On Green functions of second-order elliptic operators on Riemannian manifolds: The critical case

Let P be a second-order, linear, elliptic operator with real coefficients which is defined on a noncompact and connected Riemannian manifold M. It is well known that the equation $Pu=0$ in M admits a positive supersolution which is not a solution if and only if P admits a unique positive minimal Green function on M, and in this case, P is said to be subcritical in M. If P does not admit a positive Green function but admits a global positive (super)solution, then such a solution is called a ground state of P in M, and P is said to be critical in M.We prove for a critical operator P in M, the existence of a Green function which is dominated above by the ground state of P away from the singularity. Moreover, in a certain class of Green functions, such a Green function is unique, up to an addition of a product of the ground states of P and $P^{?}$. Under some further assumptions, we describe the behavior at infinity of such a Green function. This result extends and sharpens the celebrated result of P. Li and L.-F. Tam concerning the existence of a symmetric Green function for the Laplace–Beltrami operator on a smooth and complete Riemannian manifold M.     

Hyperfine interaction with polarized atomic environment — The nuclear tilted-foil experiment

The nuclear tilted-foil experimental field has matured from the early time-integral measurements to the current multifoil time-differential precession and polarization configurations, leading to a wide range of measurements-magnetic moments, quadrupole moments and parity-non-conservation. The physical basis of these experiments will be discussed and experimental results will be reviewed.     

Measurement of the fine structure separation in the 42 D term of lithium by the method of electric field induced level anti-crossing

An electric field is employed to convert lithium 4² D Zeeman level crossings into anti-crossings. The lithium atoms are excited by electron impact. The anti-crossings are detected by the change in polarization of the light emitted in the 4² D-2² P transition. From the corresponding values of the magnetic field, we obtain 400 (10) MHz for the zero field fine structure energy separationE(4² D _5/2)-E(4² D _3/2).     

Ozonolysis of olfefins and acetylenes adsorbed on silica gel

Ozonation of phenyl ethylenes adsorbed on untreated silica gel results in aromatic aldehydes and on dried silica gel in ozonides and aldehydes. On the other hand ozonation of alkyl ethylenes on both types of silica gel results in a mixture of ozonides or polymeric peroxides similar to that obtained in aprotic solvents.