Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

Abstract

The paper is concerned with the Bari basis property of a boundary value problem associated in 𝐿 2 ([0, 1]; ℂ 2 ) with the following 2 × 2 Dirac-type equation for 𝑦 = col(𝑦 1 , 𝑦 2 ):with a potential matrix 𝑄 ∈ 𝐿 2 ([0, 1]; ℂ 2×2 ) and subject to the strictly regular boundary conditions 𝑈𝑦 ∶= {𝑈 1 , 𝑈 2 }𝑦 = 0. If 𝑏 2 = −𝑏 1 = 1, this equation is equivalent to one-dimensional Dirac equation. We show that the normalized system of root vectors {𝑓 𝑛 } 𝑛∈ℤ of the operator 𝐿 𝑈 (𝑄) is a Bari basis in 𝐿 2 ([0, 1]; ℂ 2 ) if and only if the unperturbed operator 𝐿 𝑈 (0) is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let 𝑄 ∈ 𝐿 𝑝 ([0, 1]; ℂ 2×2 ), 𝑝 ∈ [1, 2], boundary conditions be strictly regular, and let {𝑔 𝑛 } 𝑛∈ℤ be the sequence biorthogonal to the normalized system of root vectors {𝑓 𝑛 } 𝑛∈ℤ of the operator 𝐿 𝑈 (𝑄). Then,These abstract results are applied to noncanonical initial-boundary value problem for a damped string equation.