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

Abstract

Abstract. The one-dimensional Dirac operator with periodic po-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 bc n 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

On convergence of spectral expansions of Dirac operators with regular boundary conditions

“…Dirac operators with nonsmooth potentials were considered by Burlutskaya, Kornev and Khromov [6, 19]. Regular Dirac problems with potentials $$V\in L_2(0,\pi )$$ were studied by Djakov and Mityagin [9–15], and also by Arslan [1]. It was established by Lunyov and Malamud in [22, 23] and independently by Savchuk, Sadovnichaya and Shkalikov in [29, 30] that the root function system of problem (1.2), (1.3) with strongly regular boundary conditions forms a Riesz basis in $$\mathbb {H}$$ and a Riesz basis with parentheses in $$\mathbb {H}$$ in the case of regular but not strongly regular boundary conditions.…”

“…Note that in [22, 23, 29, 30] the potential $$V\in L_1(0,\pi )$$. However, for regular but not strongly regular boundary conditions (except the special case of periodic and antiperiodic ones which was investigated in [1, 9–15, 26, 27] if the functions $$P, Q\in L_2(0,\pi )$$) all the mentioned papers remain open the question whether the root function system forms a usual Riesz basis rather than a Riesz basis with parentheses. The main purpose of the present article is to study this problem.…”