Estimates of Eigenfunctions and Localization of the Spectrum of Differential Operators

We consider a differential equation with parameter λ, (equation presented) where λ ∈ G = {λ ∈ ℂ: |λ| ≥ 1}, q_k(x) ∈ L₁(- 1, 1), and k = 0, 2n - 2. Let T = {y_λ(x)}_{λ ∈ G} be some set of solutions of (1). We show that the inequalities (equation presented) (2) can hold only if λ ∈ H. Here and below we denote by c(q) different positive constants, depending only on L₁ norms ∥q_k∥L₁(-1, 1), k = 0, 2n - 2, of coefficients of (1), (equation presented) In addition, if (2) holds, then for any λ ∈ H, |λ| ≫ c(q), there is an integer s such that λ = |sπ|²ⁿ[1 + O(1/|sπ|)]. Here the constant in O(·) depends only on L₁ norms of coefficients of (1), and the exponent of (1 + |Rc λ|) in the definition of H cannot be lessened. As an example, we study the problem of localization of the spectrum of a class of semiregular boundary value problems.