Asymptotic Behavior of Solutions of the Nonstationary Schrödinger Equation with Potential that Slowly Depends on Time

We study the asymptotic behavior of solutions of the Cauchy problem for the nonstationary Schrödinger equation with a potential that slowly depends on time. The construction is based on the spectral expansion of the solution at a given time. It does not use the adiabatic theorem of the scattering th...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2023-12, Vol.277 (4), p.689-697
Main Author: Sukhanov, V. V.
Format: Article
Language:English
Subjects:
Adiabatic flow
Asymptotic properties
Cauchy problems
Mathematical analysis
Mathematics
Mathematics and Statistics
Scattering
Schrodinger equation
Theorems
Time dependence
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Summary:We study the asymptotic behavior of solutions of the Cauchy problem for the nonstationary Schrödinger equation with a potential that slowly depends on time. The construction is based on the spectral expansion of the solution at a given time. It does not use the adiabatic theorem of the scattering theory. The solution does not depend on the dynamics of the potential and is completely determined by the value of the potential at initial moment of time in the main order (in accordance with the adiabatic theorem of the scattering theory). In this paper, we calculated 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.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-023-06874-4