Exponentially accurate semiclassical asymptotics of low-lying eigenvalues for 2×2 matrix Schrödinger operators

GEORGE A. HAGEDORN; JULIO H. TOLOZA

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

Elsevier

Lugar: San Diego, California; Año: 2005 vol. 312 p. 300 - 329

0022-247X

We consider a simple molecular-type quantum system in which the nuclei have one degree of freedom and the electrons have two levels. The Hamiltonian has the form H(e) = -e⁴/2 d²/dy² + h(y), where h(y) is a 2x2 real symmetric matrix. Near a local minimum of an electron 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 E_e and Psi_e, such that ||Psi_e||=O(1) and ||(H_e-E_e)Psi_e|| < Lambda*exp(-Gamma/e²) where Gamma>0.