Exponentially accurate quasimodes for the time-independent Born-Oppenheimer approximation on a one-dimensional molecular system

GEORGE A. HAGEDORN; JULIO H. TOLOZA

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY

Wiley Periodicals

Lugar: Hoboken, New Jersey; Año: 2005 vol. 105 p. 463 - 477

0020-7608

We consider the eigenvalue problem for a one-dimensional molecular-type quantum Hamiltonian that has the form H(e)=-e⁴/2 d²/dy² + h(y) where h(y) is an analytic family of self-adjoint operators that has an discrete, nondegenerate electronic level E(y) for y in some open subset of R. Near a local minimum of the electronic level E(y) that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born-Oppenheimer parameter e by optimal truncation of the Rayleigh-Schrödinger series. That is, we construct an energy E_e and a wave function Xi_e, such that the L^2-norm of Xi_e is O(1) and the L^2-norm of (H(e)-E_e)Xi_e is bounded by Lambda*exp(-Gamma/e²) with Gamma>0.