Molecular predissociation resonances below an energy level crossing
We study the resonances of $2\times 2$ systems of one dimensional Schrodinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.
Estimates on Complex Interactions in Phase Space
We propose a method to study the complex interactions (or microlocal tunneling) between electronic levels that do not intersect in the real domain. The method consists in using a special kind of Fourier-Bros-Iagolnitzer transformation, and in adjoining an exponential weight to the L2-type Hilbert spaces which are associated to a complex Lagrangian manifold and have been introduced by Helffer and Sjostrand for the study of resonances.
The splitting theorem in non-smooth context
We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.