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

TL;DR

This paper establishes simple criteria for 1D and 2D localization of quantum states caused by weak defects in periodic potentials, applicable across various dimensions and band structures, using an elementary variational approach.

Contribution

It introduces new sufficient conditions for localization in multiple dimensions and for defects with arbitrary weakness, extending previous results to higher dimensions and degenerate band edges.

Findings

Localization occurs for weak defects within band gaps.

Conditions also apply below the ground state energy.

Results extend to higher dimensions and degenerate bands.

Abstract

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.