Eigenvalues of limit-point Sturm–Liouville problems

The dependence of the eigenvalues of self-adjoint Sturm–Liouville problems on the boundary conditions when each endpoint is regular or in the limit-circle case is now, due to some surprisingly recent results, well understood. Here we study this dependence for singular problems with one endpoint in the limit-point case.     

Extremal Functions on Sturm–Liouville Hypergroups

We give an application of the theory of reproducing kernels to the Tikhonov regularization on the Sobolev spaces associated with a singular second-order differential operator. Next, we come up with some results regarding the multiplier operators for the Sturm–Liouville transform.     

Spectral corrections for Sturm–Liouville problems

The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented.     

Sturm–Liouville operators on time scales

We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection, we generalize several results for Sturm–Liouville equations on time scales, which have been obtained by various authors in the past.     

Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations:${\mathrm{\Delta }}^m{u}_i(k)+P_i(k,u_1(k),u_2(k),\dots ,u_n(k))=0,k\in \{0,1,\dots ,N\},i=1,2,\dots ,n$together with Sturm–Liouville boundary conditions${\mathrm{\Delta }}^j{u}_i(0)=0,0⩽j⩽m-3\text{,}$$\zeta {\mathrm{\Delta }}^{m-2}{u}_i(0)-\eta {\mathrm{\Delta }}^{m-1}{u}_i(0)=0,\omega {\mathrm{\Delta }}^{m-2}{u}_i(N+1)+\delta {\mathrm{\Delta }}^{m-1}{u}_i(N+1)=0\text{,}$where $m⩾2,N⩾m-1,\zeta >0,\omega >0,\eta ⩾0,\delta ⩾\omega ,\zeta \omega (N+1)+\zeta \delta +\eta \omega >0$. By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on $\{0,1,\dots ,N+m\}$. Examples are also included to illustrate the results obtained.     

Infinite periodic solutions to a class of second-order Sturm–Liouville neutral differential equations

By means of variational structure and _Z2$Z_2$-group index theory, we obtain infinite periodic solutions to a class second-order Sturm–Liouville neutral delay equations

(p(t)x^′(t−sτ))^′−q(t)x(t−sτ)+f(t,x(t),x(t−τ),x(t−2τ),…,x(t−2sτ))=0.${\left(p\left(t\right)x^{\prime }(t-s\tau )\right)}^{\prime }-q\left(t\right)x(t-s\tau )+f(t,x\left(t\right),x(t-\tau ),x(t-2\tau ),\dots ,x(t-2s\tau ))=0\text{.}$

Dissipative Sturm–Liouville Operators in Limit-Point Case

Dissipative singular Sturm–Liouville operators are studied in the Hilbert space L_w²[a,b) (–<a<b), that the extensions of a minimal symmetric operator in Weyls limit-point case. We construct a selfadjoint dilation of the dissipative operator and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix of the dilation. We also construct a functional model of the dissipative operator and define its characteristic function in terms of the Titchmarsh–Weyl function of a selfadjoint operator. Finally, in the case when the Titchmarsh–Weyl function of the selfadjoint operator is a meromorphic in complex plane, we prove theorems on completeness of the system of eigenfunctions and associated functions of the dissipative Sturm–Liouville operators. Mathematics Subject Classifications (2000) 47A20, 47A40, 47A45, 34B20, 34B44, 34L10.     

Critical higher order Sturm–Liouville difference operators

We investigate the so-called critical 2nth-order Sturm–Liouville difference operators and associated symmetric banded matrices. We show that arbitrarily small (in a certain sense) negative perturbation of a non-negative critical operator leads to an operator which is no longer non-negative.     

Singular Sturm–Liouville Operators with Distribution Potentials

This paper is devoted to the calculation of the deficiency index of a differential operator. In particular, we present sufficient conditions under which the operator with homogeneous boundary condition at zero is self-adjoint.     

Non-real eigenvalues of indefinite Sturm–Liouville problems

The present paper deals with non-real eigenvalues of regular indefinite Sturm–Liouville problems. A priori bounds and sufficient conditions of the existence for non-real eigenvalues are obtained under mild integrable conditions of coefficients.     

L p uncertainty principles on Sturm–Liouville hypergroups

We use the Fourier analysis associated to a singular second-order differential operator Δ, and prove a continuous-time principle for the L ^p theory.     

Sufficient oscillation conditions for the Sturm–Liouville equation

We use the variational method to establish criteria for the existence of conjugate points and for the oscillation property of the linear differential Sturm–Liouville equation.     

An Inverse Problem for the Sturm–Liouville Operator

Mathematical Notes - The existence of real solutions of a nonlinear equation in a neighborhood of an abnormal (degenerate) point is studied. We prove that if the mapping describing this equation...     

Decent flow methods for inverse Sturm–Liouville problem

In this paper, we propose three numerical methods for the inverse Sturm–Liouville operator in impedance form. We use a finite difference method to discretize the Sturm–Liouville operator and expand the impedance function with some basis functions. The correction technique is discussed. By solving an un-weighted least squares problem, we find an approximation to the impedance function. Numerical experiments are presented to show the accuracy and stability of the numerical methods.     

Inverse Sturm–Liouville problem on equilateral regular tree

In this article, we consider a spectral problem generated by the Sturm–Liouville equation on the edges of an equilateral regular tree. It is assumed that the Dirichlet boundary conditions are imposed at the pendant vertices and continuity and Kirchhoff's conditions at the interior vertices. The potential in the Sturm–Liouville equations, the same on each edge, is real, symmetric with respect to the middle of an edge and belongs to L ₂(0,?a) where a is the length of an edge. Conditions are obtained on a sequence of real numbers necessary and sufficient to be the spectrum of the considered spectral problem.