Quantum mechanics with a ghost: Counterexamples to spectral denseness
Cédric Deffayet, Atabak Fathe Jalali, Aaron Held, Shinji Mukohyama, Alexander Vikman
TLDR
This paper shows that ghostly quantum systems can have discrete, non-dense energy spectra, challenging common assumptions.
Key contributions
- Quantizes integrable point-particle systems with opposite-sign kinetic terms.
- Shows classical stability conditions imply discrete, separated eigenvalue spectra.
- Demonstrates energy spectra are unbounded but not necessarily dense.
- Establishes conditions for one or zero accumulation points in the spectrum.
Why it matters
This paper challenges the widespread notion that ghostly quantum systems must have continuous or dense energy spectra. It provides counterexamples, showing discrete separated eigenvalue spectra are possible, even with unbounded energies.
Original 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.
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