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arXiv 2402.06148quant-ph
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具有反转势阱的厄米哈密顿量的虚本征值及通过非厄米相互作用在例外点向实谱的跃迁

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

Ni Liu, Meng Luo, J. -Q. Liang

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AI总结：

本文研究 SU(1; 1) 系统中具有反转势阱的厄米哈密顿量，发现其可存在虚本征值，且通过引入非厄米相互作用可利用相似变换在例外点实现从虚谱到实谱的跃迁，揭示了厄米性与实谱之间的非充分关系及量子-经典对应。

摘要：

我们在本文中研究 SU(1; 1) 系统的哈密顿量厄米性和能谱。与厄米性是实谱的充分条件这一普遍看法相反，厄米哈密顿量可以具有虚本征值。对于反转势阱的哈密顿量，这些虚本征值是通过虚频玻色子算符的代数方法推导出来的。需要双重相互正交的本征态集合，分别对应于复共轭本征值。任意阶本征函数（被视为虚频多项式）由归一化的基态波函数生成，这些基态波函数在空间上是非局域的。包含非厄米相互作用项的哈密顿量可以通过相似变换转化为具有减小斜率的有效势的厄米哈密顿量，该斜率可由相互作用常数调节。变换算符不应是幺正的，而是厄米的，这与普通量子力学中的幺正变换不同。有效势在称为例外点的耦合强度临界值处消失，此时所有本征态简并且本征值为零，并出现从虚谱到实谱的跃迁。然而，由算符对易关系确定具有实本征值的 SU(1; 1) 生成元 $\widehat{S}_{z}$ 在虚频玻色子算符的实现中是非厄米的。具有非厄米相互作用的量子哈密顿量的经典对应物是正则变量的复函数。通过变量的正则变换，它成为实函数，准确表明了哈密顿量的一一量子-经典对应关系。

英文摘要：

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.

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