标题： 具有倒置势阱的厄米算符的虚特征值以及通过非厄米相互作用在例外点处过渡到实谱 显示英文标题

标题： Imaginary eigenvalues of Hermitian Hamiltonian with an inverted potential well and transition to the real spectrum at exceptional point by a non-Hermitian interaction

摘要： 本文研究了SU(1; 1)系统哈密顿量和能量谱的厄米性。与通常认为厄米性是实谱充分条件的观念相反，厄米哈密顿量可以具有虚数本征值。对于倒势阱哈密顿量，在代数方法中使用虚频玻色算符导出了这些虚数本征值。 需要两组互为正交的本征态，分别对应复共轭本征值。任意阶本征函数表现为虚频多项式，由归一化的基态波函数生成，且这些波函数在空间上是非局域化的。 包括非厄米相互作用项的哈密顿量可以通过相似变换转换为具有降低斜率的有效势的厄米哈密顿量，这一过程可通过相互作用常数调节。变换算子不应是幺正的，而是不同于普通量子力学中的幺正变换的厄米算子。 有效势在耦合强度的一个临界值处消失，称为例外点，在这一点上所有本征态都退化为零本征值，并且从虚谱到实谱的转变出现。 SU(1; 1)生成元 $\widehat{S}_{z}$ 的本征值由算符的对易关系确定，但在虚频玻色算符实现中是非厄米的。量子哈密顿量的经典的对应物是非厄米相互作用下的正则变量的复函数。通过正则变量的变换，它变成实函数，表明了哈密顿量之间精确的一对一量子-经典对应关系。 显示英文摘要

摘要： We in this paper study the hermiticity of Hamiltonian and energy spectrum for the SU(1; 1) systems. The Hermitian Hamiltonian can possess imaginary eigenvalues in contrast with the common belief that hermiticity is a suffcient condition for real spectrum. The imaginary eigenvalues are derived in algebraic method with imaginary-frequency boson operators for the Hamiltonian of inverted potential well. Dual sets of mutually orthogonal eigenstates are required corresponding respectively to the complex conjugate eigenvalues. Arbitrary order eigenfunctions seen to be the polynomials of imaginary frequency are generated from the normalized ground-state wave functions, which are spatially non localized. The Hamiltonian including a non-Hermitian interaction term can be converted by similarity transformation to the Hermitian one with an effective potential of reduced slope, which is turnable by the interaction constant. The transformation operator should not be unitary but Hermitian different from the unitary transformation in ordinary quantum mechanics. The effective potential vanishes at a critical value of coupling strength called the exceptional point, where all eigenstates are degenerate with zero eigenvalue and transition from imaginary to real spectra appears. The SU(1; 1) generator $\widehat{S}_{z}$ with real eigenvalues determined by the commutation relation of operators, however, is non-Hermitian in the realization of imaginay-frequency boson operators. The classical counterpart of the quantum Hamiltonian with non-Hermitian interaction is a complex function of the canonical variables. It becomes by the canonical transformation of variables a real function indicating exactly the one to one quantum-classical correspondence of Hamiltonians.