arXiv 2510.024942025-10-02quant-phmath-phmath.MP

关于一类含时非厄米哈密顿量

On a Class of Time-Dependent Non-Hermitian Hamiltonians

University of Jijel（吉杰尔大学）；University Center of Mila（米拉大学中心）；University of Constantine 3 – Salah Boubnider（君士坦丁三世大学 - 萨拉·布本迪尔）

F. Kecita, B. Khantoul, A. Bounames

备注：: 16 pages

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AI总结：

本研究提出一类可通过含时幺正变换转化为不含时赝厄米哈密顿量的含时非厄米哈密顿量，借助戴森映射关联厄米哈密顿量求解初始系统，简化不确定关系计算，并以含时质量粒子模型验证了方法的有效性与一致性。

摘要：

我们研究了一类含时（TD）非厄米哈密顿量$H(t)$，这类哈密顿量可通过合适的含时幺正变换$F(t)$转化为不含时的赝厄米哈密顿量$\u27a8H_{0}^{PH}$。后者又可通过相似变换与一个厄米哈密顿量$h$关联，即$h=ρ\u27a8H_{0}^{PH} ρ^{-1}$，其中$ρ$为戴森映射。因此，一旦解出厄米哈密顿量$h$对应的薛定谔方程，即可推导出初始系统的通解。借此可定义与$H(t)$相关的希尔伯特空间的合适$\tildeη(t)$内积，其中$\tildeη(t)=F^{\u2020}(t)ηF(t)$，而$η=ρ^{\u2020}ρ$为度规算符。这极大简化了这类系统相关不确定关系的计算。作为示例，我们考虑了一个具有含时质量、受特定含时复线性势作用的粒子模型。我们由此得到两个厄米哈密顿量，即标准简谐振子的哈密顿量与倒向振子的哈密顿量。对这两种情况，辅助方程均存在解，且精确解析解为以复系数厄米多项式表示的压缩态。此外，当厄米哈密顿量为简谐振子的哈密顿量时，位置-动量不确定关系为实数且大于等于$\u210f/2$，从而证实了其一致性。

英文摘要：

We study a class of time-dependent (TD) non-Hermitian Hamiltonians $H(t)$ that can be transformed into a time-independent pseudo-Hermitian Hamiltonian $\mathcal{H}_{0}^{PH}$ using a suitable TD unitary transformation $F(t)$. The latter can in turn be related to a Hermitian Hamiltonian $h$ by a similarity transformation, $h=ρ\mathcal{H}_{0}^{PH} ρ^{-1}$ where $ρ$ is the Dyson map. Accordingly, once the Schrödinger equation for the Hermitian Hamiltonian $h$ is solved, the general solution of the initial system can be deduced. This allows to define the appropriate $\tildeη(t)$-inner product for the Hilbert space associated with $H(t)$, where $\tildeη(t)=F^{\dagger}(t)ηF(t)$ and $η=ρ^{\dagger}ρ$ is the metric operator. This greatly simplifies the computation of the relevant uncertainty relations for these systems. As an example, we consider a model of a particle with a TD mass subjected to a specific TD complex linear potential. We thus obtain two Hermitian Hamiltonians, namely that of the standard harmonic oscillator and that of the inverted oscillator. For both cases, the auxiliary equation admits a solution, and the exact analytical solutions are squeezed states given in terms of the Hermite polynomials with complex coefficients. Moreover, when the Hermitian Hamiltonian is that of the harmonic oscillator, the position-momentum uncertainty relation is real and greater than or equal to $\hbar/2$, thereby confirming its consistency.

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