Bound states of a quartic and sextic inverse-powerlaw potential for all angular momenta

拟精确可解的对称四次和六次多项式振子

Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

对称化四次多项式振子允许sl（2，$\R$）代数化.一些简单的准精确可解（QES）的解决方案。引入了一个新的对称化六次多项式振子，并借助泛函Bethe方法显式地导出了它的一些精确的封闭形式解，从而证明了它是QES。这种多项式振子包括两类QES势：第一类包含众所周知的解析六次势作为子集，第二类是新的势，在这样的类中没有对应物。

四次和六次势中的宏观量子隧穿：对相位量子比特的应用

Macroscopic quantum tunneling in quartic and sextic potentials: application to a phase qubit

相位的宏观量子隧穿是超导纳米电路量子动力学中的一个基本现象。隧穿速率可以在这样的电路中被控制，其中相位的潜在景观可以用不同的外部偏置参数来调谐。为了模拟和理解实验，需要宏观量子隧穿率的精确理论知识。我们提出了一个推导，瞬子技术的基础上，一般的四次和对称的六次潜力，包括两个逃逸路径的逃逸率的解析表达式。这些新的潜力是最近实现时，创建一个噪声不敏感的相位量子比特的骆驼背电位的直流SQUID。

Radial Anharmonic Oscillator: Perturbation Theory, New Semiclassical Expansion, Approximating Eigenfunctions. II. Quartic and Sextic Anharmonicity Cases

In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variationa…

拟精确可解六次振子代数谱的渐近性和单值性

Asymptotics and monodromy of the algebraic spectrum of quasi-exactly solvable sextic oscillator

下面我们从理论上和数值上研究拟精确可解六次势的谱的代数部分的渐近性、它的水平交叉点以及它在其参数的复平面上的单值性。本文还讨论了拟精确可解六次势与经典四次势之间的联系。

对称化四次多项式振子及其部分精确解

Symmetrized quartic polynomial oscillators and their partial exact solvability

六次多项式振子可能是最著名的部分精确{ \it alias}准精确可解（QES）的量子系统，即，其在一定的耦合和能量下具有封闭形式的基元函数束缚态$ψ(x)$。相比之下，显然更简单和唯象上更重要的四次多项式振子是{ \em not\，} QES。提出了一个解决这个悖论的方法：对四次多项式势进行违反解析对称化$V(x)=A|x|+B x^2+C|x|^3+x^4$后，将一维薛定谔方程表示为QES.

Exact solutions of the sextic oscillator from the bi-confluent Heun equation

The sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Herm…