论文标题
光谱特性的稳定性和边界价值问题的BARI基础属性$ 2 \ times 2 $ DIRAC类型系统
Stability of spectral characteristics and Bari basis property of boundary value problems for $2 \times 2$ Dirac type systems
论文作者
论文摘要
本文与$ q \ to \ widetilde在$ l^2（[0,1]; \ bbb c^2）$相关的不同频谱特征的稳定性属性有关，以以下$ 2 \ times2 $ 2 $ dirac类型$ l_u（q） b = {\ rm diag}（b_1，b_2），\ quad b_1 <0 <b_2，\ quad y = {\ rm col}（\ rm col}（y_1，y_2），\ quad（quad（1）$条件$ uy = \ {u_1，u_2 \} y = 0 $。 我们的光谱稳定性方法取决于系统（1）的三角转换操作员的存在$ k_q^\ pm $，并在我们以前的作品中建立的l^1 $中的$ q \ $ q \。我们从$ l^p $中从$ l^p $中的K_q^\ pm $的映射$ q \ to to to Special Banach Spaces $ x _ {\ infty，p}^2，x_ {1，p}^2 $，在这里自然产生的属性，并为$ k_q^\ pm $ k_q^\ pm的类似财产获得类似的财产。这些特性具有独立的兴趣，并在本文中讨论的所有稳定结果的证据中起着至关重要的作用。例如，作为直接结果，我们获得了映射$ q \toφ_q$的Lipshitz属性，其中$φ_Q$是系统的基本矩阵（1）。 假设边界条件（BC）是严格规律的，我们表明映射$ q \ toσ（l_u（q）） - σ（l_u（q））我们还在紧凑型上建立了其Lipshitz属性。我们显示了映射$ q \ to f_q-f_0 $ to $ l^{p'}（\ bbb z; c（[0,1]; \ bbb c^2））$的类似结果，其中$ f_q $是$ l_u（q）$的一系列归一化的eigenfunctions。这些结果对这些结果的某些修改被证明是$ l^p，p \ in [1,2] $中的。 如果$ q \在l^2 $中，我们为$ l_u（q）$的根矢量系统建立标准，以在$ l^2（[0,1]; \ bbb c^2）$中形成bari基础。在一个简单的额外假设下，当且仅当BC是自我伴侣时，该系统就会形成BARI基础。
The paper is concerned with the stability property under perturbation $Q\to\widetilde Q$ of different spectral characteristics of a BVP associated in $L^2([0,1];\Bbb C^2)$ with the following $2\times2$ Dirac type equation $$L_U(Q)y=-iB^{-1}y'+Q(x)y=λy,\quad B={\rm diag}(b_1,b_2),\quad b_1<0<b_2,\quad y={\rm col}(y_1,y_2),\quad(1)$$ with a potential matrix $Q\in L^p=L^p([0,1];\Bbb C^{2\times2})$ and subject to regular boundary conditions $Uy=\{U_1,U_2\}y=0$. Our approach to spectral stability relies on the existence of triangular transformation operators $K_Q^\pm$ for system (1) with $Q\in L^1$ established in our previous works. We prove the Lipshitz property of the mapping $Q\to K_Q^\pm$ from the balls in $L^p$ to the special Banach spaces $X_{\infty,p}^2,X_{1,p}^2$, naturally arising here, and obtain similar property for Fourier transforms of $K_Q^\pm$. These properties are of independent interest and play a crucial role in the proofs of all stability results discussed in the paper. For instance, as an immediate consequence we get the Lipshitz property of the mapping $Q\toΦ_Q$, where $Φ_Q$ is the fundamental matrix of the system (1). Assuming boundary conditions (BC) to be strictly regular, we show that the mapping $Q\toσ(L_U(Q))-σ(L_U(0))$ sends $L^p,p\in[1,2]$, either into $l^{p'}$ or into $l^p(\{(1+|n|)^{p-2}\})$; we also establish its Lipshitz property on compacts. We show similar result for the mapping $Q\to F_Q-F_0$ into $l^{p'}(\Bbb Z; C([0,1];\Bbb C^2))$, where $F_Q$ is a sequence of normalized eigenfunctions of $L_U(Q)$. Certain modifications of these results are proved for balls in $L^p,p\in[1,2]$. If $Q\in L^2$ we establish a criterion for the system of root vectors of $L_U(Q)$ to form a Bari basis in $L^2([0,1];\Bbb C^2)$. Under a simple additional assumption this system forms a Bari basis if and only if BC are self-adjoint.
