第一类离散量子方阱
Discrete quantum square well of the first kind
摘要：
提出了一个玩具模型量子系统。在给定的整数$N$下，它由一对$N$ × $N$实矩阵$(H,Θ)$定义，其中第一项$H$指定一个初等的、可对角化的非厄米哈密顿量$H \neq H^\dagger$，其具有由第一类的$N-$ th Chebyshev多项式的零点给出的实数和显式谱。第二项$Θ\neq I$必须（并且正在）被构造为相关的希尔伯特空间度量，该度量指定（通常，非唯一的）物理内积，并且使我们的玩具模型哈密顿量自伴，即，与Dieudonne方程$H^\dagger Θ= Θ\,H$相容。然后，以封闭的带矩阵形式构造合格度量的（原则上，完整的）集合的元素。它们随着我们对可选参数$N-$小集合的选择而变化，$Θ=Θ(\vecκ)>0$必须（并且正在）被选择为位于度量$\vecκ \in {\cal D}^{(physical)}$的正性域中。
英文摘要：
A toy-model quantum system is proposed. At a given integer $N$ it is defined by the pair of $N$ by $N$ real matrices $(H,Θ)$ of which the first item $H$ specifies an elementary, diagonalizable non-Hermitian Hamiltonian $H \neq H^\dagger$ with the real and explicit spectrum given by the zeros of the $N-$th Chebyshev polynomial of the first kind. The second item $Θ\neq I$ must be (and is being) constructed as the related Hilbert-space metric which specifies the (in general, non-unique) physical inner product and which renders our toy-model Hamiltonian selfadjoint, i.e., compatible with the Dieudonne equation $H^\dagger Θ= Θ\,H$. The elements of the (in principle, complete) set of the eligible metrics are then constructed in closed band-matrix form. They vary with our choice of the $N-$plet of optional parameters, $Θ=Θ(\vecκ)>0$ which must be (and are being) selected as lying in the positivity domain of the metric, $\vecκ \in {\cal D}^{(physical)}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
