Eur. Phys. J. D 53, 283-288 (2009)
https://doi.org/10.1140/epjd/e2009-00140-7

Doubly uniform semiclassical quantization formula for resonances

¹ Dalarna University College, 78188 Borlänge, Sweden
² KTH-Mechanics, Royal Institute of Technology, 10044 Stockholm, Sweden

Corresponding author: ^a ket@mech.kth.se

Received: 12 December 2008
Revised: 18 March 2009
Published online: 1 May 2009

Abstract

The radial Schrödinger equation with an effective potential containing a single well and a single barrier is treated with an improved uniform semiclassical method. The improved quantization formula for complex energies (or resonances) contains a correction term that originates from a uniform treatment of the classically forbidden region near the origin in addition to the more familiar uniform treatment of the barrier region. In the present case the origin has a second-order pole, due to the centrifugal barrier potential term, and/or a Coulomb-type singularity, and these terms dominate the region inside the innermost classical turning point. Numerical results for first-order and third-order approximate complex resonance energies are compared with those of a standard (first- and third-order) barrier-uniform semiclassical method and also with those of `exact' numerical computations. The improved quantization formula provides results in significantly better agreement with the exact results as the angular momentum quantum number l approaches zero.