Quantum mechanics with a ghost: Counterexamples to spectral denseness

TL;DR

This paper investigates quantum systems with ghostly kinetic terms, showing they can have non-dense spectra and challenging the assumption that such systems must have continuous spectra.

Contribution

It provides conditions under which ghostly quantum systems have discrete, non-dense spectra, countering previous beliefs about their spectral properties.

Findings

Spectra can be unbounded but not necessarily dense.

Sufficient conditions for a single accumulation point or none are established.

Ghostly quantum systems can have discrete spectra, contrary to prior assumptions.

Abstract

We quantise integrable point-particle systems with opposite-sign kinetic terms and nontrivial interactions. Using methods from separability theory, we show that previously determined classical stability conditions also imply discrete separated eigenvalue spectra. The resulting energy spectrum is unbounded above and below but not necessarily dense. We establish sufficient conditions for (i) exactly one accumulation point, or (ii) none at all. This dispels the widespread notion that ghostly quantum systems must have a continuous or dense energy spectrum.