The 1/R expansion for \(H^ +_ 2:\) analyticity, summability, and asymptotics
Consider the two-center problem of an electron in the field of two fixed point charges \(Z_ A,Z_ B\) at a distance R apart. In non-relativistic quantum mechanics its Hamiltonian is \(H(R,Z_ A,Z_ B)=-\Delta -Z_ A| x|^{-1}-Z_ B| x-R\hat e|^{-1}\) in atomic units, with \(x\in R^ 3\), \(\hat e=(1,0,0)\). If \(Z_ A=Z_ B=1\) this describes the hydrogen molecular ion \(H^+_ 2\) in the clamped nuclei approximation, which is an important double-well problem having the virtue of being separable. In the normalization of the above expression the formal limit as \(R\to \infty\) is the Hamiltonian of hydrogen. It is proved that the 1/R expansion for \(H^+_ 2\) is divergent and Borel summable to a complex eigenvalue of a non-self-adjoint operator, which has the same 1/R expansion. The Borel sum is related to the \(H^+_ 2\) system as follows: its real part agrees with the eigenvalue doublet asymptotically to all orders, and its imaginary part determines the asymptotics of the 1/R expansion coefficients via a dispersion relation. A rigorous estimate of the leading behavior of the imaginary part is obtained, and as a consequence the approximate formula of Brézin and Zinn-Justin relating the square of the eigenvalue gap of the asymptotics of the 1/R expansion is put on a rigorous basis.
two-center problem of an electron in the field of two fixed point charges
scientific article; zbMATH DE number 3994326
scientific article