Reviewing the Helmholtz Equation on Euclidean Plane and Interbasis Expansions

TL;DR

The paper analyzes the 2D Helmholtz equation on the Euclidean plane, $\Delta \Psi + k^2 \Psi = 0$ with $k>0$, and demonstrates its separability in Cartesian, polar, and parabolic coordinates along with the corresponding symmetry operators within the $e(2)$ algebra. It develops parity-filtered, normalized bases in each coordinate system and derives detailed interbasis expansion coefficients, including simple trigonometric forms for Cartesian–polar and polynomial (continuous Hahn) forms for parabolic–polar and parabolic–Cartesian connections. By establishing explicit normalization constants, completeness relations, and inverse expansions, the work provides a comprehensive framework for interbasis expansions and contraction limits toward $E_2$ from higher-curvature spaces. The results yield polynomial expressions for expansion coefficients and supply a reference for 2D Helmholtz interbasis relations, with potential implications for symmetry-based analysis in planar wave problems.

Abstract

In the present paper we revisit the Helmholtz equation on the Euclidean plane and make some remarks on normalization constants and completeness of wave function sets. The coefficients of interbasis expansions are also reconsidered.