In this paper, we focus on high-order approximate solutions to two-level systems with quasi-resonant control. Firstly, we develop a high-order renormalization group (RG) method for Schrödinger equations. By this method, we get the high-order RG approximate solution in both resonance case and out of resonance case directly. Secondly, we introduce a time transformation to avoid the invalid expansion and get the high-order RG approximate solution in near resonance case. Finally, some numerical simulations are presented to illustrate the effectiveness of our RG method. We aim to provide a mathematically rigorous framework for mathematicians and physicists to analyze the high-order approximate solutions of quasi-resonant control problems.

中文翻译：

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具有准谐振控制的两级系统的高阶逼近

在本文中，我们将重点放在具有准谐振控制的两级系统的高阶近似解上。首先，我们为Schrödinger方程开发了一种高阶重整化组（RG）方法。通过这种方法，我们可以直接获得共振情况下和共振情况外的高阶RG近似解。其次，我们引入时间变换来避免无效扩展，并在接近共振的情况下得到高阶RG近似解。最后，通过一些数值模拟来说明RG方法的有效性。我们旨在为数学家和物理学家提供数学上严格的框架，以分析拟谐振控制问题的高阶近似解。