Self-adjoint fourth order differential operators with eigenvalue parameter dependent boundary conditions

Abstract

We consider the eigenvalue problem y (4)(λ,x) − (gy′)′(λ,x) = λ 2 y(λ,x) with separated boundary conditions B j (λ)y = 0 for j = 1,…,4, where g ∈ C 1[0, a] is a real valued function, B j (λ)y = y [p j ](a j ) or B j (λ)y = y [pj](a j ) + iϵ j αλy [qj ] (aj ), aj = 0 for j = 1, 2 and a j = a for j = 3, 4, α > 0, ϵ j ∈ {−1, 1}. We will associate to the above eigenvalue problem a quadratic operator pencil L(λ) = λ 2 M − iαλK − A in the space , where and are bounded self-adjoint operators and k is the number of boundary conditions which depend on λ. We give necessary and sufficient conditions for the operator A to be self-adjoint.