On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville operators

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

“…Recently, i.e., in the 2000s, many authors [36,[44][45][46]40,47,50,67,26,49,51] focused on the problem of convergence of eigenfunction (or more generally root function) decompositions in the case of regular but not strictly regular bc.…”

On the Basis Property of the Root Functions of Sturm–Liouville Operators with General Regular Boundary Conditions

“…Hence the system of the root functions of T 1 1 (q) does not form a Riesz basis (see [20]). Note that (b) follows also from (a) and Theorem 2 of [12,13].…”

“…Case (c) The cases β 2 , β 4 = 0 and β 1 , β 3 = (−1) σ are investigated in [12,13] and [19]. We call the boundary conditions (7) and (9) for β 2 , β 4 = 0 and β 1 , β 3 = (−1) σ which are different from the special cases (a) , (b) and (c) as general regular boundary conditions that are not strongly regular.…”

“…Theorem 5 (a) If (12) holds, then the large eigenvalues of T 1 2 (q) are simple and the square roots (with nonnegative real part) of these eigenvalues consist of two sequences {ρ n,1 } and {ρ n,2 } satisfying…”

Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions

“…It is proved in that the system of root functions of the differential operator forms an unconditional basis of the space L 2 (0,1), where q ( x ) ∈ L 1 (0,1) is an arbitrary complex‐valued function, γ is an arbitrary nonzero complex constant and σ = 0,1. Under the condition γ = 0 (periodic and antiperiodic boundary conditions) in and , necessary and sufficient conditions of unconditional basicity in L 2 (0,1) of the system of root functions of the earlier differential operator were obtained in terms of the Fourier coefficients of the potential q (x) (see also ).…”