On the Spectral Properties of High-Order Differential Operators with Periodic Boundary Conditions

Abstract

We study the spectral properties of the differential operator $L_0,$generated by the differential expression $l_0(y)=(-1)^{m}y^{2m}+q(x)y,$ $0 x 1,$,, and the boundary conditions$y^{(s)}(1)-y^{(s)}(0)=0$ $(s=\overline{0,2m-1}),$, where $m\in\mathbb{N},$ $q(x) $andis an arbitrary complex-valued function in the class $L_1^{+}(0,1)=\{q(x)\in L_1(0,1):\int_0^1q(t)e^{-2\pi ikt} dt=0,$ $k\le0\}.$,.