Derivation and Correlation between Pythagorus (560-479BC) and Mathieu's (1868) equation: Spectral nature between Mathieu's and Modified Mathieu's equation

TL;DR

This paper explores the mathematical relationship between Pythagoras theorem, Mathieu's equation, and modified Mathieu's equation, revealing spectral similarities and quantum bound states in a novel theoretical framework.

Contribution

It introduces a derivation linking Pythagoras theorem to Mathieu's equations and compares their spectral properties, including quantum bound states, which is a novel theoretical insight.

Findings

Spectral comparison between Mathieu's and modified Mathieu's equations

Derivation of Mathieu's equation from Pythagoras theorem

Identification of discrete quantum bound states

Abstract

We derive Pythagoras theorem. From the Pythagoras theorem, we also derive Mathieu's equation via modified Mathieu's equation. A spectral com- parison has been carried out between modified Mathieu's equation and Mathieu's equation. Apart from this, we also present discrete bound states corresponding to modified Mathieu's equation of a quantum rectangular type of model potential.