Riesz basis property of Hill operators with potentials in weighted spaces

Abstract

Consider the Hill operator L(v) = −d 2 /dx 2 +v(x) on [0, π] with Dirichlet, periodic or antiperiodic boundary conditions; then for large enough n close to n 2 there are one Dirichlet eigenvalue μ n and two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λ − n , λ + n (counted with multiplicity). We describe classes of complex potentials v(x) = 2Z V (k)e ikx in weighted spaces (defined in terms of the Fourier coefficients of v) such that the periodic (or antiperiodic) root function system of L(v) contains a Riesz basis if and only ifThe theory of self-adjoint ordinary differential operators (o.d.o.) is well-developed, and the spectral decompositions play a central role in it [24,29,26].Convergence of the spectral decompositions of non-self-adjoint o.d.o., considered on a finite interval I and subject to strictly regular boundary conditions (see [29, § 4.8]), has been understood completely in the early 1960's [27,23,17]. In this case, we not only have convergence, but the system of eigenfunctions (SEF) is a Riesz basis in L 2 (I). However, in the case of regular but not strictly regular boundary conditions-even in the case of periodic or antiperiodic boundary conditions-complete understanding appeared only in the 2000s as a result of the interaction of two lines of research.One stems from a question raised by A. Shkalikov in 1996A. Shkalikov in /1997 in the Kostyuchenko-Shkalikov seminar on Spectral Analysis at Moscow State University. Shkalikov formulated the following assertion and sketched an approach to its proof.Consider the Hill operatorwith a smooth potential q such that for some s ≥ 0 q (k) (0) = q (k) (π), 0 ≤ k ≤ s − 1,2010 Mathematics Subject Classification. Primary 47E05, 34L40, 34L10.

The matrix analysis of spectral projections for the perturbed self-adjoint operators

“…Если v = 0, то соответствующий оператор, который далее будет играть роль невозмущенного оператора, будем обозначать через L 0 bc , bc ∈ {per, ap, dir}. Исследованию спектральных свойств оператора L bc посвящено множество работ, соответствующую подробную библиографию можно найти в [29], а также в более поздней работе этих же авторов [30]. Обзор работ по базисности Рисса собственных и присоединенных функций имеется в [31], [32].…”

“…Это, в свою очередь, является основным результатом данной работы. В работе [30] также не ставился вопрос принадлежности спектральных проекторов весовым операторным пространствам, хотя в ней рассматривались весовые пространства функций.…”

Asymptotic formulas for spectral gaps and deviations of Hill and 1D Dirac operators

“…In [20], we study systematically classes of Hill potentials v such that the corresponding functionals β ± n satisfy β ± n (z) ∼ V (±2n). Lemma 33 follows from our results in [20].…”

On a new class of boundary value problems for the Sturm-Liouville operator

“…Recently, (see [26][27][28][29][30][31][32][33][34][35][36][37] and their extensive reference lists) by a number of authors, a very nice theory of the problems of type I was built.…”