Markov constant and quantum instabilities

TL;DR

This paper introduces a novel application of number theory, including a generalized Markov constant, to analyze spectral anomalies in two-dimensional quantum systems, offering new insights into quantum stability and related phenomena.

Contribution

It generalizes the Markov constant concept from Diophantine approximation to quantum spectral analysis, providing a new mathematical framework for understanding spectral anomalies.

Findings

Number theory methods effectively analyze quantum spectra.

Generalized Markov constant offers new insights into spectral anomalies.

Potential applications in metamaterials and quantum cosmology.

Abstract

For a qualitative analysis of spectra of certain two-dimensional rectangular-well quantum systems several rigorous methods of number theory are shown productive and useful. These methods (and, in particular, a generalization of the concept of Markov constant known in Diophantine approximation theory) are shown to provide a new mathematical insight in the phenomenologically relevant occurrence of anomalies in the spectra. Our results may inspire methodical innovations ranging from the description of the stability properties of metamaterials and of certain hiddenly unitary quantum evolution models up to the clarification of the mechanisms of occurrence of ghosts in quantum cosmology.