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

Abstract

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,

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

“…The lemma is proved. ⊓ ⊔ Lemma 3 For all large m, we have the following estimates (see, respectively, (19), (26) and (21), (27))…”

“…Moreover, for the other interesting results about the Riesz basis property of root functions of the periodic and anti-periodic problems, we refer in particular to [5,7,10,13] and [19,20].…”

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

“…We substitute these expressions in the equation (28) (refer to (27)) and divide out the common factors 3 , 2 , of the rows and also the common factor e w 4 of the last column of the determinant . /.…”

“…We refer to where spectral properties of boundary value problems for ordinary differential operators with regular boundary conditions (but not strongly regular) are studied.…”

“…Mityagin [16] proved a general criterion for basicity in terms of the Fourier coefficients of the potential. We refer to [23][24][25][26][27][28] where spectral properties of boundary value problems for ordinary differential operators with regular boundary conditions (but not strongly regular) are studied.…”