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1D Dirac operators with special periodic potentials

For 1D Dirac operators Ly= i J y' + v y, where J is a diagonal 2x2 matrix with entrees 1,-1 and v(x) is an off-diagonal matrix with L^2 [0,π]-entrees P(x), Q(x) we characterize the class X of pi-periodic potentials v such that: (i) the smoothness of potentials v is determined only by the rate of decay of related spectral gaps gamma (n) = | λ(n,+) - λ(n,-)|, where λ(..) are the eigenvalues of L=L(v) considered on [0,π] with periodic (for even n) or antiperiodic (for odd n) boundary conditions (bc); (ii) there is a Riesz basis which consists of periodic (or antiperiodic) eigenfunctions and (at most finitely many) associated functions. In particular, X contains symmetric potentials X_{sym} (\overline{Q} =P), skew-symmetric potentials X_{skew-sym} (\overline{Q} =-P), or more generally the families X_t defined for real nonzero t by \overline{Q} =t P. Finite-zone potentials belonging to X_t are dense in X_t. Another example: if P(x)=a exp(2ix)+b exp(-2ix), Q(x)=Aexp(2ix)+Bexp(-2ix) with complex a, b, A, B \neq 0, then the system of root functions of L consists eventually of eigenfunctions. For antiperiodic bc this system is a Riesz basis if |aA|=|bB| (then v \in X), and it is not a basis if |aA| \neq |bB|. For periodic bc the system of root functions is a Riesz basis (and v \in X) always.