Variational collocation on finite intervals

Abstract

In this paper we study a new family of sinc-like functions, defined on an interval of finite width. These functions, which we call "little sinc", are orthogonal and share many of the properties of the sinc functions. We show that the little sinc functions supplemented with a variational approach enable one to obtain accurate results for a variety of problems. We apply them to the interpolation of functions on finite domain and to the solution of the Schrödinger equation, and compare the performance of present approach with others.

Accurate calculation of the eigenvalues of non-uniform strings and membranes

“…When ρ( ) = 1 we are led to the particular case of constant grid spacing C N = 2L/N discussed earlier [19]. Actually, C ( ) N is roughly the grid spacing even in the general case.…”

“…In this section we explicitly show how to build a set of "little sinc functions" (LSF) [19] that satisfy Dirichlet boundary conditions at the endpoints of a given finite coordinate interval. The generalization of these results to other boundary conditions is straightforward.…”

“…The simplest example is provided by a constant ρ( ), which corresponds to the uniform grid already discussed in ref. [19].…”

PT-symmetric strings

“…Figure 3 shows the estimate obtained with the Shanks transformations and the actual value of α 1 (vertical line). We have also calculated the eigenvalues of the PT string by means of a collocation method developed some time ago [11].…”

“…A useful strategy to obtain approximate solutions to the string with density (39) is to apply the Rayleigh-Ritz method as indicated in section 4 or the collocation approach to the operatorÔ. In Figure 4 we show the numerical results for the real and imaginary parts of the first eight eigenvalues of the string with density (39) for −10 ≤ α ≤ 10: these results are obtained using a collocation approach with a grid with 2000 points [11]. Looking at the right plot we see that the eigenvalues are real when −2 α 2.…”