Broyden method for inverse non-symmetric Sturm-Liouville problems

In this paper, we propose a derivative-free method for recovering symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. A class of boundary value methods obtained as an extension of Numerov’s method is the major tool for approximating the eigenvalues in each Broyden iteration step. Numerical examples demonstrate that the method is able to reduce the number of iteration steps, in particular for non-symmetric potentials, without accuracy loss.     

Boundary value method for inverse Sturm-Liouville problems

In this paper we present a method to recover symmetric and non-symmetric potential functions of inverse Sturm-Liouville problems from the knowledge of eigenvalues. The linear multistep method coupled with suitable boundary conditions known as boundary value method (BVM) is the main tool to approximate the eigenvalues in each iteration step of the used Newton method. The BVM was extended to work for Neumann-Neumann boundary conditions. Moreover, a suitable approximation for the asymptotic correction of the eigenvalues is given. Numerical results demonstrate that the method is able to give good results for both symmetric and non-symmetric potentials.     

On a class of Sturm-Liouville operators which are connected to PT symmetric problems

A prominent class of -symmetric Hamiltonians is for some nonnegative number N. The associated eigenvalue problem is defined on a contour Γ in a specific area in the complex plane (Stokes wedges), see [3, 5]. In this short note we consider the case N = 2 only. Here we elaborate the relationship between Stokes lines and Stokes wedges and well-known limit point/limit circle criteria from [6,10,11]. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Finite difference methods for half inverse Sturm-Liouville problems

We investigate finite difference solution of the Hochstadt-Lieberman problem for a Sturm-Liouville operator defined on (0, π): given the value of the potential q on (c, π), where 0 < c < π, use eigenvalues to estimate q on (0, c). Our methods use an asymptotic correction technique of Paine, de Hoog and Anderssen, and its extension to Numerov’s method for various boundary conditions. In the classical case c = π/2, Numerov’s method is found to be particularly effective. Since eigenvalue data is scarce in applications, we also examine stability problems associated with the use of the extra information on q when c < π/2, and give some suggestions for further research.     

Geometric properties of the self-adjoint Sturm-Liouville problems

In this paper, geometric properties of the self-adjoint Sturm-Liouville problems are investigated. It is proved that for linear self-adjoint Sturm-Liouville problems, the eigenfunctions correspond exactly to the projections of the curvature lines on the energy functional surface with an appropriate metric and that the eigenvalues correspond exactly to the principal curvatures (at the origin) of the same energy functional surface.     

Equivalence of inverse Sturm-Liouville problems with boundary conditions rationally dependent on the eigenparameter

Three inverse problems for a Sturm-Liouville boundary value problem −y″+qy=λy, y(0)cosα=y′(0)sinα and y′(1)=f(λ)y(1) are considered for rational f. It is shown that the Weyl m-function uniquely determines α, f, and q, and is in turn uniquely determined by either two spectra from different values of α or by the Prüfer angle. For this it is necessary to produce direct results, of independent interest, on asymptotics and oscillation.     

Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.     

On an inverse problem for the Sturm-Liouville operator with degenerate boundary conditions

We consider the spectral problem generated by the Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We derive sufficient conditions for an entire analytic function to be the characteristic determinant of this boundary value problem.     

On a local uniqueness result for the inverse Sturm-Liouville problem

A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associatedm-function near −∞. The proof given is based on relations between the classical transformation operators and them-function.     

On sampling theory associated with the resolvents of singular Sturm-Liouville problems

This paper is concerned with the sampling theory associated with resolvents of eigenvalue problems. We introduce sampling representations for integral transforms whose kernels are Green's functions of singular Sturm-Liouville problems provided that the singular points are in the limit-circle situation, extending the results obtained in the regular problems.

     

On the inverse Sturm-Liouville problem for spatially symmetric operators,III

In this note it is proved that x(·) a boundary trajectory of a Lipschitz-continuous differential inclusion ? ? F(t, x), x(0) = 0, the tangent cone to F(t, x(t)) at ?(t) that of attainable set E(t) at x(t) coincide for almost every t provided that ?F(t, x) is smooth (similar results with more stringent assumptions were obtained by H. Hermes (J. Differential Equations3 (1967), 256–270) and S. ?ojasiewicz, Jr. (Asterisque75–76 (1980), 187–197)). It is also proved that the outward normal to these cones along the trajectory is Lipschitz-continuous (in t). Moreover, using the lower, one-side, directional derivative instead of F. H. Clarke's generalised gradient, first-order necessary conditions are obtained, which can be stronger than those of Clarke (in “International Symposium on the Calculus of Variation and Optimal Control, University of Wisconsin, Madison, Wisconsin, September 1975”). The main ideas of this paper were presented in J. Hale's seminar at Brown University (March 1976).