The nonlocal problem for the differential-operator equation of the even order with the involution

Abstract

In this paper, the problem with boundary nonself-adjoint conditions for a differential-operator equations of the order $2n$ with involution is studied. Spectral properties of operator of the problem is investigated.

By analogy of separation of variables the nonlocal problem for the differential-operator equation of the even order is reduced to a sequence $ \{L_{k}\}_{k=1}^{\infty}$ of operators of boundary value problems for ordinary differential equations of even order. It is established that each element $L_{k}$, of this sequence, is an isospectral perturbation of the self-adjoint operator $L_{0,k}$ of the boundary value problem for some linear differential equation of order 2n.

We construct a commutative group of transformation operators whose elements reflect the system $V(L_{0,k})$ of the eigenfunctions of the operator $L_{0,k}$ in the system $V(L_{k})$ of the eigenfunctions of the operators $L_{k}$. The eigenfunctions of the operator $L$ of the boundary value problem for a differential equation with involution are obtained as the result of the action of some specially constructed operator on eigenfunctions of the sequence of operators $L_{0,k}.$

The conditions under which the system of eigenfunctions of operator $L$ the studied problem is a Riesz basis is established.