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The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms
1.  University of WisconsinMadison, Mathematics Department, 480 Lincoln Dr. Madison, WI 537061388, United States 
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), 315348. doi: 10.1007/BF02791265. 
[4] 
M. Christ and A. Kiselev, Scattering and wave operators for onedimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12 (2002), 11741234. doi: 10.1007/s0003900211749. 
[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 longtime asymptotics of Schrödinger dynamics, Comm. Partial Differential Equations, 33 (2008), 307319. doi: 10.1080/03605300701249655. 
[7] 
O. Jørsboe and L. Mejlbro, "The CarlesonHunt 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), 619646. doi: 10.1215/S001270949809425X. 
[9] 
G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411424. 
[10] 
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205244. 
[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), 29262946. doi: 10.1016/j.jfa.2007.11.012. 
show all references
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), 315348. doi: 10.1007/BF02791265. 
[4] 
M. Christ and A. Kiselev, Scattering and wave operators for onedimensional Schrödinger operators with slowly decaying nonsmooth potentials, Geom. Funct. Anal., 12 (2002), 11741234. doi: 10.1007/s0003900211749. 
[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 longtime asymptotics of Schrödinger dynamics, Comm. Partial Differential Equations, 33 (2008), 307319. doi: 10.1080/03605300701249655. 
[7] 
O. Jørsboe and L. Mejlbro, "The CarlesonHunt 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), 619646. doi: 10.1215/S001270949809425X. 
[9] 
G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411424. 
[10] 
B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205244. 
[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), 29262946. doi: 10.1016/j.jfa.2007.11.012. 
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