Riesz bases generated by the spectra of Sturm-Liouville problems (original) (raw)
The Arabian Journal for Science and Engineering, Volume 33, Number 1A, January, 2008
We study basisness of root functions of Sturm–Liouville problems with a boundary condition depending quadratically on the spectral parameter. We determine the explicit form of the biorthogonal system. Using this we prove that the system of root functions, with arbitrary two functions removed, form a minimal system in L2, except some cases where this system is neither complete nor minimal. For the basisness in we prove that the part of the root space is quadratically close to systems of sines and cosines. We also consider these basis properties in the context of general L_p
On the stability of basisness in L_p(1 p +\infty) of cosines and sines
Turkish Journal of Mathematics, 2011
We study the basis properties in Lp(0, π) (1 < p < ∞) of the solution system of Sturm-Liouville equations with different types of initial conditions. We first establish some results on the stability of the basis property of cosines and sines in Lp(0, π) (1 < p < ∞) and then show that the solution system above forms a basis in Lp(0, π) if and only if certain cosine system (or sine system, depending on type of initial conditions) forms a basis in Lp(0, π).
The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function
Integral Equations and Operator Theory, 2008
We consider a regular indefinite Sturm-Liouville eigenvalue problem −f ′′ + qf = λrf on [a, b] subject to general self-adjoint boundary conditions and with a weight function r which changes its sign at finitely many, so-called turning points. We give sufficient and in some cases necessary and sufficient conditions for the Riesz basis property of this eigenvalue problem. In the case of separated boundary conditions we extend the class of weight functions r for which the Riesz basis property can be completely characterized in terms of the local behavior of r in a neighborhood of the turning points. We identify a class of non-separated boundary conditions for which, in addition to the local behavior of r in a neighborhood of the turning points, local conditions on r near the boundary are needed for the Riesz basis property. As an application, it is shown that the Riesz basis property for the periodic boundary conditions is closely related to a regular HELP-type inequality without boundary conditions.
On the stability of basisness in L p ( 1 < p < + ∞ ) of cosines and sines
2011
We study the basis properties in Lp(0, π) (1 < p < ∞) of the solution system of Sturm–Liouville equations with different types of initial conditions. We first establish some results on the stability of the basis property of cosines and sines in Lp(0, π) (1 < p < ∞) and then show that the solution system above forms a basis in Lp(0, π) if and only if certain cosine system (or sine system, depending on type of initial conditions) forms a basis in Lp(0, π) .
Hacettepe Journal of Mathematics and Statistics, 2019
The spectral problem\[-y''+q(x)y=\lambda y,\ \ \ \ 0<x<1\]\[y(0)=0, \quad y'(0)=\lambda(ay(1)+by'(1)),\]is considered, where lambda\lambdalambda is a spectral parameter, q(x)inL1(0,1)q(x)\in{{L}_{1}}(0,1)q(x)inL1(0,1) is a complex-valued function, aaa and bbb are arbitrary complex numbers which satisfy the condition ∣a∣+∣b∣ne0|a|+|b|\ne 0∣a∣+∣b∣ne0. We study the spectral properties (existence of eigenvalues, asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions in Lp(0,1){{L}_{p}}(0,1)Lp(0,1)) of the above-mentioned Sturm-Liouville problem.
On the spectrum of an irregular Sturm-Liouville problem
Doklady Mathematics, 2010
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator.
The basis property in L p of the boundary value problem rationally dependent on the eigenparameter
Studia Mathematica, 2006
We consider a Sturm-Liouville operator with boundary conditions rationally dependent on the eigenparameter. We study the basis property in L p of the system of eigenfunctions corresponding to this operator. We determine the explicit form of the biorthogonal system. Using this we establish a theorem on the minimality of the part of the system of eigenfunctions. For the basisness in L 2 we prove that the system of eigenfunctions is quadratically close to trigonometric systems. For the basisness in L p we use F. Riesz's theorem.
Boundary Value Problems, Integral Equations and Related Problems, 2010
We consider the spectral problem generated by the Sturm-Liouville equation with an arbitrary complex-valued potential q(x) ∈ L 2 (0, π) and irregular boundary conditions. We establish necessary and sufficient conditions for a set of complex numbers to be the spectrum of such an operator. In the present paper, we consider the eigenvalue problem for the Sturm-Liouvulle equation u ′′ − q(x)u + λu = 0 (1) on the interval (0, π) with the boundary conditions u ′ (0) + (−1) θ u ′ (π) + bu(π) = 0, u(0) + (−1) θ+1 u(π) = 0, (2) where b is a complex number, θ = 0, 1, and the function q(x) is an arbitrary complex-valued function of the class L 2 (0, π). Denote by c(x, µ), s(x, µ) (λ = µ 2) the fundamental system of solutions to (1) with the initial conditions c(0, µ) = s ′ (0, µ) = 1, c ′ (0, µ) = s(0, µ) = 0. The following identity is well known c(x, µ)s ′ (x, µ) − c ′ (x, µ)s(x, µ) = 1. (3) Simple calculations show that the characteristic equation of (1), (2) can be reduced to the form ∆(µ) = 0, where