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| Name: | ||
| Miloslav Znojil | ||
| Reviewer number: | ||
| 9689 | ||
| Email: | ||
| znojil@ujf.cas.cz | ||
| Item's zbl-Number: | ||
| DE 018 217 653 | ||
| Author(s): | ||
| Shterenberg, R. G.: | ||
| Shorttitle: | ||
| Absolute continuity of the spectrum of two-dimensional periodic Schroedinger operators | ||
| Source: | ||
| St. Petersbg. Math. J. 13, No. 4, 659-683 (2002); transl. from Algebra Anal. 13, No. 4, 196-228 (2001) | ||
| Classification: | ||
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| Primary Classification: | ||
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| Secondary Classification: | ||
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| Keywords: | ||
| Schroedinger operator in two dimensions; density-like periodic potential; coordinate-dependent mass-term; absolutely continous spectrum; | ||
| Review: | ||
The general coordinate-dependent two-by-two metric g(x) is admitted in the kinetic energy of the Schroedinger operator in two dimensions with periodic boundary conditions for the the density-like class of potentials. The absolute continuity of the spectrum is proved. The paper is a continuation of the author's study of a similar problem in 2000 (paper cited as [Sh]). Now, the so called subordination condition imposed upon the class of potentials (cf. eq. (0.11)) is ``maximally" weakened (basically, to eqs. (1.8) and (1.9) which just ensure the boundedness from below of the form m of the Schroedinger operator M) at a cost of the absence of the magnetic field (to be incorporated in a subsequent paper). The method which worked in the pioneering 1973 paper [T] (by L. Thomas, on the scattering from impurities in a crystal), based on a complexification of quasi-momenta (their large imaginary values prove most relevant) is still in use here. The difficulties with the metric g(x) (which is, by the way, constrained just by a single condition (0.13)) are being avoided in two steps, via a scalar case and starting from the current trivial g=1. | ||
| Remarks to the editors: | ||