A new solvable hyperbolic single wave potential is found by expanding the regular solution of the 1D Schrödinger equation in terms of square integrable basis. The main characteristic of the basis is in supporting an infinite tridiagonal matrix representation of the wave operator. However, the eigen-energies associated with this potential cannot be obtained using traditional procedures. Hence, a new approach (the "potential parameter" approach) has been adopted for this eigenvalue problem. For a fixed energy, the problem is solvable for a set of values of the potential parameters (the "parameter spectrum"). Subsequently, the map that associates the parameter spectrum with the energy is inverted to give the energy spectrum. The bound states wavefunction is written as a convergent series involving products of the ultra-spherical Gegenbauer polynomial in space and a new polynomial in energy, which is a special case of the "dipole polynomial" of the second kind.
Heun-Polynomial Representation of Regular-at-Infinity Solutions for the Basic SUSY Ladder of Hyperbolic Pöschl-Teller Potentials Starting from the Reflectionless Symmetric Potential Well
Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well
本文提出了一种计算任意高度或宽度的双曲型双势阱本征谱和相应本征态的解析方法，它超越了通常应用于准精确可解模型的方法。我们将与时间无关的薛定谔方程映射到Heun合流微分方程上，并利用无穷幂级数求解。该级数的系数是量化参数的多项式，其根对应于系统的本征能量。这导致了一个量化条件，使我们能够确定整个频谱，而不是个别的本征能量。这种方法被用来进行电子波包动力学的深入分析，重点是阱内隧穿和干涉诱导的量子桥在以前的出版物中报道[H. Chomet等人，New J. Phys. 21，123004（2019）]。考虑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Wigner准概率分布。我们的结果表现出良好的协议与数值计算，并允许我们解开不同的本征频率，管理相空间动力学。
