Parabolic Connection Formulae in Quantum Mechanics

On the Nature of Energy Levels of Anharmonic Oscillators

“…Second, Eq. (6.6) of [31] was itself obtained on retaining only the leading term in the asymptotic expansion of an exact solution of the Schro dinger equation in terms of the parabolic cylinder functions. Higher-order terms of this expansion, if retained, would also produce corrections that in the final analysis may affect the term Rg(_) in Eqs.…”

“…The effect of the pair (z 0 , z 0 *) on the quantum motion, that is, the reflection of the particle on the pair of complex-conjugate turning points, is then described by the following asymptotic expressions for the particle's wave function (x) [31] (x)t 1…”

“…The integration path in (2.2) runs in the complex z plane from the lower turning point z 0 * to the upper one z 0 . The function Rg(_) in (2.1) is defined for every real _>0 by either one of the two equivalent expressions [31,32] …”

Resonant Delay in Tunnelling through Smooth Double and Triple Barriers

“…Successive application of the parabolic connection formulae (in the notation of equations (4.9) and (4.10) of [17]) to local single barriers in a given double-barrier structure, results in the following expression for the transmission amplitude t k :…”

“…Triple barriers are investigated in Section 3. The derivation has been made by using the parabolic connection formulae [13,17]. The results are applicable to optical waveguides directly [18].…”