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- Title
Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions.
- Authors
Lunyov, Anton A.
- Abstract
The paper is concerned with the Bari basis property of a boundary value problem associated in L2([0,1];C2)$L^2([0,1]; \mathbb {C}^2)$ with the following 2 × 2 Dirac‐type equation for y=col(y1,y2)$y = \operatorname{col}(y_1, y_2)$: LU(Q)y=−iB−1y′+Q(x)y=λy,B=b100b2,b1<0<b2,$$\begin{equation*} L_U(Q) y = -i B^{-1} y^{\prime } + Q(x) y = \lambda y , \quad B = \def\eqcellsep{&}\begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad b_1 < 0 < b_2, \end{equation*}$$with a potential matrix Q∈L2([0,1];C2×2)$Q \in L^2([0,1]; \mathbb {C}^{2 \times 2})$ and subject to the strictly regular boundary conditions Uy:={U1,U2}y=0$Uy :=\lbrace U_1, U_2\rbrace y=0$. If b2=−b1=1$b_2 = -b_1 =1$, this equation is equivalent to one‐dimensional Dirac equation. We show that the normalized system of root vectors {fn}n∈Z$\lbrace f_n\rbrace _{n \in \mathbb {Z}}$ of the operator LU(Q)$L_U(Q)$is a Bari basis inL2([0,1];C2)$L^2([0,1]; \mathbb {C}^2)$ if and only if the unperturbed operator LU(0)$L_U(0)$ 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∈Lp([0,1];C2×2)$Q \in L^p([0,1]; \mathbb {C}^{2 \times 2})$, p∈[1,2]$p \in [1,2]$, boundary conditions be strictly regular, and let {gn}n∈Z$\lbrace g_n\rbrace _{n \in \mathbb {Z}}$ be the sequence biorthogonal to the normalized system of root vectors {fn}n∈Z$\lbrace f_n\rbrace _{n \in \mathbb {Z}}$ of the operator LU(Q)$L_U(Q)$. Then, {∥fn−gn∥2}n∈Z∈(ℓp(Z))∗⇔LU(0)=LU(0)∗.$$\begin{equation*} \lbrace \Vert f_n - g_n\Vert _2\rbrace _{n \in \mathbb {Z}} \in (\ell ^p(\mathbb {Z}))^* \quad \Leftrightarrow \quad L_U(0) = L_U(0)^*. \end{equation*}$$ These abstract results are applied to noncanonical initial‐boundary value problem for a damped string equation.
- Subjects
BARI (Italy); U2 (Performer); BOUNDARY value problems; BIORTHOGONAL systems; DIRAC equation; HEAT equation; VALUATION of real property
- Publication
Mathematische Nachrichten, 2023, Vol 296, Issue 9, p4125
- ISSN
0025-584X
- Publication type
Article
- DOI
10.1002/mana.202200095