# Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides
## 2 The General Equations
### Variable Orientation  
We now allow the orientation of the cross section to vary relative to the minimal rotating frame. More precisely, consider a general frame \(\mathbf{t},\mathbf{n}_{1},\mathbf{n}_{2}\) where  
\[\frac{\mathrm{d}}{\mathrm{d}s}\begin{pmatrix}\mathbf{t}\\ \mathbf{n}_{1}\\ \mathbf{n}_{2}\end{pmatrix}=\begin{pmatrix}0&\kappa_{1}&\kappa_{2}\\ -\kappa_{1}&0&\omega\\ -\kappa_{2}&-\omega&0\end{pmatrix}\begin{pmatrix}\mathbf{t}\\ \mathbf{n}_{1}\\ \mathbf{n}_{2}\end{pmatrix}\,. \tag{16}\]  
We then have  
\[\mathbf{n}_{1}=\cos\vartheta\,\mathbf{p}+\sin\vartheta\,\mathbf{q}\,, \tag{17}\]where \(\omega=\vartheta^{\prime}\). By following steps similar to the analysis in Refs. [15; 19], the resulting 1D equation, accounting for a varying thickness and curvature effects, becomes,  
\[-\left(\frac{\mathrm{d}}{\mathrm{d}s}-\frac{h^{\prime}}{h}\langle \mathcal{R}\rangle-i\omega\langle\mathcal{L}\rangle\right)^{2}\phi\\ -\left(\left(\frac{h^{\prime}}{h}\right)^{2}\left(\langle \mathcal{R}^{2}\rangle-\langle\mathcal{R}\rangle^{2}\right)-\omega^{2}\left( \langle\mathcal{L}^{2}\rangle-\langle\mathcal{L}\rangle^{2}\right)+\frac{ \kappa^{2}}{4}-\frac{\lambda_{0}}{\epsilon^{2}h^{2}}\right)\phi=\lambda\phi\,, \tag{18}\]  
where \(\mathcal{L}\) is the \(\theta\)-component of the orbital angular moment operator, i.e.,  
\[\mathcal{L}=-i\left(v\frac{\partial}{\partial w}-w\frac{\partial}{\partial v} \right)=-i\frac{\partial}{\partial\theta}\,, \tag{19}\]  
and \(\theta\) is the polar angle.  
Let us perform a gauge transformation, \(\phi\to\varphi\),  
\[\phi(s)=e^{\mathcal{S}(s)}\varphi(s)\,,\quad\text{where}\quad g(s)=\langle \mathcal{R}\rangle\log h(s)+i\langle\mathcal{L}\rangle\vartheta(s)\,. \tag{20}\]  
Then  
\[\frac{\mathrm{d}}{\mathrm{d}s}-\frac{h^{\prime}}{h}\langle\mathcal{R}\rangle-i \omega\langle\mathcal{L}\rangle=\frac{\mathrm{d}}{\mathrm{d}s}-g^{\prime}\,,\]  
and  
\[\left(\frac{\mathrm{d}}{\mathrm{d}s}-\frac{h^{\prime}}{h}\langle \mathcal{R}\rangle-i\omega\langle\mathcal{L}\rangle\right)\phi=\left(\frac{ \mathrm{d}}{\mathrm{d}s}-g^{\prime}\right)e^{\mathcal{S}(s)}\varphi(s)\\ =g^{\prime}(s)e^{\mathcal{S}(s)}\varphi(s)+e^{\mathcal{S}(s)} \varphi^{\prime}(s)-g^{\prime}(s)e^{\mathcal{S}(s)}\varphi(s)=e^{\mathcal{S}(s )}\varphi^{\prime}(s)\,.\]  
Hence, Equation (18) is equivalent to  
\[-\frac{\mathrm{d}^{2}\varphi}{\mathrm{d}s^{2}}-\left(\left(\frac{h^{\prime}}{ h}\right)^{2}\left(\langle\mathcal{R}^{2}\rangle-\langle\mathcal{R}\rangle^{2} \right)-\omega^{2}\left(\langle\mathcal{L}^{2}\rangle-\langle\mathcal{L} \rangle^{2}\right)+\frac{\kappa^{2}}{4}-\frac{\lambda_{0}}{\epsilon^{2}h^{2}} \right)\varphi=\lambda\varphi\,. \tag{21}\]  
A similar transformation was used in Ref. [15] in the case without a thickness variation.  
We see that all geometric effects: curvature, thickness variation, and frame rotation appear in the form of a geometric potential only. It is hard to say something general about the significance, except that the last term \(\lambda_{0}/(\epsilon h)^{2}\) dominates when \(\epsilon\to 0\) and unless \(h\) is constant the eigenstates will concentrate at the maxima of \(h\). In Appendix B, we examine examples where either the first or the last term is most important.