Stability of the classical catenoid and Darboux-P\"oschl-Teller potentials

TL;DR

This paper analyzes the stability of catenoids connecting concentric rings, providing explicit unstable modes for the inner catenoid through spectral analysis of a Schr"odinger operator with Darboux-P"oschl-Teller potential.

Contribution

It offers a new explicit construction of the unstable mode of the inner catenoid using exactly solvable Schr"odinger operators with Darboux-P"oschl-Teller potentials.

Findings

Explicit unstable mode for the inner catenoid derived.

Spectral analysis of Schr"odinger operator with Darboux-P"oschl-Teller potential.

Revisits classical stability problem with exact solutions.

Abstract

We revisit the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings and give an explicit new construction of the unstable mode of the inner catenoid by studying the spectrum of an exactly solvable one-dimensional Schr\"odinger operator with an asymmetric Darboux-P\"oschl-Teller potential.