The generalized Fourier transform associated with a selfadjoint Sturm–Liouville operator is a unitary transformation which converts the action of this operator into a simple product by a spectral variable. For a particular operator defined on the half-line and which involves a step function, we show how to extend such a transformation to generalized functions, or distributions, with a suitable ...

If f (x, k) is the Jost solution and f (x) = f (0, k), then the I-function is I(k) := f ′ (0,k) f (k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), k j , s j , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = − d 2 dx 2 + q(x) on (0, ∞) with the Dirichlet condition at x = 0 and q(x) ∈ L 1,1 := {q : q = q, ∞ 0 (1 ...

We classify the general linear boundary conditions involving u′′, u′ and u on the boundary {a, b} so that a Sturm-Liouville operator on [a, b] has a unique self-adjoint extension on a suitable Hilbert space.

We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator ?4 d 2 ds 2 + 2 (s) with potential given by the curvature of a closed curve.

If f(x, k) is the Jost solution and f(k) = f(0, k), then the I-function is I(k) := f ′(0, k)/f(k). It is proved that I(k) is in one-to-one correspondence with the scattering triple S := {S(k), kj , sj , 1 ≤ j ≤ J} and with the spectral function ρ(λ) of the Sturm-Liouville operator l = −d/dx + q(x) on (0,∞) with the Dirichlet condition at x = 0 and q(x) ∈ L1,1 := {q | q = q, ∫∞ 0 (1 + x)|q(x)dx ...

This paper examines and develops matrix methods to approximate the eigenvalues of a fourth order Sturm-Liouville problem subjected to a kind of fixed boundary conditions, furthermore, it extends the matrix methods for a kind of general boundary conditions. The idea of the methods comes from finite difference and Numerov’s method as well as boundary value methods for second order regular Sturm-L...

The inverse Sturm-Liouville problem finds its applications in the identification of mechanical properties and/or geometrical configurations of a vibrating continuous medium; however, this problem is hard to solve, either theoretically or numerically. Previously, Liu (2008a) has constructed a Lie-group shooting method to determine the eigenvalues, and the corresponding eigenfunctions, for the di...

In this paper we prove an analogue of the Ramanujan’s master theorem in setting Sturm Liouville operator $$\begin{aligned} \mathcal L=\frac{d^2}{dt^2} + \frac{A'(t)}{A(t)} \frac{d}{dt}, \end{aligned}$$on \((0,\infty )\), where \(A(t)=(\sinh t)^{2\alpha +1}(\cosh t)^{2\beta +1}B(t); \alpha ,\beta > -\frac{1}{2}\) with suitable conditions on B. When \(B\equiv 1\) get back Master for Jacobi operat...

In this work, we have estimated nodal points and nodal lengths for the diffusion operator. Furthermore, by using these new spectral parameters, we have shown that the potential function of the diffusion operator can be established uniquely. An analogous inverse problem was solved for the Sturm–Liouville problem in recent years. c © 2005 Elsevier Ltd. All rights reserved.