## 8 The Right Reflection Matrix. The Spectral Asymptotics
\begin{split} b(\lambda)=& i\lambda^{2}\alpha S \rho\bigg{[}i\lambda\sqrt{\frac{\rho}{K}}+\frac{i\sqrt{K}\gamma}{2\lambda \sqrt{\rho}}+O\bigg{(}\frac{1}{\lambda^{3}}\bigg{)}\bigg{]}+i\alpha EI\bigg{[} -i\lambda^{3}\bigg{(}\frac{\rho}{K}\bigg{)}^{3/2}-\lambda\frac{3i\sqrt{\rho} \gamma}{2\sqrt{K}}+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{]}\\ =&\frac{\lambda^{3}\alpha\rho^{3/2}}{\gamma\sqrt{K} }+\frac{\lambda\alpha\gamma}{2\sqrt{K}}\sqrt{\rho}\Big{[}3 EI-SK\Big{]}+O \bigg{(}\frac{1}{\lambda}\bigg{)}.\end{split}  \tag{224}\]  
\[ \begin{split}\widetilde{b}(\lambda)=& i\lambda^{2}\alpha S\rho\bigg{[}i\lambda\sqrt{\frac{S\rho}{EI}}- \frac{i\sqrt{EI}\gamma}{2\lambda\sqrt{EI}}+O\bigg{(}\frac{1}{\lambda^{3}} \bigg{)}\bigg{]}+i\alpha EI\bigg{[}-i\lambda^{3}\bigg{(}\frac{S\rho}{EI}\bigg{)} ^{3/2}+\lambda\frac{3i\sqrt{S\rho}\gamma}{2\sqrt{EI}}+O\bigg{(}\frac{1}{ \lambda}\bigg{)}\bigg{]}\\ =&\frac{1}{2}\alpha\gamma\sqrt{S\rho}\sqrt{EI}- \frac{3}{2}\alpha\gamma\sqrt{S\rho}\sqrt{EI}+O\bigg{(}\frac{1}{\lambda}\bigg{)} =-\alpha\gamma\sqrt{S\rho}\sqrt{EI}+O\bigg{(}\frac{1}{\lambda}\bigg{)}.\end{split}  \tag{225}\]  
**Asymptotic approximations for the entries of the matrix \(\mathbb{A}(\lambda)\) defined in Equation (215)**  
Using Equations (222) and (224), we get  
\[ \begin{split} a(\lambda)\pm b(\lambda)&=\Bigg{[}- \frac{\lambda^{3}\rho^{2}}{\gamma}\pm\frac{\lambda^{3}\alpha\rho^{3/2}}{\gamma \sqrt{K}}\Bigg{]}+\Bigg{[}-\lambda\gamma\rho EI\pm\frac{\lambda\alpha\gamma \sqrt{\rho}}{2\sqrt{K}}\Big{(}3 EI-SK\Big{)}\Bigg{]}+O\bigg{(}\frac{1}{\lambda} \bigg{)}\\ &=\lambda^{3}\delta_{\pm}+\lambda\sigma_{\pm}+O\bigg{(}\frac{1}{ \lambda}\bigg{)},\end{split}  \tag{226}\]  
where  
\[\delta_{\pm}=\frac{\rho^{3/2}}{\gamma}\Bigg{[}-\sqrt{\rho}\pm\frac{\alpha}{ \sqrt{K}}\Bigg{]}, \tag{227}\]  
\[\sigma_{\pm}=\gamma\sqrt{\rho}\Bigg{[}-\sqrt{\rho}EI\pm\frac{\alpha\big{(}3 EI- SK\big{)}}{2\sqrt{K}}\Bigg{]}. \tag{228}\]  
Using Equations (223) and (225), we get  
\[\widetilde{a}(\lambda)\pm\widetilde{b}(\lambda)=\lambda\gamma\sqrt{EI\rho} \Big{(}\sqrt{EI\rho}\mp\alpha\sqrt{S}\Big{)}+O\bigg{(}\frac{1}{\lambda}\bigg{)}. \tag{229}\]It is convenient to collect all ingredient for the spectral equation (Equation (219)). Namely,  
\[a(\lambda)\pm b(\lambda) =\Big{(}\lambda^{3}\delta_{\pm}+\lambda\sigma_{\pm}\Big{)}\bigg{(}1+ O\bigg{(}\frac{1}{\lambda^{4}}\Big{)}\bigg{)},\] (230) \[\widetilde{a}(\lambda)\pm\widetilde{b}(\lambda) =\gamma\lambda\sqrt{EI\cdot\rho}\Big{(}\sqrt{EI\cdot\rho}\mp\alpha \sqrt{S}\Big{)}\bigg{(}1+O\bigg{  
Then, the spectral equation can be written in the asymptotical form as follows:  
\[\text{det}\,\Big{[}\mathcal{A}(\lambda)\mathcal{R}_{l}(\lambda)-\mathcal{B}( \lambda)\Big{]}=O\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}. \tag{241}\]  
Let \(\mathcal{D}(\lambda)\) be the notation for \(\mathcal{A}(\lambda)\mathcal{R}_{l}(\lambda)-\mathcal{B}(\lambda)\), i.e., let \(\mathcal{D}(\lambda)=\Big{[}d_{ij}\Big{]}_{i=1,\ j=1}^{i=2,\ j=2}\). Let us evaluate the asymptotic approximation for each entry of the matrix \(\mathcal{D}(\lambda)\) with the accuracy \(\mathcal{O}\Big{(}\lambda^{-2}\Big{)}\). We have  
\[ \begin{split} d_{11}&=\bigg{(}\delta_{-}+\frac{ \sigma_{-}}{\lambda^{2}}\bigg{)}\bigg{(}1-\frac{2\mathbb{R}}{\lambda^{2}} \bigg{)}e^{2}(\lambda)-\frac{2\gamma}{\lambda^{2}}\sqrt{EI\cdot\rho}\Big{(} \sqrt{EI\cdot\rho}+\alpha\sqrt{S}\Big{)}\bigg{(}1-\frac{\mathbb{R}}{\lambda^{ 2}}\bigg{)}e(\lambda)\widehat{e}(\lambda)+\\ &\quad\bigg{(}\delta_{+}+\frac{\sigma_{+}}{\lambda^{2}}\bigg{)}= \bigg{(}\delta_{-}+\frac{\sigma_{-}}{\lambda^{2}}-\frac{2\mathbb{R}\delta_{-}} {\lambda^{2}}\bigg{)}e^{2}(\lambda)-\frac{2\gamma\sqrt{EI\cdot\rho}\Big{(} \sqrt{EI\cdot\rho}+\alpha\sqrt{S}\Big{)}}{\lambda^{2}}e(\lambda)\widehat{e}( \lambda)+\\ &\quad\bigg{(}\delta_{+}+\frac{\sigma_{+}}{\lambda^{2}}\bigg{)}+O \bigg{(}\frac{1}{\lambda^{4}}\bigg{)}=\delta_{-}e(\lambda)+\delta_{+}+O \bigg{(}\frac{1}{\lambda^{2}}\bigg{)},\end{split}  \tag{242}\]  
\[