Regular Boundary Value Problems for the Dirac Operator
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Regular Boundary Value Problems for the Dirac Operator A. S. Makin Presented by Academician of the RAS E.I. Moiseev January 13, 2020 Received January 14, 2020; revised March 31, 2020; accepted March 31, 2020
Abstract—Spectral problems for the Dirac operator specified on a finite interval with regular, but not strongly regular boundary conditions and a complex-valued integrable potential are studied. This work is aimed at finding the conditions under which the root function system forms a common Riesz basis rather than a Riesz basis with parentheses. Keywords: Dirac operator, spectral expansion, regular boundary conditions DOI: 10.1134/S106456242003014X
1. INTRODUCTION We consider the Dirac system
By' + Vy = λ y,
(1)
where y = col( y1( x), y2 ( x )) ,
−i 0 0 P ( x) B= , V = , 0 i Q( x) 0 P ( x), Q( x ) ∈ L1(0, π) , with two-point boundary conditions (2) U (y ) = Cy(0) + Dy(π) = 0, where C =
a 11 a 21
a12 , a22
D=
a 13 a 23
a14 , a24
the coefficients aij are arbitrary complex numbers, and the rows of the matrix
A=
a 11 a 21
a12 a13 a14 a22 a23 a24
are linearly independent. The operator Ly = By' + Vy is regarded as a linear operator in the space H = L2(0, π) ⊕ L2(0, π) with the domain D(L) = {y ∈ W11[0, π] : Ly ∈ H, U j (y ) = 0 ( j = 1, 2)} . Eigenvalue problems for the operator L with boundary conditions (2) have been studied in many
Russian Technological University (MIREA), Moscow, 119454 Russia e-mail: [email protected]
works. It was established in [1] that the root function system of problem (1), (2) with regular boundary conditions is complete in H . Regular Dirac problems with potentials V ∈ L2(0, π) were treated by Djakov and Mityagin [2–10]; in [8], they proved a theorem on the unconditional basis property with parentheses for the corresponding root function system. A much more complicated case when V ( x) ∈ L1(0, π) was considered by Savchuk and Shkalikov in [11], where they established that the root function system of problem (1), (2) with strongly regular boundary conditions forms a Riesz basis in H and a Riesz basis with parentheses in H in the case of regular, but not strongly regular boundary conditions. By another method, analogous results on the Riesz basis property of the root function system of problem (1), (2) with strongly regular boundary conditions for V ( x) ∈ L1(0, π) were deduced by Lunyov and Malamud in [12, 13] and on the Riesz basis property with parentheses in the case of regular, but not strongly regular boundary conditions in [13], as well as by Savchuk and Sadovnichaya in [14] (see [13, p. 778] for more detailed references). However, in the case of regular, but not strongly regular boundary conditions (except for the particular case of periodic and antiperiodic conditions, which was studied in [2–10] when P ( x), Q( x) ∈ L2(0, π) ), all the above works leave open the question of whether the root function system forms a common Riesz basis rather than a Riesz basis with parentheses. The main purpose
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