A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Abstract

Under the assumption that V ∈ L 2 ([0, π]; dx), we derive necessary and sufficient conditions for (non-self-adjoint) Schrödinger operators −d 2 /dx 2 + V in L 2 ([0, π]; dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators −d 2 /dx 2 + V in L p ([0, π]; dx), p ∈ (1, ∞).

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

“…In this connection we also mention the recent papers by F. Gesztesy and V. Tkachenko [16,17]. In particular, in [17], as well as in the recent preprint by P. Djakov and B. Mityagin [11], the authors established a criterion for eigenfunctions and associated functions to form a Riesz basis for periodic (resp., antiperiodic) Sturm-Liouville operators on [0, 1].…”

“…In particular, in [17], as well as in the recent preprint by P. Djakov and B. Mityagin [11], the authors established a criterion for eigenfunctions and associated functions to form a Riesz basis for periodic (resp., antiperiodic) Sturm-Liouville operators on [0, 1]. The criterion is formulated directly in terms of periodic (resp., antiperiodic) and Dirichlet eigenvalues.…”

Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

“…Criterion for L p -spaces, 1 < p < ∞, given in [27,Theorem 1.4] can be essentially improved and extended as well. We take any separable rearrangement invariant function space E on [0, π] (see [39,43]) squeezed between L a and L b , 1 < a b < ∞.…”

Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday

“…In their recent paper [77], under the assumption that q ∈ L 2 (0, 1), necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators in L 2 (0, 1) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors were derived. Without entering details, we mention that this problem generated an enormous amount of interest and remained open for a long time.…”