Instability Zones of a Periodic 1d Dirac Operator and Smoothness of Its Potential

Let L be the differential operator Ly = i ( 1 0 0 −1 ) dy dx + ( 0 P (x) Q(x) 0 ) y, y = ( y1 y2 ) , where P (x), Q(x) are 1-periodic functions such that Q(x) = P (x). The operator L, considered on [0, 1] with periodic (y(0) = y(1)), or antiperiodic (y(0) = −y(1)) boundary conditions, is self-adjoint, and moreover, for large |n| it has, close to nπ, a pair of periodic (if n is even), or antiperiodic (if n is odd) eigenvalues λn , λ − n . We study the relationship between the decay rate of instability zone sequence γn = λn − λn , n→ ±∞, and the smoothness of the potential function P (x).