The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms

Sergey A. Denisov

doi:10.3934/dcds.2011.30.77

Article Contents

2011, Volume 30, Issue 1: 77-113. Doi: 10.3934/dcds.2011.30.77

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The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms

Sergey A. Denisov^1,

1.
University of Wisconsin-Madison, Mathematics Department, 480 Lincoln Dr. Madison, WI 53706-1388

Received: November 30, 2009

Revised: April 30, 2010

Published: December 2010

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Abstract

In this paper, we study the generic behavior of the solutions to a large class of evolution equations. The Schrödinger evolution is considered as an application.

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Mathematics Subject Classification: Primary 34G10; secondary 34E10.

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References

[1]	M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables," National Bureau of Standards Applied Mathematics Series, 55.
[2]	A. Böttcher and B. Silbermann, "Introduction to Large Truncated Toeplitz Matrices," Springer. 1998.
[3]	J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential, J. Anal. Math., 77 (1999), 315-348.doi: 10.1007/BF02791265.
[4]	M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12 (2002), 1174-1234.doi: 10.1007/s00039-002-1174-9.
[5]	S. Denisov, Continuous analogs of polynomials orthogonal on the unit circle. Krein systems, Int. Math. Res. Surveys, Vol. 2006 (2006).
[6]	S. Denisov, An evolution equation as the WKB correction in long-time asymptotics of Schrödinger dynamics, Comm. Partial Differential Equations, 33 (2008), 307-319.doi: 10.1080/03605300701249655.
[7]	O. Jørsboe and L. Mejlbro, "The Carleson-Hunt Theorem on Fourier Series," Lecture Notes in Mathematics 911, Springer, 1982.
[8]	A. Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators, Duke Math. J., 94 (1998), 619-646.doi: 10.1215/S0012-7094-98-09425-X.
[9]	G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411-424.
[10]	B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205-244.
[11]	B. Simon, "Orthogonal Polynomials on the Unit Circle," Parts 1 and 2. American Mathematical Society Colloquium Publications, 54, American Mathematical Society, Providence, RI, 2005.
[12]	W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations, J. Funct. Anal., 254 (2008), 2926-2946.doi: 10.1016/j.jfa.2007.11.012.