## 4 Non-Selfadjoint Operator Pencil Generated by the Harvester System
Reflection Matrices at the Ends of the Beam  
In this section, we derive asymptotic approximations for two special matrices that we call _the left-reflection matrix_ and _the right-reflection matrix_. We apply the method of reflection matrices developed in a series of our earlier papers: [PERSON] [41], [PERSON] [25; 38; 39]. This method is very efficient in completing mathematical calculations which are extremely lengthy when using direct approach. Using these matrices, we obtain two equations (see Equations (157) and (158) below) generating two branches of the eigenvalues. Since both the operator \(\mathbb{L}\) and the pencil \(\mathcal{P}(\lambda)\) have the same spectra, we will derive the aforementioned equations for the pencil. To this end, we look for a solution of Equation (70) in the form of a linear combination of the fundamental set of solutions in Equations (75)-(78)  
\[\Psi(\lambda,x)=\sum_{j=1}^{2}C_{j}(\lambda)\psi_{j}(\lambda,x)+\sum_{j=1}^{2} D_{j}(\lambda)\psi_{j+2}(\lambda,x), \tag{106}\]  
where \(C_{j}(\lambda)\), and \(D_{j}(\lambda)\), \(j=1,2\) are arbitrary functions of \(\lambda\). Note that physically, in the first sum of Equation (106), there are the Fourier representations for the terms corresponding to the waves moving from right to left and in the second sum--the Fourier representations for the terms corresponding to the waves moving from left to right (see Equations (75)-(78)). **Theorem 3**. : _For the solution \(\Psi(\lambda,x)\) of Equation (106) to satisfy the left-end boundary conditions in Equations (62) and (65), the coefficients \(C_{j}(\lambda)\), \(D_{j}(\lambda)\), \(j=1,2\), must be related by_  
\[\left[ \begin{array}{c}C_{1}(\lambda)\\ C_{2}(\lambda)\end{array} \right]=\left[ \begin{array}{cc}1+O\big{(}\lambda^{- 1}\big{)}&O\big{(}\lambda^{-1}\big{)}\\ -2\Big{(}\frac{\mathrm{S}K(0)}{EI(0)}\Big{)}^{1/4}\Big{(}1+O\big{(}\lambda^{- 1}\big{)}\Big{)}&-\Big{(}1+O\big{(}\lambda^{-1}\big{)}\Big{)}\end{array}  \right]\left[ \begin{array}{c}D_{1}(\lambda)\\ D_{2}(\lambda)\end{array} \right]. \tag{107}\]  
_The matrix at the right-hand side of Equation (107) is called the left reflection matrix as in [PERSON] and Peterson [35], [PERSON] [36; 37; 38] and denoted as \(\mathfrak{R}_{l}(\lambda)\)._  
Proof. : We look for a solution \(\Psi(\lambda,x)\), which satisfies two boundary conditions in Equations (62) and (65). Substituting Equation (106) into the boundary conditions in Equations (62) and (65) related to the end \(x=0\), we obtain two equations for four unknown coefficients \(C_{j}(\lambda)\), and \(D_{j}(\lambda)\), \(j=1,2\). It is convenient to represent the asymptotical approximations for the derivatives of the function \(\Psi(\lambda,x)\)  
\[ \begin{split}\Psi_{x}(\lambda,x)=& i\lambda\Bigg{(} \frac{\rho(x)}{K(x)}\Bigg{)}^{1/4}\Bigg{\{}C_{1}(\lambda)e^{i\lambda\eta(x)} \bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}+\\ & C_{2}(\lambda)\bigg{(}\frac{SK(x)}{EI(x)}\bigg{)}^{1/4}e^{i \lambda\bar{\eta}(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}-D_ {1}(\lambda)e^{-i\lambda\eta(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)} \bigg{)}-\\ & D_{2}(\lambda)\bigg{(}\frac{SK(x)}{EI(x)}\bigg{)}^{1/4}e^{-i \lambda\bar{\eta}(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)} \Bigg{\}},\end{split}  \tag{108}\]  
\[ \begin{split}\Psi_{xx}(\lambda,x)=-&\lambda^{2} \Bigg{(}\frac{\rho(x)}{K(x)}\Bigg{)}^{3/4}\Bigg{\{}C_{1}(\lambda)e^{i\lambda \eta(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}+\\ & C_{2}(\lambda)\bigg{(}\frac{SK(x)}{EI(x)}\bigg{)}^{3/4}e^{i \lambda\bar{\eta}(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)}+D_ {1}(\lambda)e^{-i\lambda\eta(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)} \bigg{)}+\\ & D_{2}(\lambda)\bigg{(}\frac{SK(x)}{EI(x)}\bigg{)}^{3/4}e^{-i \lambda\bar{\eta}(x)}\bigg{(}1+O\bigg{(}\frac{1}{\lambda}\bigg{)}\bigg{)} \Bigg{\}}.\end{split}  \tag{109}\]  
Taking into account the assumption in Equation (74) \(\Big{(}SK(0)\
eq EI(0)\Big{)}\), we combine Equations (111) and (113) in a system and have  
\[