幂律势：从玻尔-索末菲能量到精确能量
Power-like potentials: from the Bohr-Sommerfeld energies to exact ones
AI总结：
该研究比较幂律势的玻尔-索末菲能量与精确能量，通过引入WKB修正重现精确能谱，并首次给出四次和六次振子的高精度解析解。
摘要：
对于一维幂律势 $|x|^m, m > 0$，从玻尔-索末菲量子化条件中显式提取的玻尔-索末菲能量（BSE）与精确能量进行了比较。结果表明，对于基态以及所有正宇称态，BSE 总是高于精确能量，而对于负宇称态，当 $m>2$ 时 BSE 仍高于精确能量，但当 $m < 2$ 时 BSE 低于精确能量。作为 $m$ 函数的基态 BSE 与线性（$m=1$）、四次（$m=4$）和六次（$m=6$）振子的精确能量具有相同的数量级，但相对偏差随 $m$ 增大而增大，在 $m=\infty$ 时达到值 4。对于物理上重要的情形 $m=1,4,6$，第 $100$ 个激发态的 BSE 与精确能量在 5-6 位有效数字上一致。\n 我们证明了通过引入所谓的 WKB 修正 $\gamma$（来自在精确能量处计算的高阶 WKB 项之和）来修改玻尔-索末菲量子化条件的右侧，即所谓的精确 WKB 条件，可以重现精确能量。结果表明，WKB 修正是小的有界函数，对于所有 $m \geq 1$，$|\gamma| < 1/2$；对于固定量子数，它随 $m$ 增大而缓慢增长，而在固定 $m$ 下，它随量子数增大而衰减。首次对于四次和六次振子，WKB 修正和能谱（以及本征函数）以闭合解析形式显式写出，具有高相对精度 $10^{-9 \\ -11}$（以及 $10^{-6}$）。
英文摘要：
For one-dimensional power-like potentials $|x|^m, m > 0$ the Bohr-Sommerfeld Energies (BSE) extracted explicitly from the Bohr-Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones contrary to the negative parity states where BSE remain above the exact ones for $m>2$ but they are below them for $m < 2$. The ground state BSE as the function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but relative deviation grows with $m$ reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$ for the $100$th excited state BSE coincide with exact ones in 5-6 figures. It is demonstrated that modifying the right-hand-side of the Bohr-Sommerfeld quantization condition by introducing the so-called {\it WKB correction} $γ$ (coming from the sum of higher order WKB terms taken at the exact energies) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is small, bounded function $|γ| < 1/2$ for all $m \geq 1$, it is slow growing with increase in $m$ for fixed quantum number, while it decays with quantum number growth at fixed $m$. For the first time for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are written explicitly in closed analytic form with high relative accuracy $10^{-9 \ -11}$ (and $10^{-6}$).
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