Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

Official URL: http://risc01.sabanciuniv.edu/record=b1620363 (Table of Contents)

The one-dimensional Dirac operators with periodic potentials subject to periodic, antiperiodic and a special family of general boundary conditions have discrete spectrums. It is known that, for large enough ∣n∣ in the disc centered at n of radius 1/4, the operator has exactly two eigenvalues (counted according to multiplicity) which are periodic (for even n) or antiperiodic (for odd n) and one eigenvalue derived from each general boundary condition. These eigenvalues construct a deviation which is the sum of the distance between two periodic (or antiperiodic) eigenvalues and the distance between one of the periodic (or antiperiodic) eigenvalues and one eigenvalue from the general boundary conditions. We show that the smoothness of the potential could be characterized by the decay rate of this spectral deviation. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if the absolute value of the ratio of these deviations is bounded.