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

2016/02/03 by Arslan, İlker
#34L40 #47E05 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1602.01290

Abstract

The one-dimensional Dirac operator with periodic potential V=\beginpmatrix 0 & P(x) Q(x) & 0 \endpmatrix, where P,Q∈ L²([0,π]) subject to periodic, antiperiodic or a general strictly regular boundary condition (bc) has 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 (periodic if n is even or antiperiodic if n is odd) eigenvalues λ_n⁺ and λ_n^- (counted according to multiplicity) and one eigenvalue μ_n^bc corresponding to the boundary condition (bc). We prove that the smoothness of the potential could be characterized by the decay rate of the sequence |δ_n^bc|+|γ_n|, where δ_n^bc=μ_n^bc-λ_n⁺ and γ_n=λ_n⁺-λ_n^-. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if sup_γn≠0 \frac|δ_n^bc||γ_n| is finite.

Related

• 1D Dirac operators with special periodic potentials
• Riesz bases consisting of root functions of 1D Dirac operators
• Bari-Markus property for Riesz projections of 1D periodic Dirac operators
• Asymptotics of spectral gaps of 1D Dirac operator with two exponential terms potential
• Unconditional convergence of spectral decompositions of 1D Dirac operators with regular boundary conditions