Asymptotics of eigenvalues of differential operator with alternating weight function (Q1795325)

The author considers a boundary-value problem which consists of the sixth-order differential equations \[ \begin{alignedat}{2} y_{1}^{\left( 6\right) }\left( x\right) +q_{1}\left( x\right) y_{1}\left( x\right) &=\lambda a^{6}y_{1}\left( x\right), \quad &0\leq {x}<{x_{1}},\;&a>0 , \\ y_{2}^{\left( 6\right) }\left( x\right) +q_{2}\left( x\right) y_{2}\left( x\right) &=-\lambda b^{6}y_{2}\left( x\right), \quad &{x_{1}}< {x}\leq {\pi},\;&b>0 , \end{alignedat} \] with the gluing conditions at the point \(x_{1}\) \[ y_{1}\left( x_{1}-0\right) =y_{2}\left( x_{1}+0\right) ,~~\quad y_{1}^{\left( m\right) }\left( x_{1}-0\right) =y_{2}^{\left( m\right) }\left( x_{1}+0\right) ,\;m=1,2,\dots,5, \] and the boundary conditions \ \[ y_{1}^{\left( m_{1}\right) }\left( 0\right) =y_{1}^{\left( m_{2}\right) }\left( 0\right) =\dots =y_{1}^{\left( m_{5}\right) }\left( 0\right) =y_{2}^{\left( n_{1}\right) }\left( \pi\right) =0, \] \[ {m_{1}}<{m_{2}}<\dots <{m_{5}} ,\;m_{k},n_{1}\in\left\{ 0,1,2,\dots,5\right\} ,\;k=1,2,\dots,5, \] where the potential \(q\left( x\right) \) is a piecewise smooth function in \(\left[ 0,\pi\right] \). The author obtains asymptotic formulas for the eigenvalues and the eigenfunctions of the problem.