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Averaging of linear operators, adiabatic approximation, and pseudodifferential operators. (English. Russian original) Zbl 1282.47079

Math. Notes 92, No. 2, 151-165 (2012); translation from Mat. Zametki 92, No. 2, 163-180 (2012).
The authors deal with the Schrödinger and Klein-Gordon equations with fast oscillating coefficients (velocity and potential), using the averaging methods (the adiabatic approximation) based on V. P. Maslov’s operator method.
In the averaging methods, operators appearing in the Schrödinger and Klein-Gordon equations are expanded in a certain parameter. This parameter is something like \(\hbar\), Planck’s constant divided by \(2\pi\), in quantum mechanics.
The authors construct asymptotic solutions of the Schrödinger and Klein-Gordon equations as the parameter goes to zero in the averaging methods.

MSC:

47N20 Applications of operator theory to differential and integral equations
35Q40 PDEs in connection with quantum mechanics
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