Premium
Reality and complexity in asymptotic expansions for eigenvalues and eigenfunctions, with application to the JWKB connection‐formula problem
Author(s) -
Silverstone Harris J.
Publication year - 1986
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560290216
Subject(s) - eigenfunction , eigenvalues and eigenvectors , asymptotic expansion , connection (principal bundle) , simple (philosophy) , quantum field theory , quantum mechanics , quantum , perturbation theory (quantum mechanics) , path integral formulation , mathematics , physics , statistical physics , mathematical analysis , philosophy , geometry , epistemology
Asymptotic expansions occur widely in quantum physics. The Rayleigh‐Schrödinger perturbation theory for hydrogen in an electrostatic field (the LoSurdo—Stark effect) is one example. The 1/ R expansion for the hydrogen molecule ion H + 2is a second. The quantum defect theory and the JWKB method are two more. It is not so widely known that the sum of such real asymptotic expansions may be complex , while the sum of complex asymptotic expansions may be real . The key to this nonintuitive behavior is Borel summation. By examining a simple example related to the exponential integral, the nature of this real‐iscomplex, complex‐is‐real phenomenon is made simple. Then special application is made to derive and clarify the connection formulas (to all orders) in the JWKB method.