I present an example of a discrete Schrödinger operator that shows that it is possible to have embedded singular spectrum and, at the same time, discrete eigenvalues that approach the edges of the essential spectrum (much) faster than exponentially. This settles a conjecture of Simon (in the negative). The potential is of von Neumann-Wigner type, with careful navigation around a previously iden...

Since 2001, Laplace decomposition algorithm (LDA) has been one of the reliable mathematical methods for obtaining exact or numerical approximation solutions for a wide range of nonlinear problems. The Laplace decomposition algorithm was developed by Khuri in [2] to solve a class of nonlinear differential equations. The basic idea in Laplace decomposition algorithm, which is a combined form of t...

For simplicity, we follow the rules: x, y are sets, N is an element of N, c, i, j, k, m, n are natural numbers, D is a non empty set, s is an element of 2Set Seg(n + 2), p is an element of the permutations of n-element set, p1, q1 are elements of the permutations of (n+ 1)-element set, p2 is an element of the permutations of (n+ 2)-element set, K is a field, a, b are elements of K, f is a finit...

other fields. However, the spectral properties of the Laplace Transform tend to complicate its numerical treatment; therefore, the closely related “Truncated” Laplace Transforms are often used in applications. We have constructed efficient algorithms for the evaluation of the left singular functions and singular values of the Truncated Laplace Transform. Together with the previously introduced ...

February 2, 2008 Abstract. Recently, Breiter et al.(2004) reported the computation of Hansen coefficients X k for non integer values of γ. In fact, the Hansen coefficients are closely related to the Laplace b (m) s , and generalized Laplace coefficients b (m) s,r (Laskar and Robutel, 1995) that do not require s, r to be integers. In particular, the coefficients X 0 have very simple expressions ...

In this article, we show that Laplace transform can be applied to fractional system. To this end, solutions of linear fractional-order equations are first derived by a direct method, without using Laplace transform. Then the solutions of fractional-order differential equations are estimated by employing Gronwall and Hölder inequalities. They are showed be to of exponential order, which are nece...