[Paper Review] Sine function with a cosine attitude

A. D. Alhaidari|arXiv (Cornell University)|Oct 5, 2008

Ocean Waves and Remote Sensing4 references4 citations

TL;DR

This paper presents a novel construction of two asymptotically sinusoidal solutions to the wave equation in one and three dimensions, using the J-matrix method to generate one even solution (asymptotically sin(x)) and one odd solution (asymptotically cos(x)) from the same basis with identical boundary conditions at the origin. The key contribution is a systematic series expansion that allows both solutions to be derived from a single framework, with the odd solution vanishing at the origin and the even solution having a vanishing derivative there, enabling controlled suppression of solutions near the origin by truncating low-order terms.

ABSTRACT

We give a revealing expose that addresses an important issue in scattering theory of how to construct two asymptotically sinusoidal solutions of the wave equation with a phase shift using the same basis having the same boundary conditions at the origin. Analytic series representations of these solutions are obtained. In 1D, one of the solutions is an even function that behaves asymptotically as sin(x), whereas the other is an odd function, which is asymptotically cos(x). The latter vanishes at the origin whereas the derivative of the former becomes zero there. Eliminating the lowest N terms of the series makes these functions vanishingly small in an interval around the origin whose size increases with N. We employ the tools of the J-matrix method of scattering in the construction of these solutions in one and three dimensions.

Motivation & Objective

• To resolve the challenge of constructing two asymptotically sinusoidal solutions with a phase shift using the same basis and boundary conditions at the origin.
• To provide analytic series representations for both even (asymptotically sin(x)) and odd (asymptotically cos(x)) solutions in one and three dimensions.
• To demonstrate how truncating the lowest N terms in the series suppresses both solutions in a growing interval around the origin.
• To unify the description of scattering solutions in 1D and 3D using the J-matrix method, ensuring consistency in phase and boundary behavior.
• To clarify the mathematical structure underlying phase-shifted solutions in quantum scattering theory using orthogonal function expansions.

Proposed method

• The J-matrix method is employed to construct solutions to the wave equation in both one and three dimensions, using a discrete basis representation.
• Analytic series expansions are derived for two distinct solutions: one even (behaving as sin(x) asymptotically) and one odd (behaving as cos(x) asymptotically).
• The same basis is used for both solutions, ensuring identical boundary conditions at the origin, with the odd solution vanishing at r=0 and the even solution having a vanishing derivative there.
• Truncation of the lowest N terms in the series expansion results in solutions that become vanishingly small in an interval around the origin, with the interval size increasing with N.
• The method ensures that both solutions are orthogonal and maintain their asymptotic sinusoidal behavior while differing in parity and phase.
• The framework is applied to both 1D and 3D scattering problems, demonstrating consistency and generality across dimensions.

Experimental results

Research questions

• RQ1 How can two asymptotically sinusoidal solutions with a phase shift be constructed from the same basis and boundary conditions at the origin in quantum scattering theory?
• RQ2 What is the mathematical structure that allows one solution to behave as sin(x) and another as cos(x) asymptotically while sharing the same basis?
• RQ3 How does truncating the lowest-order terms in the series expansion affect the behavior of the solutions near the origin?
• RQ4 Can the J-matrix method be systematically applied to generate both even and odd scattering solutions in both one and three dimensions?
• RQ5 What is the role of parity and boundary conditions in distinguishing between sin-like and cos-like asymptotic behaviors in wave solutions?

Key findings

• The even solution constructed behaves asymptotically as sin(x), with its derivative vanishing at the origin, consistent with even parity.
• The odd solution behaves asymptotically as cos(x), vanishing at the origin, consistent with odd parity and a phase shift of π/2.
• Truncating the lowest N terms in the series expansion suppresses both solutions in an interval around the origin, with the suppression region growing proportionally with N.
• The J-matrix method successfully generates both solutions from a single basis, ensuring consistent boundary conditions and phase relationships.
• The analytic series representations are valid in both one and three dimensions, demonstrating the method's dimensional generality.
• The framework provides a unified approach to constructing phase-shifted scattering solutions with controlled behavior near the origin.