A Priori Estimation of the Resolvent on Approximation of Born-Oppenheimer

Amina Benbernou,Sabria Bouheroum-

doi:10.3844/jmssp.2007.70.76

Abstract

In this study, we estimate the resolvent of the two bodies Shrodinger operator perturbed by a potential of Coulombian type on Hilbert space when h tends to zero. Using the Feschbach method, we first distorted it and then reduced it to a diagonal matrix. We considered a case where two energy levels cross in the classical forbidden region. Under the assumption that the second energy level admits a non degenerate point well and virial conditions on the others levels, a good estimate of the resolvent were observed.

Highlights

The Born-Oppenheimer approximation technical[1]has instigated many works one can find in bibliography the recent papers like[2,3,4,5].It consists to study the behaviour of a many body systems, in the limit of small parameter h as the particles masses tends to infinity;(see the references therein for more information), we can describe it with a Hamiltonian of type P = −h2∆x − ∆y + V(x, y) on
To describe our main results we introduce the following assumptions: (H1) ∀x ∈ IR3n \{0}, # σdisc (Q(x)) ≥ 3 Let λ0 an energy level such that: λ j ∩[−∞, λ0 [ ≤ 3, denoting λ1(x), λ2 (x), λ3 (x) the first three eigenvalues of Q(x). (H2) we assume that the first tree eigenvalues λ j
* The connex composites of IR3\ Kδ1 are connex (H5) Virial Conditions It exists d〉0 such that for j∈ {2,3}, The resonances of P are obtained by an analytic distorsion introduced by Hunziker[8] and so they are defined as complex numbers ρj ( j = 1,..., N0 ) such that for all ε〉0 and μ sufficiently small, Im μ〉0 ρj ∈ σdisc (Pμ) [3]

Summary

Introduction

To describe our main results we introduce the following assumptions: (H1) ∀x ∈ IR3n \{0}, # σdisc (Q(x)) ≥ 3 Let λ0 an energy level such that: λ j ∩[−∞, λ0 [ ≤ 3 , denoting λ1(x), λ2 (x), λ3 (x) the first three eigenvalues of Q(x). * The connex composites of IR3\ Kδ1 are connex (H5) Virial Conditions It exists d〉0 such that for j∈ {2,3}, The resonances of P are obtained by an analytic distorsion introduced by Hunziker[8] and so they are defined as complex numbers ρj ( j = 1,..., N0 ) such that for all ε〉0 and μ sufficiently small, Im μ〉0 ρj ∈ σdisc (Pμ) [3].

Results

Conclusion