Zig-zag-matrix 代数与可解准厄米量子模型
Zig-zag-matrix algebras and solvable quasi-Hermitian quantum models
AI总结：
本文研究非厄米量子系统，提出在 $Θ-$准厄米量子力学中，可用哈密顿量的极大稀疏“zig-zag-matrix”表示替代对角化矩阵解来处理量子束缚态问题。
摘要：
众所周知，封闭 $M-$ 能级量子系统的酉演化可由具有实谱的非厄米哈密顿量 $H$ 生成。其厄米性可通过修正的内积度量 $Θ$ 恢复。在厄米情形中，谱（即束缚态能量）的评估通常通过哈密顿量的对角化实现。在非厄米（或更准确地说，在 $Θ-$准厄米）量子力学中，我们推测量子束缚态问题的对角化矩阵解的作用，可转移到哈密顿量的极大稀疏“zig-zag-matrix”表示上。
英文摘要：
It is well known that the unitary evolution of a closed $M-$level quantum system can be generated by a non-Hermitian Hamiltonian $H$ with real spectrum. Its Hermiticity can be restored via an amended inner-product metric $Θ$. In Hermitian cases the evaluation of the spectrum (i.e., of the bound-state energies) is usually achieved by the diagonalization of the Hamiltonian. In the non-Hermitian (or, more precisely, in the $Θ-$quasi-Hermitian) quantum mechanics we conjecture that the role of the diagonalized-matrix solution of the quantum bound-state problem could be transferred to a maximally sparse ``zig-zag-matrix'' representation of the Hamiltonians.
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