In this work, we introduce a new realization of exactly-solvable time-dependent Hamiltonians based on the solutions of the fourth Painlev and the Ermakov equations. The latter is achieved by introducing a shape-invariant condition between an unknown quantum invariant and a set of third-order intertwining operators with time-dependent coefficients. New quantum invariants are constructed after adding a deformation term to the well-known quantum invariant of the parametric oscillator. Such a deformation depends explicitly on time through solutions of the Ermakov equation, a property that simultaneously ensures the regularity of the new time-dependent potentials at each time. The fourth Painlev equation appears after introducing an appropriate reparametrization of the spatial coordinate and the time parameter, where the parameters of the fourth Painlev equation dictate the spectral information of the quantum invariant. In this form, the eigenfunctions of the third-order ladder operators lead to several sequences of solutions to the Schrdinger equation, which are determined in terms of the solutions of the Riccati equation, Okamoto polynomials, and nonlinear bound states of the derivative nonlinear Schrdinger equation. Remarkably, it is noticed that the solutions in terms of the nonlinear bound states lead to a quantum invariant with equidistant eigenvalues, which contains both an finite-dimensional and an infinite-dimensional sequences of eigenfunctions. The resulting family of time-dependent Hamiltonians is such that, to the authors’ knowledge, have been unnoticed in the literature of stationary and nonstationary systems.

在这项工作中，我们基于第四个Painlev方程和Ermakov方程的解，介绍了完全可解的时变哈密顿量的新实现。后者是通过在未知的量子不变性和一组具有时间相关系数的三阶交织算子之间引入形状不变条件而实现的。将变形项添加到参数振荡器的众所周知的量子不变量后，即可构造新的量子不变量。这种变形通过Ermakov方程的解显式地取决于时间，该属性同时确保每次新的时间相关电位的规律性。引入空间坐标和时间参数的适当重新参数化之后，出现第四个Painlev方程，其中第四个Painlev方程的参数决定了量子不变量的光谱信息。以这种形式，三阶梯形算子的本征函数导致了薛定equation方程的多个解序列，这些序列是根据Riccati方程，冈本多项式和导数非线性薛定equation方程的非线性界态确定的。值得注意的是，注意到根据非线性束缚态的解导致具有等距特征值的量子不变量，其既包含有限维序列又包含无限维本征函数序列。由此产生的时变哈密顿算子族，就作者所知，在平稳和非平稳系统的文献中并未引起注意。