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We consider sub-Laplacians in open bounded sets in a homogeneous Carnot groupand study their spectral properties. We prove that these operators have a purepoint spectrum, and prove the existence of the spectral gap. In addition, wegive applications to the small ball problem for a hypoelliptic Brownian motionand the large time behavior of the heat content in a regular domain.

Dirichlet sub-Laplacians on homogeneous Carnot groups: spectral properties, asymptotics, and heat content