Physics – Mathematical Physics
Scientific paper
2011-01-29
Physics
Mathematical Physics
36 pages, 10 figures, A part of text missed in the previous version was added
Scientific paper
In general, the motion of charged particle in the external electromagnetic field is described by the full 4D Dirac equation (for the spin equal to 1/2). For time-independent fields Pauli obtained a famous beautiful nonrelativistic approximation. We are going to consider a special 2D case where the electric field is equal to zero. The remarkable theory of this operator starts in 1979--80. In the present work we found unification of the Theory of Ground States for the Pauli Operator with the Algebro-Geometric Spectral Theory of periodic scalar Schr\"odinger operators and corresponding theory of soliton nonlinear systems. Especially analog of the 2D Burgers hierarchy plays a key role here. The Aharonov--Bohm type terms in magnetic field like the quantized delta function also play important role in our construction. This work extends results published in arXiv:1001.4300, arXiv:1004.1157, arXiv:1005.0612. We pay special attention to the boundary problems allowing to reveal physical meaning of the complex Bloch functions.
Grinevich Piotr
Mironov Aleksej
Novikov Sergey
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