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Spectra of Sub-Dirac Operators on Certain Nilmanifolds

Ines Kath, Oliver Ungermann


We study sub-Dirac operators associated to left-invariant bracket-generating sub-Riemannian structures on compact quotients of nilpotent semi-direct products $G=\mathsf{R}^n\rtimes_A\mathsf{R}$. We prove that these operators admit an $L^2$-basis of eigenfunctions. Explicit examples of this type show that the spectrum of these operators can be non-discrete and that eigenvalues may have infinite multiplicity. In this case the sub-Dirac operator is neither Fredholm nor hypoelliptic.

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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