## 8 The Right Reflection Matrix. The Spectral Asymptotics
If \(\Re\alpha\geq 0\), then the \(\alpha\)-branch is asymptotically close to a horizontal line in the upper half-plane and the second branch is asymptotically close to the real axis._  
_(3) The following asymptotic approximations are valid for the spectral branches:_  
\[\lambda_{n}^{\alpha} =\frac{\pi n}{\rho L/K}+\frac{i}{2\rho L/K}\ln\frac{\alpha+\sqrt{ \rho K}}{\alpha-\sqrt{\rho K}}+O\bigg{(}\frac{1}{n^{2}}\bigg{)},\qquad n \rightarrow\infty,\qquad\lambda_{-n}^{\alpha}=-\overline{\lambda_{n}^{ \alpha}}, \tag{217}\] \[\lambda_{m}^{\theta} =\frac{\pi\big{(}m+1/2\big{)}}{S\rho L/EI}-\frac{\theta^{2}\sqrt {EI/S\rho}}{\pi mCL}+O\bigg{(}\frac{1}{m^{2}}\bigg{)},\qquad m\rightarrow\infty,\qquad\lambda_{-m}^{\theta}=-\overline{\lambda_{m}^{\theta}}; \tag{218}\]  
Proof. : It is convenient to introduce a new structural parameter  
\[\gamma=\frac{1}{EI/K-S}. \tag{219}\]  
**Asymptotic approximations for the functions \(a(\lambda)\), \(\widetilde{a}(\lambda)\), \(b(\lambda)\), and \(\widetilde{b}(\lambda)\). ** Using the definition in Equation (177) and Equations (175) and (190), we obtain the following formulas as \(|\lambda|\rightarrow\infty\):  
\[ \begin{split}(i)\qquad\Lambda(\lambda)&=-\lambda^{2 }\ \frac{\rho}{K}-\frac{1}{EI/K-S}+O\bigg{(}\frac{1}{\lambda^{2}}\bigg{)}=- \lambda^{2}\ \frac{\rho}{K}-\gamma+O\bigg{(}\frac{1}{\lambda^{2}}\bigg{)},\\ (ii)\qquad\sqrt{\Lambda(\lambda)}&=i\lambda\sqrt{ \frac{\rho}{K}}\ \left[1+\frac{1}{2\lambda^{2}\big{(}\rho/K\big{)}\Big{(}EI/K-S \Big{)}}+O\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}\right]\\ &=i\lambda\sqrt{\frac{\rho}{K}}+\frac{i\sqrt{K}\gamma}{2\lambda \sqrt{\rho}}+O\bigg{(}\frac{1}{\lambda^{3}}\bigg{)},\\ (iii)\qquad\big{(}\Lambda(\lambda)\big{)}^{3/2}&=-i \lambda^{3}\bigg{(}\frac{\rho}{K}\bigg{)}^{3/2}\ \left[1+\frac{1}{\lambda^{2}\big{(}\rho/K\big{)}\Big{(}EI/K-S \Big{)}}+O\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}\right]^{3/2}\\ &=-i\lambda^{3}\bigg{(}\frac{\rho}{K}\bigg{)}^{3/2}-\lambda\ \frac{3i\sqrt{\rho}\gamma}{2\sqrt{K}}+O\bigg{(}\frac{1}{\lambda}\bigg{)}. \end{split}  \tag{220}\]  
Using the definition in Equation (178), and Equations (176) and (191), we obtain:  
\[ \begin{split}(i)\qquad\widetilde{\Lambda}(\lambda)&= -\lambda^{2}\frac{S\rho}{EI}+\frac{1}{EI/K-S}+O\bigg{(}\frac{1}{\lambda^{2}} \bigg{)}=-\lambda^{2}\frac{S\rho}{EI}+\gamma+O\bigg{(}\frac{1}{\lambda^{2}} \bigg{)},\\ (ii)\qquad\sqrt{\widetilde{\Lambda}(\lambda)}&=i \lambda\sqrt{\frac{S\rho}{EI}}\ \left[1-\frac{EI}{2S\rho\Big{(}EI/K-S \Big{)}\lambda^{2}}+O\bigg{(}\frac{1}{\lambda^{4}}\bigg{)}\right]\\ &=i\lambda\sqrt{\frac{S\rho}{EI}}-\frac{i\sqrt{EI}\gamma}{2 \lambda\sqrt{S\rho}}+O\bigg{(}\frac{1}{\lambda^{3}}\bigg{)},\end{split}  \tag{221}\]\[(iii)\qquad\Big{(}\widetilde{\Lambda}(\lambda)\Big{)}^{3/2} =-i\lambda^{3}\bigg{(}\frac{S\rho}{EI}\bigg{)}^{3/2}\ \left[1-\frac{3 EI}{2S\rho\Big{(}EI/K-S\Big{)}\lambda^{2}}+O\bigg{(}\frac{1}{ \lambda^{4}}\bigg{)}\right]\] \[=-i\lambda^{3}\bigg{(}\frac{S\rho}{EI}\bigg{)}^{3/2}+\lambda\ \frac{3i\sqrt{S\rho}\gamma}{2\sqrt{EI}}+O\bigg{(}\frac{1}{\lambda}\bigg{)}.\]  
The following asymptotic approximations hold for the functions \(a(\lambda)\), \(\widetilde{a}(\lambda)\), \(b(\lambda)\), \(\widetilde{b}(\lambda)\). Using Equations (220)(i) and (221)(i), we obtain from Equation (208)  
\[a(\lambda)=\lambda^{3}\rho^{2}\bigg{(}S-\frac{EI}{K}\bigg{)}-\lambda\gamma\rho EI +O\bigg{(}\frac{1}{\lambda}\bigg{)}=-\frac{\lambda^{3}\rho^{2}}{\gamma}- \lambda\gamma\rho EI+O\bigg{(}\frac{1}{\lambda}\bigg{)}. \tag{222}\]  
\[\widetilde{a}(\lambda)=\lambda^{3}S\rho^{2}+\lambda\rho EI\bigg{[}-\lambda^{ 2}\frac{S\rho}{EI}+\gamma+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{]}=\lambda \gamma\rho EI+O\bigg{(}\frac{1}{\lambda}\bigg{)}. \tag{223}\]  
Using Equations (220)(ii), (220)(iii), (221)(ii), and (221)(iii), we obtain from Equation (208)  
\[