论文标题
关于一类非分解振荡器的经典和量子动力学
On the classical and quantum dynamics of a class of nonpolynomial oscillators
论文作者
论文摘要
我们考虑了两个一维非线性振荡器，即（i）Higgs振荡器和（ii）$ k $依赖性的非物质理性潜力，其中$ k $是riemannian歧管的持续曲率。这两个系统都是依赖位置的质量形式的系统，$ {\ displayStyle m（x）= \ frac {1} {（1 + k x^2）^2}}} $，属于二次li $ \ actute $ \ actute {e} $ nard类型非线性振荡器。他们承认在古典层面上的各种动作。在求解系统的量子版本的同时，我们考虑了依赖于大量哈密顿的广义位置，其中质量项的排序参数被视为任意。我们观察到，HIGGS振荡器的量子版本在适当的订购参数限制下是可以解决的，而第二个非线性系统则显示为准确解决的准质量可解决的方法，其中使用bethe ansatz方法，其中订购参数的任意性也起着quasi-polynomial lightomial solotister的重要作用。我们将研究扩展到这些非线性振荡器的三维概括，并获得三维HIGGS振荡器的经典和量子版本的精确解。 $ k $依赖性非分解电位的量子对应物的三维概括被认为是准确解决的。
We consider two one dimensional nonlinear oscillators, namely (i) Higgs oscillator and (ii) a $k$-dependent nonpolynomial rational potential, where $k$ is the constant curvature of a Riemannian manifold. Both the systems are of position dependent mass form, ${\displaystyle m(x) = \frac{1}{(1 + k x^2)^2}}$, belonging to the quadratic Li$\acute{e}$nard type nonlinear oscillators. They admit different kinds of motions at the classical level. While solving the quantum versions of the systems, we consider a generalized position dependent mass Hamiltonian in which the ordering parameters of the mass term are treated as arbitrary. We observe that the quantum version of the Higgs oscillator is exactly solvable under appropriate restrictions of the ordering parameters, while the second nonlinear system is shown to be quasi exactly solvable using the Bethe ansatz method in which the arbitrariness of ordering parameters also plays an important role to obtain quasi-polynomial solutions. We extend the study to three dimensional generalizations of these nonlinear oscillators and obtain the exact solutions for the classical and quantum versions of the three dimensional Higgs oscillator. The three dimensional generalization of the quantum counterpart of the $k$-dependent nonpolynomial potential is found out to be quasi exactly solvable.
