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The 1/R expansion for \(H^ +_ 2:\) analyticity, summability, and asymptotics. (English) Zbl 0614.46068

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.
Reviewer: P.Doktor

MSC:

46N99 Miscellaneous applications of functional analysis
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
81V10 Electromagnetic interaction; quantum electrodynamics
47A55 Perturbation theory of linear operators
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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