Spectral problem for an equation of high even order (Q309438)

The author studies the following higher order eigenvalue problem in a rectangular domain: \(\{(x,y):0<x<a,0<y<b\}\). \[ (-1)^nD_x^{2n}u+(-1)^mD_y^{2m}u+ q(x,y)=\lambda u\quad q(x,y)\geq 0 \] with the boundary conditions \[ \begin{aligned} & \frac{\partial^{2i}u}{\partial x^{2i}}(0,y)=\frac{\partial^{2i}u}{\partial x^{2i}}(a,y)\\ & \frac{\partial^{2i}u}{\partial x^{2i}}(x,0)=\frac{\partial^{2i}u}{\partial x^{2i}}(x,b).\end{aligned} \] In classical spaces of differentiable functions all eigenvalues are searched. Moreover, interesting results were obtained on the distribution and asymptotics of eigenvalues and properties of eigenfunctions by using classical methods.