According to the Maupertuis principle, the movement of a classical particle in an external potential V (x) can be understood as the movement in a curved space with the metric gμν(x) = 2M [V (x) − E]δμν . We show that the principle can be extended to the quantum regime, i.e., we show that the wave function of the particle follows a Schrödinger equation in curved space where the kinetic operator ...

Various geometric operators have been playing an important role in surface processing. For example, many shape analysis algorithms have been developed based on eigenfunctions of the Laplace–Beltrami operator (LBO), which is defined based on the first fundamental form of the surface. In this paper, we introduce two new geometric operators based on the second fundamental form of the surface, name...

In this paper a non-relativistic particle moving on a hypersurface in a curved space and the multidimensional rotator are investigated using the Hamilton-Jacobi formalism. The equivalence with the Dirac Hamiltonian formalism is demonstrated in both Cartesian and curvilinear coordinates. The energy spectrum of the multi-dimensional rotator is equal to that of a pure Laplace-Beltrami operator wit...

It is shown that eigenvalues of the Laplace–Beltrami operator on a compact Riemannian manifold can be determined as limits of eigenvalues of certain finite-dimensional operators in spaces of polyharmonic functions with singularities. In particular, a bounded set of eigenvalues can be determined using a space of such polyharmonic functions with a fixed set of singularities. It also shown that co...

We use the so-called spectral embedding to give upper bounds on the spectral function of the Laplace-Beltrami operator on homogeneous spaces in terms of the volume growth of balls. In the case of compact manifolds, our bounds extend the 1980 lower bound of Peter Li [Li80] for the smallest positive eigenvalue to all eigenvalues. We also improve Li’s bound itself. Our bounds translate to explicit...

This paper is devoted to various considerations on a family of sharp interpolation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincaré, logarithmic Sobolev and critical Sobolev (Onofri in dimension two) inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors ad...

We derive an implicit-explicit (IMEX), realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms. In contrast to the fully-explicit schemes in [3, 42] the time step does not depend on the physical parameters, removing the stiffness from the system. Furthermore, a wider class of collision operators (e.g. the Laplace-Beltrami operator) can be used. The d...

We determine the L spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces M whose universal covering X is a symmetric space of non-compact type with rank one. More precisely, we show that the L spectra of M and X coincide if the fundamental group of M is small and if the injectivity radius of M is bounded away from zero. In the L case, the restriction on the inje...