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Uspekhi Mat. Nauk, 2015, Volume 70, Issue 2(422), Pages 109–140 (Mi umn9650)  

This article is cited in 1 scientific paper (total in 1 paper)

On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator

P. G. Grinevicha, A. E. Mironovbc, S. P. Novikovdae

a L.D. Landau Institute for Theoretical Physics RAS
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
c Novosibirsk State University
d Steklov Mathematical Institute of Russian Academy of Sciences
e University of Maryland, College Park, MD, USA

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.

Funding Agency Grant Number
Russian Foundation for Basic Research 13-01-12469-офи-м2
Ministry of Education and Science of the Russian Federation НШ-4833.2014.1
Russian Academy of Sciences - Federal Agency for Scientific Organizations
Russian Science Foundation 14-11-00441
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).


DOI: https://doi.org/10.4213/rm9650

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English version:
Russian Mathematical Surveys, 2015, 70:2, 299–329

Bibliographic databases:

UDC: 517.958+517.929.7+517.984.5
MSC: 81Q05, 35C05, 35J10, 35J67
Received: 07.01.2015

Citation: P. G. Grinevich, A. E. Mironov, S. P. Novikov, “On the non-relativistic two-dimensional purely magnetic supersymmetric Pauli operator”, Uspekhi Mat. Nauk, 70:2(422) (2015), 109–140; Russian Math. Surveys, 70:2 (2015), 299–329

Citation in format AMSBIB
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    This publication is cited in the following articles:
    1. 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.  mathnet  crossref
  • Успехи математических наук Russian Mathematical Surveys
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