论文标题
$ 2 \ times 2 $ dirac-type操作员的标准Bari基础财产的标准严格规则边界条件
Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions
论文作者
论文摘要
本文与$ l^2（[0,1]; \ Mathbb {C}^2）$相关的边界价值问题的BARI基础属性以及以下$ 2 \ times 2 $ dirac-type方程，for $ y = {\ rm col}（\ rm col}（\ rm col}（y_1，y_1，y_2） \ quad b = \ begin {pmatrix} b_1＆0 \\ 0＆b_2 \ end {pmatrix}，\ quad b_1 <0 <0 <b_2，$ $ p in l^2 in l^2（[0,1] ：= \ {u_1，u_2 \} y = 0 $。如果$ b_2 = -b_1 = 1 $，则此方程等于一维dirac方程。我们表明，均方根的系统$ \ {f_n \} _ {n \ in \ Mathbb {z}} $的运算符$ l_u（q）$的$ l^2（[0,1]; \ mathbb {c}^2）$ if和仅iS unferterbed oper $ l_u（$ l_u l_U（n ofterbe）$ l_u（n lund_e）$ l_u（n n ofterbb）$ l_u（n n ofterbb inforbed optor $ l_u（untor）$ l_u（在边界条件下的系数方面，我们还为此提供了明确的条件。 BARI基准标准是我们更一般结果的结果：让$ q \ in l^p（[0,1]; \ Mathbb {c}^{2 \ times 2}）$，$ p \ in [1,2] $，边界条件是严格规律的，让$ \ \ {g_n \} $ be} BiorThogonal to root vectors $ \ {f_n \} _ {n \ in \ Mathbb {z}} $ ofer $ l_u（q）$然后$ \ {\ | f_n -g_n \ | _2 \} _ {n \ in \ Mathbb {z}}} \ in（\ ell^p（\ ell^p（\ althbb {z}}） $$这些抽象结果应用于抑制字符串方程的非规范初始值问题。
The paper is concerned with the Bari basis property of a boundary value problem associated in $L^2([0,1]; \mathbb{C}^2)$ with the following $2 \times 2$ Dirac-type equation for $y = {\rm col}(y_1, y_2)$: $$L_U(Q) y =-i B^{-1} y' + Q(x) y = λy , \quad B = \begin{pmatrix} b_1 & 0 \\ 0 & b_2 \end{pmatrix}, \quad b_1 < 0 < b_2, $$ with a potential matrix $Q \in L^2([0,1]; \mathbb{C}^{2 \times 2})$ and subject to the strictly regular boundary conditions $Uy :=\{U_1, U_2\}y=0$. If $b_2 = -b_1 =1$ this equation is equivalent to one dimensional Dirac equation. We show that the system of root vectors $\{f_n\}_{n \in \mathbb{Z}}$ of the operator $L_U(Q)$ forms a Bari basis in $L^2([0,1]; \mathbb{C}^2)$ if and only if the unperturbed operator $L_U(0)$ is self-adjoint. We also give explicit conditions for this in terms of coefficients in the boundary conditions. The Bari basis criterion is a consequence of our more general result: Let $Q \in L^p([0,1]; \mathbb{C}^{2 \times 2})$, $p \in [1,2]$, boundary conditions be strictly regular, and let $\{g_n\}_{n \in \mathbb{Z}}$ be the sequence biorthogonal to the system of root vectors $\{f_n\}_{n \in \mathbb{Z}}$ of the operator $L_U(Q)$. Then $$ \{\|f_n - g_n\|_2\}_{n \in \mathbb{Z}} \in (\ell^p(\mathbb{Z}))^* \quad\Leftrightarrow\quad L_U(0) = L_U(0)^*. $$ These abstract results are applied to non-canonical initial-boundary value problem for a damped string equation.
