论文标题
块雅各比矩阵和狄拉克运算符的缺陷指数和离散属性具有点相互作用
Deficiency indices and discreteness property of block Jacobi matrices and Dirac operators with point interactions
论文作者
论文摘要
本文涉及无限对称块jacobi矩阵$ \ bf j $，$ p \ times p $ -matrix条目。我们提出了新的条件，使雅各比矩阵成为自我之间并具有离散频谱的新条件。 在我们以前的论文中，建立了此类矩阵类别与对称$ 2P \ times 2p $ 2p $ dirac运算符$ \ mathrm {\ bf d} _ {x，α} $与$ l^2（\ bb r; \ bb r; \ bbb c^{2pp}）$的点交互。特别是，他们的缺陷索引与$ n_ \ pm（\ mathrm {\ bf d} _ {x，α}）= n_ \ pm（{\ bf j} _ {x，x，α}）$相关。对于此类的jacobi矩阵，我们提供几个条件，以确保平等$ n_ \ pm（{\ bf j} _ {x，α}）= k $，并使用任何$ k \ le p $。给出了具有点相互作用的矩阵Schrodinger和Dirac运算符的应用。值得一提的是，狄拉克（Dirac）和雅各比（Jacobi）操作员之间的联系首次在这两个方向上使用。特别是，要证明$ n_ \ pm（{\ bf j} _ {x，α}）= p $ for $ {\ bf j} _ {x，α，α} $，我们首先为dirac Operator $ \ mathrm {\ bf d} _ {x，x，α，α，α} $建立了它。
The paper concerns with infinite symmetric block Jacobi matrices $\bf J$ with $p\times p$-matrix entries. We present new conditions for general block Jacobi matrices to be selfadjoint and have discrete spectrum. In our previous papers there was established a close relation between a class of such matrices and symmetric $2p\times 2p$ Dirac operators $\mathrm{\bf D}_{X,α}$ with point interactions in $L^2(\Bbb R; \Bbb C^{2p})$. In particular, their deficiency indices are related by $n_\pm(\mathrm{\bf D}_{X,α})= n_\pm({\bf J}_{X,α})$. For block Jacobi matrices of this class we present several conditions ensuring equality $n_\pm({\bf J}_{X,α})=k$ with any $k \le p$. Applications to matrix Schrodinger and Dirac operators with point interactions are given. It is worth mentioning that a connection between Dirac and Jacobi operators is employed here in both directions for the first time. In particular, to prove the equality $n_\pm({\bf J}_{X,α})=p$ for ${\bf J}_{X,α}$ we first establish it for Dirac operator $\mathrm{\bf D}_{X,α}$.
