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Peierls substitution and the Maslov operator method. (English. Russian original) Zbl 1197.35086

Math. Notes 87, No. 4, 521-536 (2010); translation from Mat. Zametki 87, No. 4, 554-571 (2010).
Summary: We consider a periodic Schrödinger operator in a constant magnetic field with vector potential \(A(x)\). A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: \(\mathcal E^nu \left(-i \mu \partial_x , x \right) \varphi = E \varphi\), where \(\mathcal E^\nu\) is the corresponding energy level of some auxiliary Schrödinger operator assumed to be nondegenerate, and \(\mu \) is a small parameter. In the present paper, we use V. P. Maslov’s operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation \(\mathcal E^\nu \left(\hat P , \mu \right) \varphi = E \varphi\) with the operator \(\mathcal E^\nu \left(\hat P , \mu \right)\) represented as a function depending only on the operators of kinetic momenta \(\hat P_j = - i\mu \partial _{x_j } + A_j \left( x \right)\).

MSC:

35J10 Schrödinger operator, Schrödinger equation
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
47N20 Applications of operator theory to differential and integral equations
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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