# Quantum Eigenstates of Curved and Varying Cross-Sectional Waveguides
## 2 The General Equations
### Constant Orientation  
We assume that we have a curve \(\mathbf{r}\) parametrised by arc length with tangent \(\mathbf{t}=\mathbf{r}^{\prime}\). We first choose a minimal rotating frame, i.e., an orthonormal frame \(\mathbf{t}\), \(\mathbf{p}\), \(\mathbf{q}\) (see Figure 1) along the curve satisfying,  
\[\frac{\mathrm{d}}{\mathrm{d}s} \begin{pmatrix}\mathbf{t}\\ \mathbf{p}\\ \mathbf{q}\end{pmatrix} = \begin{pmatrix}0&\kappa_{1}&\kappa_{2}\\ -\kappa_{1}&0&0\\ -\kappa_{2}&0&0\end{pmatrix}\begin{pmatrix}\mathbf{t}\\ \mathbf{p}\\ \mathbf{q}\end{pmatrix} \,. \tag{1}\]  
Then, the curvature of the curve is \(\kappa=\sqrt{\kappa_{1}^{2}+\kappa_{2}^{2}}\) and the torsion is \(\tau=\frac{\kappa_{1}\kappa_{2}^{2}-\kappa_{1}^{\prime}\kappa_{2}}{\kappa_{1} ^{2}+\kappa_{2}^{2}}\). We now consider the following parametrisation of a neighbourhood of \(\mathbf{r}\), see Figure 1:  
\[\mathbf{x}(s,v,w)=\mathbf{r}(s)+eh(s)\left(v\mathbf{p}(s)+w\mathbf{q}(s) \right),\qquad s\in[0,L]\,,\quad(v,w)\in\Omega_{0}\subseteq\mathbb{R}^{2}\,, \tag{2}\]  
where \((v,w)\) belongs to a fixed domain \(\Omega_{0}\) in the \(vw\)-plane and \(L\) is the length of the curve. If we fix \(s\) then we obtain a scaled (by \(eh(s)\)) and rotated copy of \(\Omega_{0}\), in Figure 1 it is the unit disk. The scale is allowed to vary from point to point, this is encoded in the function \(h\) and the overall scaling is given by the positive number \(\epsilon\). We will expand the Laplace operator in powers of \(\epsilon\) and as we consider \(\epsilon\) as a small number we will disregard terms in \(\epsilon\) of degree one or higher. The partial derivatives are  
\[\frac{\partial\mathbf{x}}{\partial s} =\mathbf{r}^{\prime}+\epsilon\left(h^{\prime}(v\mathbf{p}+w \mathbf{q})+h(v\mathbf{p}^{\prime}+w\mathbf{q}^{\prime})\right)\] \[=(1-eh(\kappa_{1}v+\kappa_{2}w))\mathbf{t}+\epsilon\,h^{\prime} (v\mathbf{p}+w\mathbf{q})\,, \tag{3}\] \[\frac{\partial\mathbf{x}}{\partial v} =\epsilon h\mathbf{p}\,,\] (4) \[\frac{\partial\mathbf{x}}{\partial w} =\epsilon h\mathbf{q}\,. \tag{5}\]  
We let  
\[F=1-\epsilon h\left(\kappa_{1}v+\kappa_{2}w\right), \tag{6}\]  
and find the metric tensor by taking the inner products between the partial derivatives,  
\[G_{k\ell}= \begin{pmatrix}F^{2}+(\epsilon h^{\prime})^{2}(v^{2}+w^{2})& \epsilon^{2}hh^{\prime}v&\epsilon^{2}hh^{\prime}w\\ \epsilon^{2}hh^{\prime}v&\epsilon^{2}h^{2}&0\\ \epsilon^{2}hh^{\prime}w&0&\epsilon^{2}h^{2}\end{pmatrix} \,. \tag{7}\]  
The determinant is  
\[G=\epsilon^{4}h^{4}F^{2}\,, \tag{8}\]and the inverse is  
\[G^{kl}= \begin{pmatrix}\frac{1}{F^{2}}&\frac{h^{\prime}v}{hF^{2}}&\frac{h^{\prime}w }{hF^{2}}\\ \frac{h^{\prime}v}{hF^{2}}&\frac{1}{e^{2}h^{2}}+\left(\frac{h^{\prime}v}{hF} \right)^{2}&\frac{(h^{\prime})^{2}vw}{h^{2}F^{2}}\\ \frac{h^{\prime}w}{hF^{2}}&\frac{(h^{\prime})^{2}vw}{h^{2}F^{2}}&\frac{1}{e^{ 2}h^{2}}+\left(\frac{h^{\prime}w}{hF}\right)^{2}\end{pmatrix} \,. \tag{9}\]  
We let  
\[\mathcal{R}=v\frac{\partial}{\partial v}+w\frac{\partial}{\partial w}=r\frac{ \partial}{\partial r}\,, \tag{10}\]  
where \(r=\sqrt{v^{2}+w^{2}}\).