Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

Details Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps

2010-02-24

arxiv.org/abs/1002.4426v3

Phys. Rev. B 81, 155324 (2010)

Physics

Condensed Matter

Other Condensed Matter

9 pages, 3 figures

Scientific paper

10.1103/PhysRevB.81.155324

We prove, via an elementary variational method, 1d and 2d localization within the band gaps of a periodic Schrodinger operator for any mostly negative or mostly positive defect potential, V, whose depth is not too great compared to the size of the gap. In a similar way, we also prove sufficient conditions for 1d and 2d localization below the ground state of such an operator. Furthermore, we extend our results to 1d and 2d localization in d dimensions; for example, a linear or planar defect in a 3d crystal. For the case of D-fold degenerate band edges, we also give sufficient conditions for localization of up to D states.

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