On the Riesz basis property of the root functions in certain regular boundary value problems

Abstract

ABSTRACT. The differential operator ly = y" + q(z)y with periodic (antiperiodic) boundary conditions that are not strongly regular is studied. It is assumed that q(z) is a complex-valued function of class C(4) [0,1] and q(0) # q(1). We prove that the system of root functions of this operator forms a Riesz basis in the space L2(0, 1).KEY WORDS: second-order differential operator, Riesz basis, periodic (antiperiodic) boundary condition.It is well known [1][2][3] that the system of root functions of an ordinary differential operator of arbitrary order with strongly regular boundary conditions forms a Riesz basis in L2. An example of a differential operator with regular boundary conditions (but not strongly regular) whose root functions do not form a basis was given in [2].It was proved in [4,5] that the system of root functions of a differential operator with not strongly regular boundary conditions forms a Riesz basis with parentheses.Consider the differential operator ly = y" -4-q(z)yeither with the periodic boundary conditions y(1)=y(0), r162or with the antiperiodic boundary conditions y(1)=-y(0), r162 (3)A. A. Shka!ikov conjectured that the root functions of the problem (i), (2) or (I), (3) form an ordinary Riesz basis rather than a Riesz basis with parentheses despite the fact that these problems are only regular and not strongly regular.In the present paper we show that for such problems the basis properties are determined in some cases by the values of the potential at the endpoints of the closed interval.The following result is valid.

On the Riesz Basisness of Systems Composed of Root Functions of Periodic Boundary Value Problems

“…In this way one can easily write a similar result for the anti-periodic problem. In addition to all the above results, we note that if either the first condition of (8) and (9), or the second condition of (8) and (9) hold then all the periodic eigenvalues are asymptotically simple. We can write a similar result for the anti-periodic problem.…”

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 nonself-adjoint ordinary differential operators with periodic boundary conditions

“…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.…”