A gap in the energy spectrum of the one-dimensional Dirac operator

Abstract

One considers the one-dimensional Dirac operator with a slowly oscillating potential

$$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$

((1))

where

. The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (−∞,−¦q¦), (¦q¦, ∞). The interval (−¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C₊=sign C₋, situated either at the point (under the condition C₊>0) or at the point λ=−¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.