## 8 The Right Reflection Matrix. The Spectral Asymptotics
\begin{split} d_{12}=&-\delta_{-}\frac{2\mathbb{R}} {\lambda^{2}}e(\lambda)\widehat{e}(\lambda)-\frac{\gamma}{\lambda^{2}}\sqrt{ EI\cdot\rho}\Big{(}\sqrt{EI\cdot\rho}+\alpha\sqrt{S}\Big{)}\widehat{e}^{2}( \lambda)+\\ &\quad\frac{\gamma}{\lambda^{2}}\sqrt{EI\cdot\rho}\Big{(}\sqrt{ EI\cdot\rho}-\alpha\sqrt{S}\Big{)}+O\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}=O \bigg{(}\frac{1}{\lambda^{2}}\bigg{)}.\end{split}  \tag{243}\]  
\[ \begin{split} d_{21}=&\Bigg{[}\sqrt{\frac{\rho}{K} }+\frac{\gamma}{2\lambda^{2}}\sqrt{\frac{K}{\rho}}-\frac{\Theta(\lambda)}{ \lambda EI}-\frac{2\mathbb{R}}{\lambda^{2}}\sqrt{\frac{\rho}{K}}\bigg{]}e^{2} (\lambda)-\\ &\quad 2\Bigg{[}\sqrt{\frac{S\rho}{EI}}-\frac{\gamma}{2\lambda^{2}} \sqrt{\frac{EI}{S\rho}}-\frac{\Theta(\lambda)}{\lambda EI}-\frac{\mathbb{R}}{ \lambda^{2}}\sqrt{\frac{S\rho}{EI}}\bigg{]}e(\lambda)\widehat{e}(\lambda)-\\ &\quad\bigg{(}\sqrt{\frac{\rho}{K}}+\frac{\gamma}{2\lambda^{2}} \sqrt{\frac{K}{\rho}}+\frac{\Theta(\lambda)}{\lambda EI}\bigg{)}+O\bigg{(} \frac{1}{\lambda^{3}}\bigg{)}=\Bigg{(}\sqrt{\frac{\rho}{K}}-\frac{\Theta( \lambda)}{\lambda EI}\bigg{)}e^{2}(\lambda)-\\ &\quad 2\Bigg{(}\sqrt{\frac{S\rho}{EI}}-\frac{\Theta(\lambda)}{ \lambda EI}\bigg{)}e(\lambda)\widehat{e}(\lambda)-\Bigg{(}\sqrt{\frac{\rho}{K} }+\frac{\Theta(\lambda)}{\lambda EI}\bigg{)}+O\bigg{(}\frac{1}{\lambda^{2}} \bigg{)},\end{split}  \tag{244}\]  
\[ \begin{split} d_{22}=&-\frac{2\mathbb{R}}{\lambda^{2 }}\sqrt{\frac{\rho}{K}}e(\lambda)\widehat{e}(\lambda)-\Bigg{[}\sqrt{\frac{S \rho}{EI}}-\frac{\gamma}{2\lambda^{2}}\sqrt{\frac{EI}{S\rho}}-\frac{\Theta( \lambda)}{\lambda EI}-\frac{2\mathbb{R}}{\lambda^{2}}\sqrt{\frac{S\rho}{EI}} \bigg{]}\widehat{e}^{2}(\lambda)-\\ &\quad\Bigg{[}\sqrt{\frac{S\rho}{EI}}-\frac{\gamma}{2\lambda^{2}} \sqrt{\frac{EI}{S\rho}}+\frac{\Theta(\lambda)}{\lambda EI}\bigg{]}+O\bigg{(} \frac{1}{\lambda^{3}}\bigg{)}\\ =&-\Bigg{(}\sqrt{\frac{S\rho}{EI}}-\frac{\Theta( \lambda)}{\lambda EI}\bigg{)}\widehat{e}^{2}(\lambda)-\Bigg{(}\sqrt{\frac{S \rho}{EI}}+\frac{\Theta(\lambda)}{\lambda EI}\bigg{)}+O\bigg{(}\frac{1}{\lambda^ {2}}\bigg{)}.\end{split}  \tag{245}\]  
Using these formulas, we obtain the following approximation for the determinant of Equation (241):  
\[d_{11}(\lambda)d_{22}(\lambda)-d_{12}(\lambda)d_{21}(\lambda)=d_{11}(\lambda)d_ {22}(\lambda)+O\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}.\]  
Therefore, the asymptotic form for the spectral equation becomes  
\[\Big{(}\delta_{-}e^{2}(\lambda)+\delta_{+}\Big{)}\Bigg{[}\Bigg{(}\sqrt{\frac{S \rho}{EI}}-\frac{\Theta(\lambda)}{\lambda EI}\Bigg{)}\widehat{e}^{2}(\lambda)+ \Bigg{(}\sqrt{\frac{S\rho}{EI}}+\frac{\Theta(\lambda)}{\lambda EI}\bigg{)} \Bigg{]}=O\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}, \tag{246}\]which yields the following two equations:  
\[e^{2}(\lambda)=-\frac{\delta_{+}}{\delta_{-}}+\mathcal{O}\bigg{(}\frac{1}{ \lambda^{2}}\bigg{)}=-\frac{\sqrt{\rho K}-\alpha}{\sqrt{\rho K}+\alpha}+\mathcal{ O}\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}. \tag{247}\]  
\[ \begin{split}\widehat{e}^{\,2}(\lambda)&=-\frac{ \sqrt{\frac{S\rho}{EI}}+\frac{\Theta(\lambda)}{\lambda EI}}{\sqrt{\frac{S\rho }{EI}}-\frac{\Theta(\lambda)}{\lambda EI}}+\mathcal{O}\bigg{(}\frac{1}{ \lambda^{2}}\bigg{)}=-1-\frac{2\Theta(\lambda)}{\lambda\sqrt{S\rho\cdot EI}} \bigg{[}1+\frac{\Theta(\lambda)\sqrt{EI}}{\lambda EI\sqrt{S\rho}}+\mathcal{O} \bigg{(}\frac{1}{\lambda^{2}}\bigg{)}\bigg{]}\\ &=-1-\frac{2\Theta(\lambda)}{\lambda\sqrt{S\rho\cdot EI}}+\mathcal{ O}\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}.\end{split}  \tag{248}\]  
Using Equation (210) for \(\Theta\), we get  
\[ \begin{split}\Theta(\lambda)&=\frac{\lambda\theta^ {2}}{C\Big{(}\frac{1}{CR}+i\lambda\Big{)}}=\frac{\lambda\theta^{2}}{iC\Lambda \Big{(}\frac{1}{iCR\lambda}+1\Big{)}}=\frac{-i\theta^{2}}{C}\bigg{[}1-\frac{1 }{iCR\lambda}+\mathcal{O}\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}\bigg{]}\\ &=-\frac{i\theta^{2}}{C}+\frac{\theta^{2}}{\lambda C^{2}R}+ \mathcal{O}\bigg{(}\frac{1}{\lambda^{2}}\bigg{)},\end{split}