Quantum mechanics with a ghost: Counterexamples to spectral denseness

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.