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

We prove, via an elementary variational method, one-dimensional (1D) and two-dimensional (2D) localization within the band gaps of a periodic Schrödinger 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, by a linear or planar defect in a three-dimensional crystal. For the case of $D$-fold degenerate band edges, we also give sufficient conditions for localization of up to $D$ states.

Figure 1

(Color online) Example of a periodic potential $V_0$ and defect potential $V$ with the spectrum $\sigma \left(H_0\right)$ located at the right. The spectrum of the unperturbed Schrödinger operator has an infimum $E_0$ and may also have a gap in the set $\left(E_1,E_2\right)$.

Figure 2

(Color online) Linear defect for a periodic lattice (red circles) in two dimensions is created by adding a defect potential (blue circles) that is periodic along the $\mathbf{z}$ direction, commensurate with the periodicity of the underlying lattice. This will localize a state with respect to the perpendicular directions, denoted $\mathbf{r}$.

Figure 3

The leftmost plot is a schematic picture of the spectrum $\sigma \left(H\right)$ (with some arbitrary origin 0). The first step is to shift the center of the gap to the zero position by subtracting $E_g$. The middle plot shows $\sigma \left(H-E_g\right)$. Then, the operator is squared and all energies become positive. The final plot shows $\sigma \left[{\left(H-E_g\right)}^2\right]$: any eigenvalue introduced into the gap of the original operator will now be an extremal eigenvalue of the shifted and squared operator.