The relevance in Physics of non-Hermitian operators with real eigenvalues is being widely recognized not only in quantum mechanics but also in other areas, such as quantum optics, quantum fluid dynamics and quantum field theory. %stochastic processesand so on. In this note, a quantum system described by a non-Hermitian Hamiltonian, which is constituted by two types of interacting bosons, is investigated. The real eigenvalues of the Hamiltonian are explicitly determined, as well as complete biorthogonal sets of eigenfunctions of the Hamiltonian and its adjoint. The diagonal representation of $H$ is obtained using pseudo-bosonic operators.

中文翻译：

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具有非厄米哈密顿量的量子系统

具有实特征值的非厄米算符在物理学中的相关性不仅在量子力学中被广泛认可，而且在量子光学、量子流体动力学和量子场论等其他领域也得到了广泛认可。%随机过程等等。在本笔记中，研究了由非厄米哈密顿量描述的量子系统，该系统由两种相互作用的玻色子构成。哈密​​顿量的实特征值以及哈密顿量及其伴随的特征函数的完整双正交集是明确确定的。$H$ 的对角线表示是使用伪玻色算子获得的。