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Regular version of the site

Article

Adiabatic approximation via hodograph translation and zero-curvature equations

Russian Journal of Mathematical Physics. 2014. Vol. 21. No. 2. P. 197-218.

For quantum, as well classical slow-fast systems, we develop a general method which allows one to compute the adiabatic invariant (approximate integral of motion), its symmetries, the adiabatic guiding center coordinates and the effective scalar Hamiltonian in all orders of a small parameter. The scheme does not exploit eigenvectors or diagonalization but is based on using the ideas of isospectral deformation and zero-curvature equations, where the role of "time" is played by the adiabatic (quantization) parameter. The algorithm includes the construction of the zero-curvature adiabatic connection and its splitting generated by averaging up to an arbitrary order in the small parameter.