On the basis property of the root functions of some class of non-self-adjoint Sturm-Liouville operators
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RESEARCH
Open Access
On the basis property of the root functions of some class of non-self-adjoint Sturm-Liouville operators Cemile Nur* and Oktay A Veliev *
Correspondence: [email protected] Department of Mathematics, Dogus University, Acıbadem, Kadiköy, Istanbul, Turkey
Abstract We obtain the asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators with some regular boundary conditions. Using these formulas, we find sufficient conditions on the potential q such that the root functions of these operators do not form a Riesz basis. MSC: 34L05; 34L20 Keywords: asymptotic formulas; regular boundary conditions; Riesz basis
1 Introduction and preliminary facts Let T , T , T , and T be the operators generated in L [, ] by the differential expression l(y) = –y + q(x)y
()
and the following boundary conditions: y + βy = ,
y – y = ,
()
y + βy = ,
y + y = ,
()
y – y = ,
y + αy =
()
y + y = ,
y + αy = ,
()
and
respectively, where q(x) is a complex-valued summable function on [, ], β = ± and α = ±. In conditions (), (), (), and () if β = , β = –, α = , and α = –, respectively, then any λ ∈ C is an eigenvalue of infinite multiplicity. In () and () if β = – and α = – then they are periodic boundary conditions; in () and () if β = and α = then they are antiperiodic boundary conditions. These boundary conditions are regular but not strongly regular. Note that, if the boundary conditions are strongly regular, then the root functions form a Riesz basis (this result was proved independently in [, ] and []). In the case when an operator is associated © 2014 Nur and Veliev; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Nur and Veliev Boundary Value Problems 2014, 2014:57 http://www.boundaryvalueproblems.com/content/2014/1/57
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with the regular but not strongly regular boundary conditions, the root functions generally do not form even a usual basis. However, Shkalikov [, ] proved that they can be combined in pairs, so that the corresponding -dimensional subspaces form a Riesz basis of subspaces. In the regular but not strongly regular boundary conditions, periodic and antiperiodic boundary conditions are the ones more commonly studied. Therefore, let us briefly describe some historical developments related to the Riesz basis property of the root functions of the periodic and antiperiodic boundary value problems. First results were obtained by Kerimov and Mamedov []. They established that, if q ∈ C [, ],
q() = q(),
then the root functions of the operator L(q) form a Riesz basis in L [, ], where L(q) denotes the operator generated by () and the periodic boundary conditions. The first result in terms of the Fourier coefficients of the potential q wa
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