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This paper further improves the Lie group method with Magnus expansion proposed in a previous paper by the authors, to solve some types of direct singular Sturm&ndash, Liouville problems. Next, a concrete implementation to the inverse Sturm&ndash, Liouville problem algorithm proposed by Barcilon (1974) is provided. Furthermore, computational feasibility and applicability of this algorithm to solve inverse Sturm&ndash, Liouville problems of higher order (for n=2,4) are verified successfully. It is observed that the method is successful even in the presence of significant noise, provided that the assumptions of the algorithm are satisfied. In conclusion, this work provides a method that can be adapted successfully for solving a direct (regular/singular) or inverse Sturm&ndash, Liouville problem (SLP) of an arbitrary order with arbitrary boundary conditions.

In this paper, we study an inverse scattering problem on Liouville surfaces having two asymptotically hyperbolic ends. The main property of Liouville surfaces consists in the complete separability of the Hamilton-Jacobi equations for the geodesic flow. An important related consequence is the fact that the stationary wave equation can be separated into a system of a radial and angular ODEs. The full scattering matrix at fixed energy associated to a scalar wave equation on asymptotically hyperbolic Liouville surfaces can be thus simplified by considering its restrictions onto the generalized harmonics corresponding to the angular separated ODE. The resulting partial scattering matrices consists in a countable set of $2 \times 2$ matrices whose coefficients are the so called transmission and reflection coefficients. It is shown that the reflection coefficients are nothing but generalized Weyl-Titchmarsh functions for the radial ODE in which the generalized angular momentum is seen as the spectral parameter. Using the Complex Angular Momentum method and recent results on 1D inverse problem from generalized Weyl-Titchmarsh functions, we show that the knowledge of the reflection operators at a fixed non zero energy is enough to determine uniquely the metric of the asymptotically hyperbolic Liouville surface under consideration., 39 pp

A characterization of weighted L 2 ( I ) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley–Wiener-type theorems for these spaces. Unlike the classical Paley–Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm–Liouville boundary-value problems on a half line and on the whole line.

The problem of computing eigenvalues of a singular Sturm–Liouville problem is reduced to the computation of eigenvalues of a Hilbert–Schmidt infinite matrix. The uniform convergence of the generalized determinant allows for the approximation of eigenvalues by the finite section. A key feature of the method that leads to a fast algorithm is to combine generating functions with the Laplace transform to compute explicitly the entries of the matrix without numerical integration.

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy,x∈(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.