Analogy with a quantum harmonic oscillator {#subsec:Analytic_oscillator}
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As seen in figure \[fig:TI\_eigenvalue\], tertiary modes are centered at the phase-space equilibria. Based on this, let us expand the Hamiltonian up to the second order both in $x$ and in $\hat{k}_x$. Specifically, we approximate the ZF velocity with a parabola: $$U\approx U_0+\frac{1}{2}\,\mathcal{C}x^2,\label{eq:AnalyticTI_oscillator_Uexpand}$$ where $U_0$ is the local ZF velocity and $\mathcal{C}\doteq U''(0)$ is the local ZF curvature. For the sinusoidal velocity (\[eq:Linear\_Tertiary\_sinusoidalU\]), this corresponds to $U_0=\pm u$ and $\mathcal{C}=\mp q_x^2 u$. We also make the approximation that $\hat{D}\approx D_0\doteq D_{\boldsymbol{k}=(0,k_y)}$ and $$\quad \hat{k}^{-2}\approx k_0^{-2}+k_0^{-4}\frac{\rm d^2}{{\rm d}x^2},\quad k_0^2\doteq 1+k_y^2-{\rm i}\delta_0 k_y.\label{eq:AnalyticTI_oscillator_Kexpand}$$ Then, the Hamiltonian operator $\hat{H}$ (\[eq:Linear\_Tertiary\_eigen\]) is approximated as $$\hat{H}\approx k_y U_0 +\frac{1}{2}k_y\mathcal{C}\hat{x}^2+k_y(\beta+\mathcal{C})\left(k_0^{-2}+k_0^{-4}\frac{\rm d^2}{{\rm d}x^2}\right)-{\rm i}D_0,\label{eq:AnalyticTI_oscillator_Hexpand}$$ and the corresponding eigenmode equation (\[eq:Linear\_Tertiary\_eigen\]) becomes $$\left(-\tau^2\frac{\rm d^2}{{\rm d}x^2}+x^2\right)w=\varepsilon w.\label{eq:AnalyticTI_oscillator}$$ It is the same equation that describes a quantum harmonic oscillator, except that here the coefficients are complex; specifically, $$\tau^2\doteq-\frac{2}{k_0^4}\left(1+\frac{\beta}{\mathcal{C}}\right),\quad\varepsilon\doteq\frac{2[\omega-k_y U_0+{\rm i}D_0-k_y(\beta+\mathcal{C})/k_0^2]}{k_y\mathcal{C}}.\label{eq:AnalyticTI_oscillator_coefficients}$$ Note that the coefficients are different at minima and maxima of $U$, as they depend on the sign of $\mathcal{C}$. Also note that for runaway modes, we have shifted the coordinate as $x\to x+\upi/q_{x}$ to recenter the ZF minimum at $x=0$.

Following the standard procedure known from quantum mechanics [@Sakurai94book], one can 