## 4 Non-Selfadjoint Operator Pencil Generated by the Harvester System
\begin{split}& EI(L)\big{(}i\lambda\big{)}\bigg{(}\frac{\rho(L)}{K( L)}\bigg{)}^{1/4}\Bigg{\{}C_{1}(\lambda)e(\lambda)\Big{(}1+O\big{(}1/\lambda \big{)}\Big{)}+C_{2}(\lambda)\bigg{(}\frac{SK(L)}{EI(L)}\bigg{)}^{1/4}\widehat {e}(\lambda)\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}-\\ & D_{1}(\lambda)\big{(}e(\lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/ \lambda\big{)}\Big{)}-D_{2}(\lambda)\bigg{(}\frac{SK(L)}{EI(L)}\bigg{)}^{1/4} \big{(}\widehat{e}(\lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/\lambda\big{)} \Big{)}\Bigg{\}}-\\ &\left[\frac{EI(L)K^{\prime}(L)}{K(L)}-\frac{i\lambda\theta^{2} }{C\Big{(}\frac{1}{C\!R}+i\lambda\Big{)}}\right]\bigg{(}\frac{K(L)}{\rho(L)} \bigg{)}^{1/4}\Bigg{\{}C_{1}(\lambda)e(\lambda)\Big{(}1+O\big{(}1/\lambda \big{)}\Big{)}+\\ & C_{2}(\lambda)\bigg{(}\frac{EI(L)}{SK(L)}\bigg{)}^{1/4}\widehat {e}(\lambda)\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}+D_{1}(\lambda)\big{(}e( \lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}+\\ & D_{2}(\lambda)\bigg{(}\frac{EI(L)}{SK(L)}\bigg{)}^{1/4}\big{(} \widehat{e}(\lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)} \Bigg{\}}=0.\end{split}  \tag{135}\]  
Collecting coefficients for \(C_{1}(\lambda)e(\lambda)\), we get  
\[\left\{\big{(}i\lambda\big{)}EI(\lambda)\bigg{(}\frac{S\rho(L)}{EI(L)}\bigg{)} ^{1/4}-\bigg{(}\frac{EI(L)}{S\rho(L)}\bigg{)}^{1/4}\left[\frac{EI(L)K^{\prime} (L)}{K(L)}-\frac{i\lambda\theta^{2}}{C\Big{(}\frac{1}{C\!R}+i\lambda\Big{)}} \right]\right\}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}. \tag{136}\]  
Collecting coefficients for \(C_{2}(\lambda)\widehat{e}(\lambda)\), we get  
\[\left\{\big{(}i\lambda\big{)}EI(\lambda)\bigg{(}\frac{S\rho(L)}{EI(L)}\bigg{)} ^{1/4}-\bigg{(}\frac{EI(L)}{S\rho(L)}\bigg{)}^{1/4}\left[\frac{EI(L)K^{\prime} (L)}{K(L)}-\frac{i\lambda\theta^{2}}{C\Big{(}\frac{1}{C\!R}+i\lambda\Big{)}} \right]\right\}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}. \tag{137}\]  
Collecting coefficients for \(D_{1}(\lambda)e^{-1}(\lambda)\), we get  
\[\left\{-\big{(}i\lambda\big{)}EI(L)\bigg{(}\frac{S\rho(L)}{EI(L)}\bigg{)}^{1/4} -\bigg{(}\frac{K(L)}{S\rho(L)}\bigg{)}^{1/4}\left[\frac{EI(L)K^{\prime}(L)}{K( L)}-\frac{i\lambda\theta^{2}}{C\Big{(}\frac{1}{C\!R}+i\lambda\Big{)}} \right]\right\}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}. \tag{138}\]  
Collecting coefficients for \(D_{2}(\lambda)\widehat{e}^{-1}(\lambda)\), we get  
\[\left\{-\big{(}i\lambda\big{)}EI(\lambda)\bigg{(}\frac{S\rho(L)}{EI(L)}\bigg{)} ^{1/4}-\bigg{(}\frac{EI(L)}{S\rho(L)}\bigg{)}^{1/4}\left[\frac{EI(L)K^{\prime} (L)}{K(L)}-\frac{i\lambda\theta^{2}}{C\Big{(}\frac{1}{C\!R}+i\lambda\Big{)}} \right]\right\}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}. \tag{139}\]Collecting together Equations (136)-(139) and using Equation (118), we obtain the following asymptotic approximation for the second right end boundary condition:  
\[\left\{\left(i\lambda\right)\left(\frac{\rho(L)}{K(L)}\right)^{1/4} -\left(\frac{K(L)}{\rho(L)}\right)^{1/4}\Theta(\lambda)\right\}C_{1}(\lambda)e( \lambda)\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}+ \tag{140}\] \[= \left\{\left(i\lambda\right)\left(\frac{\rho(L)}{K(L)}\right)^{1/ 4}+\left(\frac{K(L)}{\rho(L)}\right)^{1/4}\Theta(\lambda)\right\}D_{1}(\lambda) \big{(}e(\lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}+\] \[\left\{\left(i\lambda\right)\left(\frac{S\rho(L)}{EI(L)}\right)^{1 /4}+\left(\frac{EI(L)}{S\rho(L)}\right)^{1/4}\Theta(\lambda)\right\}D_{2}( \lambda)\big{(}\widehat{e}(\lambda)\big{)}^{-1}\Big{(}1+O\big{(}1/\lambda \big{)}\Big{)}.\]  
Let  
\[A=\left(\frac{\rho(L)}{K(L)}\right)^{1/4},\qquad B=\left(\frac{S\rho(L)}{EI(L) }\right)^{1/4},\qquad F=(\rho(L)K(L))^{-1/2}. \tag{141}\]  
Using Notations (141), we rewrite the right-end boundary conditions in Equations (133) and (140) as the following system:  
\[\left(1+\alpha F\right)\Big{(}1+O\big{(}1/\lambda\big{)}\Big{)}C_ {1}e(\lambda)+C_{2}\widehat{e}(\lambda)\;\;O\big{(}1/\lambda\big{)}\] (142) \[= -\Big{(}1-\alpha F\Big{)}\Big{(}1+O\big{(}1/\lambda\big{  
From Equation (149), it follows that  
\[\left[