We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrdinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system’s eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet etal (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

我们提出了一种分析方法，用于计算任意高度或宽度的双曲线双势势的本征谱和相应本征态，这超出了应用于拟精确可求解模型的常规技术。我们将与时间无关的Schrdinger方程映射到Heun汇合微分方程，这可以通过使用无限次幂级数来解决。该级数的系数是量化参数中的多项式，其根对应于系统的本征能。这导致了一个量化条件，该条件使我们能够确定整个光谱而不是单个本征能。然后采用此方法对电子波包动力学进行深入分析，等人（2019新J.物理学。 21 123004）。考虑到不同宽度和峰值位置的初始波包，我们计算了自相关函数和维格纳拟概率分布。我们的结果与数值计算显示出极好的一致性，并允许我们解开控制相空间动力学的不同本征频率。