Abstract:
The complete manifold of ground-state eigenfunctions for the purely magnetic two-dimensional Pauli operator is considered as a byproduct of a new reduction (found by the authors several years ago) for the algebro-geometric inverse spectral data (that is, Riemann surfaces and divisors). This reduction is associated with a (2+1)-soliton hierarchy containing a 2D analogue of the famous ‘Burgers system’. This paper also surveys previous papers since 1980, including the first topological ideas in the space of quasi-momenta, and presents new results on self-adjoint boundary-value problems for the Pauli operator. The ‘non-spectral’ Bloch–Floquet functions of zero 2D level give discrete points of additional spectrum analogous to the ‘boundary states’ of finite-gap 1D potentials in the gaps.
Bibliography: 35 titles.
Keywords:
magnetic Pauli operator, algebro-geometric solutions, ground state, Landau levels, boundary-value problems.
The work of the first and third authors was done with the support of the Russian Foundation for Basic Research (grant no. 13-01-12469-офи-м2) and the programme "Leading Scientific Schools" (grant НШ-4833.2014.1). The first author was also supported by the programme "Fundamental Problems of Non-Linear Dynamics" of the Presidium of the Russian Academy of Sciences. The investigation of the second author was carried out with the support of a grant from the Russian Science Foundation (project 14-11-00441).
Citation:
P. G. Grinevich, A. E. Mironov, S. P. Novikov, “On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator”, Russian Math. Surveys, 70:2 (2015), 299–329
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\pages 299--329
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https://doi.org/10.1070/RM2015v070n02ABEH004948
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This publication is cited in the following 2 articles:
Polina A. Leonchik, Andrey E. Mironov, “Two-dimensional discrete operators and rational functions on algebraic curves”, São Paulo J. Math. Sci., 2024
Guillaume Dhont, Toshihiro Iwai, Boris Zhilinskii, “Topological Phase Transition in a Molecular Hamiltonian with Symmetry and Pseudo-Symmetry, Studied through Quantum, Semi-Quantum and Classical Models”, SIGMA, 13 (2017), 054, 34 pp.
Citation in format AMSBIB
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