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Two-dimension almost-Riemannian structures of step 2 are naturalgeneralizations of the Grushin plane. They are generalized Riemannianstructures for which the vectors of a local orthonormal frame can becomeparallel. Under the 2-step assumption the singular set $Z$, where the structureis not Riemannian, is a 1D embedded submanifold. While approaching the singularset, all Riemannian quantities diverge. A remarkable property of thesestructures is that the geodesics can cross the singular set withoutsingularities, but the heat and the solution of the Schr\"{o}dinger equation(with the Laplace-Beltrami operator $\Delta$) cannot. This is due to the factthat (under a natural compactness hypothesis), the Laplace-Beltrami operator isessentially self-adjoint on a connected component of the manifold without thesingular set. In the literature such phenomenon is called quantum confinement.  In this paper we study the self-adjointness of the curvature Laplacian,namely $-\Delta+cK$, for $c\in(0,1/2)$ (here $K$ is the Gaussian curvature),which originates in coordinate-free quantization procedures (as for instance inpath-integral or covariant Weyl quantization). We prove that there is noquantum confinement for this type of operators.

Quantum confinement for the curvature Laplacian $-\Delta+cK$ on 2D-almost-Riemannian manifolds