Convergence of expansions in the root functions of periodic boundary value problems

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

“…Thus (38) {λ n,1,ε , λ n,2,ε } ⊂ U n,1 ∪ U n,2 , U n,1 ∩ U n,2 = ∅, ∀n ≥ N 0 , ∀ε ∈ [0, 1], where λ n,j,ε is the eigenvalue of P ε satisfying (5). First we prove that if λ n,j,0 lies in U n,1 , then the normalized eigenfunctions ϕ * n,j (x) of S * corresponding to the eigenvalue λ n,j,0 satisfies…”

“…Now let us consider the operator A generated by the antiperiodic boundary conditions (2). Instead of (3), (5) and (8), we use (4), (6) where Φ n,j (x) is a normalized eigenfunction of A corresponding to the eigenvalue µ n,j ; and instead of (36) we take a family of operators A ε = B + ε(A − B), 0 ≤ ε ≤ 1, where B is the operator generated by the differential expression y (m) + (p 2,0 + p 2,2n+1 e i2π(2n+1)x + p 2,−2n−1 e −i2π(2n+1)x )y (m −2) and boundary conditions (2). Arguing as in the proof of Theorem 2, we get Theorem 3: Let µ n,1 , µ n,2 be the eigenvalues of A satisfying (6).…”

“…Therefore, in general, the root functions of P and A do not form a Riesz basis; they form a basis with bracket (see [8,9]). In this paper we prove that if m is an even integer and The case m = 2 is investigated in [1,3,5]. In this paper we consider the more complicated case m = 2k > 2.…”

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

“…because otherwise n would be an intermediate vertex with necessity, which is not possible for admissible walks by (12). Set…”

“…For certain classes of potentials, there have been given sufficient and necessary conditions on whether blocks could be split into (one-dimensional) eigenfunction decompositions [2,13,14,26]. Maybe, in 2006 Makin [12] and the authors [3,Theorem 71] gave first examples of such potentials that SEAF for periodic or antiperiodic boundary conditions is NOT a basis in L 2 ([0, π]) even though all but finitely many eigenvalues are simple. The existence of such potentials indirectly follows from the recent results in [8] as well.…”