoudji, J. Duport-Roc, and G. Arynberg, Atom-Photon Interactions(Wiley, 1992).
5. O. Costin, On Borel summation and Stokes phenomena for rank one nonlinear systems of ODE's, Duke Math. J. 93:289–344 (1998).
O. Costin, R. D. Costin, and J. Lebowitz, Transition to the continuum of a particle in time-periodic potentials, in Advances in Differential Equations and Mathematical Physics, AMS Contemporary Mathematics series, Karpeshina, Stolz, Weikard, and Zeng, eds. (2003).
O. Costin, J. Lebowitz, and A. Rokhlenko, Exact results for the ionization of a model quantum system, J. Phys. A: Math. Gen. 33:1–9 (2000).
O. Costin, R. D. Costin, J. Lebowitz, and A. Rokhlenko, Evolution of a model quantum system under time periodic forcing: Conditions for complete ionization, Commun. Math. Phys. 221:1–26 (2001).
O. Costin, A. Rokhlenko, and J. Lebowitz, On the complete ionization of a periodically perturbed quantum system, CRM Proceedings and Lecture Notes 27:51–61 (2001).
O. Costin and A. Soffer, Resonance theory for Schrödinger operators, C ommun. Math. Phys. 224(2001).
O. Costin, R. D. Costin, and J. L. Lebowitz, in preparation.
O. Costin and R. D. Costin, Rigorous WKB for discrete schemes with smooth coefficients, SIAM J. Math. Anal. 27:110–134 (1996).
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger Operators(Springer-Verlag, 1987).
J. Écalle, Fonctions Resurgentes(Publications Mathématiques D'Orsay, 1981).
J. Écalle, Bifurcations and periodic orbits of vector fields, NATO ASI Series 408(1993).
A. Galtbayar, A. Jensen, and K. Yajima, Local time-decay of solutions to Schrödinger equations with time-periodic potentials, J. Stat. Phys. 116:231–282 (2004).
L. Hörmander, Linear Partial Differential Operators(Springer, 1963).
J. S. Howland, Stationary scattering theory for time dependent Hamiltonians, Math. Ann. 207:315–335 (1974).
H. R. Jauslin and J. L. Lebowitz, Spectral and stability aspects of quantum chaos, Chaos 1:114–121 (1991).
T. Kato, Perturbation Theory for Linear Operators(Springer-Verlag, 1995).
P. D. Miller, A. Soffer, and M. I. Weinstein, Metastability of breather modes of time dependent potentials, Nonlinearity 13:507–568 (2000).
C. Miranda, Partial Differential Equations of Elliptic Type(Springer-Verlag, 1970).
M. Reed and B. Simon, Methods of Modern Mathematical Physics(Academic Press, New York, 1972).
A. Rokhlenko, O. Costin, and J. L. Lebowitz, Decay versus survival of a local state subjected to harmonic forcing: Exact results, J. Phys. A: Math. Gen. 35:8943 (2002).
S. Saks and A. Zygmund, Analytic Functions(Warszawa-Wroclaw, 1952).
B. Simon, Schrödinger operators in the twentieth century, J. Math. Phys. 41:3523 (2000).
A. Soffer and M. I. Weinstein, Nonautonomous Hamiltonians, J. Stat. Phys. 93:359–391 (1998).
28. F. Treves, Basic Linear Partial Differential Equations(Academic Press, 1975).
W. Wasow, Asymptotic Expansions for Ordinary Differential Equations(Interscience, 1968).
K. Yajima, Scattering theory for Schrödinger equations with potentials periodic in time, J. Math. Soc. Japan 29:729 (1977).
K. Yajima, Existence of solutions of Schrödinger evolution equations, Commun. Math. Phys. 110:415 (1987).
A. H. Zemanian, Distribution Theory and Transform Analysis(McGraw-Hill, New York, 1965).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Costin, O., Costin, R.D. & Lebowitz, J.L. Time Asymptotics of the Schrödinger Wave Function in Time-Periodic Potentials. Journal of Statistical Physics 116, 283–310 (2004). https://doi.org/10.1023/B:JOSS.0000037244.42209.f7
Issue Date:
DOI: https://doi.org/10.1023/B:JOSS.0000037244.42209.f7