## 4 Non-Selfadjoint Operator Pencil Generated by the Harvester System
From Equation (66), we have  
\[\lambda^{2}\rho(x)\psi_{2}(\lambda,x)+\Big{(}K(x)\psi_{2x}(\lambda,x)\Big{)}_ {x}-\Big{(}K(x)\psi_{0}(\lambda,x)\Big{)}_{x}=0, \tag{47}\]7  
We use Equation (63) for \(f_{2}\) and Equation (64) for \(\psi_{2}\) to have  
\[\lambda^{2}\rho(x)\psi_{2}(\lambda,x)+\Big{(}K(x)\psi_{2x}(\lambda,x)\Big{)}_ {x}-\Big{(}K(x)\psi_{0}(\lambda,x)\Big{)}_{x}=0, \tag{47}\]8 \[\lambda^{2}\rho(x)\psi_{2}(\lambda,x)+\Big{(}K(x)\psi_{2x}(\lambda,x)\Big{)}_ {x}-\Big{(}K(x)\psi_{0}(\lambda,x)\Big{)}_{x}=0, \tag{47}\]9Modifying this equation, we get  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]0 \[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]1  
Finally, we rewrite the condition in Equation (51) in terms of \(f_{0}\). From Equation (46), it follows that  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]2  
Since \(\psi_{1}=i\lambda\psi_{0}\), we get  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]3  
Now, we take into account the condition in Equation (51) and rewrite Equation (68) as  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]4  
Using Equation (53), we modify this equation to the form  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]5  
Hence, our goal is to find the solution of the pencil Equation (60) satisfying the left-end boundary conditions in Equations (62) and (65) and the right-end boundary conditions in Equations (67) and (69). **Fundamental system of solutions of the pencil equation** (Equation (60)). Let us rewrite Equation (60) in the form of a polynomial with respect to \(\lambda\) ( Marcus [34])  
\[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]6 \[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]7 \[\lambda^{2}S\rho(x)\psi_{0}(\lambda,x)+\Big{(}EI(x)\psi_{0x}(\lambda,x)\Big{)} _{x}-K(x)\Big{(}\psi_{0}(\lambda,x)-\psi_{2x}\Big{)}=0. \tag{48}\]8  
This equation can be written in the form  
\[\sum_{ \begin{subarray}{c}n=0\\ m+n\leq 4\end{subarray} }^{4}\lambda^{n}w_{nm}(x)\frac{d^{m}\psi(\lambda,x)}{ dx^{m}}=0. \tag{71}\]  
Coefficients \(w_{nm}(x)\) can be easily found from Equation (70) (see Naimark [35]). Following the terminology of [PERSON] [36] and [PERSON] [37], the terms of Equation (70) that satisfy the condition \(n+m=4\) are called the leading terms of Equation (70) and the remaining terms are called the lower order terms. Let us rewrite Equation (60) keeping the leading terms at the left-hand side and moving the lower order terms to the right-hand side. We have  
\[\lambda^{4}S\big{(}a(x)\big{)}^{-1}\psi(\lambda,x)+\lambda^{2}\big{(}S+c(x) \big{)}\psi_{xx}(\lambda,x)+a(x)c(x)\psi_{xxxx}(\lambda,x)=\mathfrak{S}( \lambda,x), \tag{72}\]where  
\[\mathfrak{S}(\lambda,x)=\sum_{ \begin{subarray}{c}n=0\\ m+n\leq 3\end{subarray} }^{3}\sum_{ \begin{subarray}{c}m=0\\ m+n\leq 3\end{subarray} }^{3}\lambda^{n}w_{nm}(x)\frac{d^{m}\psi(\lambda,x)}{dx^{m}}. \tag{73}\]  
Now, we impose the following assumption on the structural parameters:  
\[SK(x)\
eq EI(x),\quad\text{for}\quad\text{all}\quad x\in[0,L]. \tag{74}\]  
The next statement is concerned with the fundamental system of solutions of Equation (70). To this end, we perform an asymptotic analysis when the spectral parameter \(\lambda\) tends to infinity. Similar analysis can be found in the works by [PERSON] [38; 39].