Universal Ladder Structure Across Scales: From Quantum to Black Hole Physics

TL;DR

This paper introduces a unified symmetry-based framework to identify and construct hierarchical ladder structures in second-order linear differential equations across physics, revealing connections to supersymmetric quantum mechanics and applying to quantum oscillators and black holes.

Contribution

It provides a novel criterion and explicit method for detecting and building ladder structures in diverse physical systems, linking them to supersymmetry.

Findings

Uncovered a connection to supersymmetric quantum mechanics.

Developed a litmus-test criterion for ladder structures.

Applied framework to quantum harmonic oscillator and Kerr black holes.

Abstract

Second-order ordinary linear differential equations appear ubiquitously across physics, describing the behavior of systems from the quantum world of atoms to the classical world of gravitating bodies. We present a unified symmetry-based framework that provides a ``litmus-test criterion'' to determine when such a system admits a hierarchical ladder structure, and, whenever it does, explicitly constructs the ladder. This approach uncovers a previously underappreciated connection to supersymmetric quantum mechanics and a deep commonality among diverse physical problems. Applications to the quantum harmonic oscillator and dynamical tidal response of Kerr black holes are presented to illustrate the framework.