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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.

which contradicts condition (1.3). Therefore, λ * ∈ R. The entire function occurring on the left-hand side in Eq. (2.1) does not vanish for nonreal λ. Consequently, it does not vanish identically. Therefore, its zeros form an at most countable set without finite limit points.

We consider the following boundary value problem with a spectral parameter in the equation and the boundary conditions:-u" + q(z)u = AZu, O

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