具有实谱的玩具哈密尔顿算子$H \neq H^\dagger$族的合格度量的完整菜单
The complete menu of eligible metrics for a family of toy Hamiltonians $H \neq H^\dagger$ with real spectra
摘要：
考虑了一组具有实谱的非厄米$N$乘$N$矩阵$H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\dagger$的初等集合，假设这些矩阵中的每一个表示{ \it ad hoc}希尔伯特状态空间${\cal H}^{(physical)}$中的自伴量子哈密顿量.所有这些空间的明确规范的问题（即，本质上，所有合格的{ \it adhoc}内积和度量运算符$Θ$）都被寻址。证明了该问题的精确可解性，并且对于矩阵$H^{(N)}(λ)$中的每个尺寸$N=2,4,...$和参数$λ\in (-1,1)$，由封闭公式递归地定义了完整的$N-$参数度量集$Θ^{(N)}_{α_1,α_2,...,α_N}(λ)$.
英文摘要：
An elementary set of non-Hermitian $N$ by $N$ matrices $H^{(N)}(λ) \neq [ H^{(N)}(λ)]^\dagger$ with real spectra is considered, assuming that each of these matrices represents a selfadjoint quantum Hamiltonian in an {\it ad hoc} Hilbert space of states ${\cal H}^{(physical)}$. The problem of an explicit specification of all of these spaces (i.e., in essence, of all of the eligible {\it ad hoc} inner products and metric operators $Θ$) is addressed. The problem is shown exactly solvable and, for every size $N=2,4,...$ and parameter $λ\in (-1,1)$ in matrix $H^{(N)}(λ)$, the complete $N-$parametric set of metrics $Θ^{(N)}_{α_1,α_2,...,α_N}(λ)$ is recurrently defined by closed formula.
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