1 Introduction.- 2 Graphs and associated Laplacians.- 3 Scales of Hilbert space and boundary triples.- 4 Two operators in different Hilbert spaces.- 5 Manifolds, tubular neighbourhoods and their perturbations.- 6 Plumber's shop: Estimates for star graphs and related spaces.- 7 Global convergence results.

Spectral Analysis on Graph-like Spaces

In this article we give general lower bounds of shape resonance widths in the semi-classical limit for Schrodinger operators in the exterior of smooth compact obstacles with Dirichlet or Neuman boundary conditions and with long range dilation analytic potentials. These lower bounds are exponentially small with respect to the Planck constant.

Lower bounds for shape resonances widths of long range schrödinger operators

An implementation of exterior complex scaling using the finite elements method with high degree polynomials is presented. We apply the method to find the resonances of the potential 7.5r 2e−r and of a phenomenological coupled channel model of the CaH molecule. In both cases the method is quickly convergent and extremely stable numerically. Convergence could be pushed to the point where the real parts of most resonance energies were independent of the complex scaling angle and of the exterior scaling radius within machine precision (14 significant digits). All imaginary parts were stable to at least eight significant digits. Several resonances of CaH which had evaded searches with a finite difference method could be located.

A finite element implementation of exterior complex scaling for the accurate determination of resonance energies

Semiclassical resonances generated by non-degenerate critical points

In this article we give general lower bounds of shape resonance widths in the semi- classical limit for Schrodinger operators in the exterior of smooth compact obstacles with Dirichlet or Neuman boundary conditions and with long range dilation analytic potentials. These lower bounds are exponentially small with respect to the Planck constant. 1. Introduction, hypotheses. The purpose of this article is the study of the localization of resonances for the semi-classical Schrodinger operator near positive energy levels. The shape resonances are the metastable states of a system whose evolution is described by a Hamiltonian, H, depending upon the Planck constant h. An important feature of the resonances is their lifetime (the reverse of their width) which describes the time needed by a wave packet associated to the resonance and initially localized in a compact region of the space, to develop a nonzero probality density in the outside region. In this paper we shall study the semi-classical approximation when the planck constant tends to 0. Most results

Lower bounds for shape resonances widths of long range schr ¨ odinger operators

For a class of Schrodinger operatorsH:=−(ℏ2/2m)Δ+V onL
2(ℝ
 n
 ), with potentials having minima embedded in the continuum of the spectrum and non-trapping tails, we show the existence of shape resonances exponentially close to the real axis as ℏ↘0. The resonant energies are given by a convergent perturbation expansion in powers of a parameter exhibiting the expected exponentially small behaviour for tunneling.

P. Duclos

Papers

The shape resonance