We consider the quantum completeness problem, i.e. the problem of confining quantum particles, on a non-complete Riemannian manifold M equipped with a smooth measure \omega , possibly degenerate or singular near the metric boundary of M , and in presence of a real-valued potential V\in L^2_\mathrm {loc}(M) . The main merit of this paper is the identification of an intrinsic quantity, the effective potential V_\mathrm {eff} , which allows to formulate simple criteria for quantum confinement. Let \delta be the distance from the possibly non-compact metric boundary of M . A simplified version of the main result guarantees quantum completeness if V\ge -c\delta^2 far from the metric boundary and V_\mathrm {eff}+V\ge \frac3{4\delta^2}-\frac{\kappa}{\delta}, \quad \mathrm {close \:to \:the \:metric \:boundary}. These criteria allow us to: (i) obtain quantum confinement results for measures with degeneracies or singularities near the metric boundary of M ; (ii) generalize the Kalf–Walter–Schmincke–Simon Theorem for strongly singular potentials to the Riemannian setting for any dimension of the singularity; (iii) give the first, to our knowledge, curvature-based criteria for self-adjointness of the Laplace–Beltrami operator; (iv) prove, under mild regularity assumptions, that the Laplace–Beltrami operator in almost-Riemannian geometry is essentially self-adjoint, partially settling a conjecture formulated in [9].

We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. We also show that, in the compact case, this criterion implies discreteness of the sub-Laplacian spectrum even though the total volume of the manifold is infinite. As a consequence of our result, the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp’s measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This settles a conjecture formulated by Boscain and Laurent (Ann. Inst. Fourier (Grenoble) 63(5), 1739–1770, 2013 ), under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.