Complex Higgs Oscillators

TL;DR

This paper explores the complex Higgs oscillator on hyperbolic space, analyzing eigenvalues and resonances, and proposes open problems related to complex potentials and hyperbolic analogues.

Contribution

It introduces the complex Higgs oscillator in hyperbolic geometry, computes eigenvalues and resonances, and suggests new research directions.

Findings

Eigenvalues and resonances computed for various hyperbolic examples

Identification of open problems in complex potentials on hyperbolic spaces

Proposals for future research on hyperbolic analogues of oscillators

Abstract

In this note we discuss the complex version of the Higgs oscillator on the hyperbolic space. The eigenvalues and resonances of the complex Higgs oscillator are computed in different examples in the hyperbolic setting. We also propose open problems like whether the complex absorbing potential (CAP) method works for asymptotically hyperbolic manifolds and finding hyperbolic analogues of the complex harmonic oscillator.