We find real spectral behaviour in negative potentials having PT-symmetry in nature. Apart from linear non-polynomial and polynomial operators, we also consider few nonlinear quantum operators. In the case of polynomial operators, we use appropriate supersymmetry Hamiltonians and verify the iso-spectral nature of energy eigenvalues. All the quantum operators are one-dimensional in nature. The present computation on negative quartic operator yields excellent agreement with the previously computed results (Bender and Boettcher in Phys Rev Lett 80:5243, 1998) and equivalent result of Jones and Mateo (Phys Rev D 73:085002, 2006).

我们发现自然界中具有PT对称性的负电位中的真实光谱行为。除了线性非多项式和多项式运算符，我们还考虑了很少的非线性量子运算符。在多项式算子的情况下，我们使用适当的超对称哈密顿量并验证能量特征值的等谱性质。所有的量子算子本质上都是一维的。当前对负四次算子的计算与先前计算的结果（Bender和Boettcher在Phys Rev Lett 80：5243，1998）以及Jones和Mateo的等效结果（Phys Rev D 73：085002，2006）方面取得了极好的一致性。