Resolvent growth and Birkhoff-regularity

Abstract

We prove a long standing conjecture in the theory of two-point boundary value problems that unconditional basisness implies Birkhoff-regularity. It is a corollary of our two main results: minimal resolvent growth along a sequence of points implies nonvanishing of a regularity determinant, and sparseness of nthorder roots of eigenvalues in small sectors provided that eigen and associated functions of the boundary value problem form an unconditional basis.Considerations are based on a new direct method, exploiting almost orthogonality of Birkhoff's solutions of the equation l(y) = λy. This property was discovered earlier by the author.

A criterion for Hill operators to be spectral operators of scalar type

“…A criterion for the latter system to be a Riesz basis in L 2 ([−π, π]) was found by Pavlov [49]. Most recently, Minkin [46] made essential progress in studying the same property for systems of eigenfunctions of two-point boundary problems for higher-order differential operators, and Makin [36] found sufficient conditions for the root system of a Schrödinger operator on the interval [0, 1] associated with periodic and antiperiodic boundary conditions to be (or not to be) a Riesz basis. Using well-known asymptotic formulas (see [38]) for the periodic/antiperiodic and Dirichlet spectra of the corresponding operator H restricted to the interval [0, 1], it is easy to check that under the assumptions made on V in [38], the conditions of Theorem 3.6 are either satisfied (or not satisfied), with the corresponding conclusions about the system of eigenfunctions and its property of forming a Riesz basis.…”

“…In Section 8 we will further discuss results from [17], [44], [46], [72]- [76], [79] on spectral expansions associated with a non-selfadjoint Hill operator H.…”

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

“…(18) has exactly two roots in D, counted with multiplicities. So, we have proved for n > N 1 that λ = n 2 +z, z ∈ D, is a periodic or antiperiodic eigenvalue of algebraic multiplicity 2 if and only if z is a double root of (18). Thus, the number λ = n 2 + z, z ∈ D, is a periodic or antiperiodic eigenvalue of algebraic multiplicity 2 if and only if z satisfies the system of Eq.…”

“…By (18), we have |z * n − a(n, z * n | = |B + (n, z * n )B − (n, z * n )| 1/2 . Therefore, by (24) and (25), Eq.…”

“…If the boundary conditions are strictly regular then the system of eigenfunctions and associated functions (SEAF) is a Riesz basis in L 2 ([0, π]) as it has been shown in [6,7,9,16]; see more details and history in [17,18]. However, Per + , Per − are regular but not strictly regular boundary conditions.…”