one obtains $$\begin{aligned}
        \mathcal{L}_\mu \begin{pmatrix} \phi_\alpha \\ -\phi^*_\alpha \end{pmatrix} = \mu_\alpha \begin{pmatrix} \phi \\ \phi^* \end{pmatrix}. \label{eq:1storder}
    \end{aligned}$$ The same expression follows by replacement $ \alpha\rightarrow\beta $. Thus, taking the difference of double parameter derivatives, one can obtain the following zero-energy solution: $$\begin{aligned}
        \mathcal{L}_\mu \begin{pmatrix} [\mu,\phi]_{\alpha\beta} \\ -[\mu,\phi^*]_{\alpha\beta} \end{pmatrix} = 0.
    \end{aligned}$$ It must be emphasized that this solution exists even when a localized potential barrier is added, in other words, when the fundamental equation loses a translational symmetry. What we only need is two kinds of parameter derivatives. So, this solution is not a symmetry-originated zero-mode. (For a symmetry consideration, see \[app:symmetry\].)\
Some technical (but crucially important) identities are derived in \[app:idnty\]. Equation (\[eq:1storder\]) will be used again in the process of energy expansions.

Density fluctuation and phase fluctuation {#subsec:fg}
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Here we introduce notations for the linearized density fluctuations and phase fluctuations, and rewrite the Bogoliubov equation with respect to these variables. They are convenient for both calculations and physical interpretations. Through the symbols $ (u,v)=(\delta\phi,-\delta\phi^*) $, the density and phase fluctuations are expressed as $$\begin{aligned}
        \delta \rho &= \delta(\phi\phi^*)=\delta\phi\phi^*+\phi\delta\phi^*=u\phi^*-v\phi \\
        \delta S &= \delta\left( \frac{1}{2\mathrm{i}}\log\frac{\phi}{\phi^*} \right)=\frac{1}{2\mathrm{i}}\left( \frac{\delta\phi}{\phi}-\frac{\delta\phi^*}{\phi^*} \right)=\frac{1}{2\mathrm{i}}\left( \frac{u}{\phi}+\frac{v}{\phi^*} \right)
    \end{aligned}$$ Therefore, if one defines $ f 