On the Basis Problem of the Eigenfunctions of an Ordinary Differential Operator

“…If m is an even integer, then (1) and (2) are not strongly regular 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].…”

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On the nonself-adjoint ordinary differential operators with periodic boundary conditions

Veliev

2010

Isr. J. Math.

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In this article we obtain asymptotic formulas for eigenvalues and eigenfunctions of the nonself-adjoint ordinary differential operator with periodic and antiperiodic boundary conditions, when coefficients are arbitrary summable complex-valued functions. Then using these asymptotic formulas, we obtain necessary and sufficient conditions on the coefficient for which the root functions of these operators form a Riesz basis.Let P and A be the operators generated in L 2 [0, 1] by the periodic (1) y (k) (1) = y (k) (0), k = 0, 1, 2, . . . , (m − 1) and the antiperiodic(2) y (k) (1) = −y (k) (0), k = 0, 1, 2, . . . , (m − 1) boundary conditions respectively and by the differential expressionwhere m is an even integer and p 2 , p 3 , . . . , p m are complex-valued summable functions. In this article we derive asymptotic formulas for the eigenvalues and eigenfunctions of the operators P and A. Using these asymptotic formulas, we obtain conditions on the coefficient p 2 for which the root functions of these operators form a Riesz basis in L 2 [0, 1]. Note that if m is an odd number, then the boundary conditions (1) and (2) are strongly regular and the root functions of P and A form a Riesz basis (see [4,

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“…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.…”

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

Керимов

Mamedov

1998

Math Notes

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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.

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On the Basis Problem of the Eigenfunctions of an Ordinary Differential Operator

Cited by 63 publications

References 2 publications

Bari–Markus property for Riesz projections of 1D periodic Dirac operators

Bari–Markus property for Riesz projections of 1D periodic Dirac operators

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

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

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