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Abstract: For 1D Dirac operators Ly= i J y- + v y, where J is a diagonal 2x2 matrixwith entrees 1,-1 and vx is an off-diagonal matrix with L^2 0,\pi-entreesPx, Qx 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 ofrelated spectral gaps gamma n = | \lambda n,+ - \lambda n,-|, where\lambda

are the eigenvalues of L=Lv considered on 0,\pi with periodicfor even n or antiperiodic for odd n boundary conditions bc;ii there is a Riesz basis which consists of periodic or antiperiodiceigenfunctions 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 generallythe families X t defined for real nonzero t by \overline{Q} =t P. Finite-zonepotentials belonging to X t are dense in X t.Another example: if Px=a exp2ix+b exp-2ix, Qx=Aexp2ix+Bexp-2ixwith complex a, b, A, B eq 0, then the system of root functions of L consistseventually of eigenfunctions. For antiperiodic bc this system is a Riesz basisif |aA|=|bB| then v \in X, and it is not a basis if |aA| eq |bB|. Forperiodic bc the system of root functions is a Riesz basis and v \in X always.



Author: Plamen Djakov, Boris Mityagin

Source: https://arxiv.org/







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