We consider the following question: given A ∈ SL(2,R), which potentials q for the second order Sturm-Liouville problem have A as its Floquet multiplier? More precisely, define the monodromy map μ taking a potential q ∈ L2([0, 2π]) to μ(q) = Φ̃(2π), the lift to the universal cover G = ̃ SL(2,R) of SL(2,R) of the fundamental matrix map Φ : [0, 2π] → SL(2,R), Φ(0) = I, Φ(t) = (

In this text we describe the spectral nature (pure point or continuous) of a self-similar Sturm-Liouville operator on the line or the half-line. This is motivated by the more general problem of understanding the spectrum of Laplace operators on unbounded finitely ramified self-similar sets. In this context, this furnishes the first example of a description of the spectral nature of the operator...

This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.

A new and fairly elementary proof is given of the result by B. Simon [Sim99], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated m-function near −∞. The proof given is based on relations between the classical transformation operators and the m-function.

In this paper we present a new method for proving the existence of an eigenvalue below the essential spectrum for Sturm-Liouville operators with coefficients which are L1 perturbations of periodic functions. This method combines operator theory and “standard” numerical analysis with interval arithmetic analysis. We illustrate the method by showing that the SturmLiouville problem (SLP) consistin...

The research is financed by the Natural Science Foundation of China and the National Natural Science Foundation of Neimongo. No. 10661008 and 200711020102 (Sponsoring information) Abstract In this paper, we consider an indefinite Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions. In an appropriate space K, we define a new self-adjoint operato...

In this work we study the different type of regular boundary value problems on a path associated with the Schrödinger operator. In particular, we obtain the Green function for each problem and we emphasize the case of Sturm-Liouville boundary conditions. In any case, the Green function is given in terms of second kind Chebyshev polynomials since they verify a recurrence law similar to the one v...

We present a new oscillation criterion to determine whether the number of eigenvalues below the essential spectrum of a given Jacobi operator is finite or not. As a special case we obtain a generalization of Kenser’s criterion for Sturm-Liouville operators to Jacobi operators.

In the paper, we study problem of recovering Sturm--Liouville operator with frozen argument from its spectrum and additional data. For this inverse problem, establish a substantial property uniform stability, which consists in that potential depends Lipschitz continuously on input