Geodesic
Asymptotic behavior of solutions of the nonstationary Schrodinger equation with a slowly time dependent potential
Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 50, Tome 493 (2020), pp. 323-335

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The asymptotic behavior of solutions of the Cauchy problem for the nonstationary Schrodinger equation with a rapidly decreasing potential is studied. The potential is slowly depending on time. The construction of asymptotic solutions is based on the spectral expansion of the solution at a given time. It do not use the adiabatic theorem of scattering theory. In the highest order (as in the approach associated with the adiabatic theorem of scattering theory) the solution does not depend on the dynamics of the potential and is completely determined by the value of the potential at the zero moment of time. We calculate corrections to the leading term of the solution associated with the boundary of the continuous spectrum. These corrections take into account the time dependence of the operator. 
@article{ZNSL_2020_493_a21,
     author = {V. V. Sukhanov},
     title = {Asymptotic behavior of solutions of the nonstationary {Schrodinger} equation with a slowly time dependent potential},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {323--335},
     publisher = {mathdoc},
     volume = {493},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_493_a21/}
}
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V. V. Sukhanov. Asymptotic behavior of solutions of the nonstationary Schrodinger equation with a slowly time dependent potential. Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 50, Tome 493 (2020), pp. 323-335. http://geodesic.mathdoc.fr/item/ZNSL_2020_493_a21/