The adaptive perturbation method decomposes a Hamiltonian by the diagonal elements and nondiagonal elements of the Fock state. The diagonal elements of the Fock state are solvable but can contain the information about coupling constants. We study the harmonic oscillator with the interacting potential, [Formula: see text], where [Formula: see text] and [Formula: see text] are coupling constants, and [Formula: see text] is the position operator. In this study, each perturbed term has an exact solution. We demonstrate the accurate study of the spectrum and [Formula: see text] up to the next leading-order correction. In particular, we study a similar problem of Higgs field from the inverted mass term to demonstrate the possible nontrivial application of particle physics.

中文翻译：

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自适应摄动法的精确研究

自适应扰动方法通过福克状态的对角元素和非对角元素分解哈密顿量。Fock 状态的对角元素是可解的，但可以包含有关耦合常数的信息。我们研究具有相互作用势的谐振子，[公式：见文]，其中[公式：见文]和[公式：见文]为耦合常数，[公式：见文]为位置算子。在这项研究中，每个扰动项都有一个精确解。我们展示了对频谱和 [公式：见文本] 的准确研究，直至下一次领先修正。特别是，我们从反质量项研究了类似的希格斯场问题，以证明粒子物理学可能的重要应用。