问题
将微分方程
\begin{align} \label{eq:inhomo} \mathcal{L} \psi(x) = k^2 \psi(x) \end{align}的通解 \(\phi(k, x)\) 在 \(k = 0\) 处级数展开
\begin{align} \phi(k, x) = u_0 + k^2 u_1 + \mathcal{O}(k^4) \end{align}将展开的结果代回原方程可得
\begin{align} \mathcal{L} u_0 =& 0 \\ \label{eq:step} \mathcal{L} u_1 =& u_0 \end{align}那么 \(u_{0}\) 是否是方程
\begin{align} \label{eq:homo} \mathcal{L} \psi(x) = 0 \end{align}的通解?
看起来是
解方程 \((\ref{eq:homo})\) 得到通解, 然后再解 \((\ref{eq:step})\) , 逐阶求解, 就得到 了 \((\ref{eq:inhomo})\) 的通解.
Reference
Yu, Z., Thywissen, J. H. & Zhang, S. Supplementary Material: Universal Relations for a Fermi Gas Close to a p-wave Interaction Resonance.
旧
将微分方程
\begin{align} \mathcal{L} \psi(x) = k^2 \psi(x) \end{align}的通解 \(\phi(k, x)\) , 取 \(k\to 0\) 时的极限, 能否得
\begin{align} \mathcal{L} \psi(x) = \psi(x) \end{align}的全部解?