26 March 2001 Physics Letters A 281 (2001) 213–217 www.elsevier.nl/locate/pla Quasi-bound state in supersymmetric quantum mechanics Zafar Ahmed Nuclear Physics Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India Received 23 October 2000; received in revised form 23 January 2001; accepted 29 January 2001 Communicated by P.R. Holland Abstract We point out that shape-invariant supersymmetric partner potentials discussed so far do not admit quasi-bound states. We present non-shape-invariant partner potentials which are iso-spectral and admit a quasi-bound state. We also propose a supersymmetric WKB formula for calculating the width (life-time) of the quasi-bound state. A versatile (Ginocchio) potential is shown to expose an interesting limitation of semi-classical quantizations.  2001 Elsevier Science B.V. All rights reserved. A potential well attached to a finite side barrier of finite width or a double-humped well (DHW, Figs. 1(a) and 2), i.e., a well surrounded by two shoulder barriers of finite width may hold complex energy eigenvalues: E n - iΓ n /2. If E n <V S , where V S is the height of the shoulder barrier, these states are known as quasi- bound or metastable states with life time ¯ h/Γ n . Had the barrier been infinitely thick, these states would have been the stable (bound) states at energies E = E n . These potentials may hold other complex energy states above the barrier representing scattering resonances. The scattering resonances may also exist over a highly localized single well Fig. 1(b). However, it is the quasi-bound states which have been sitting at the heart of several physical processes such as alpha decay, nuclear fission, ionization of atoms and emission of electrons from a metal. The last two decades have witnessed a remarkable development in the super symmetry inspired quan- tum mechanics [1]. The fundamental aspects such as exact solvability of Schrödinger equation, phase- equivalent potentials, semiclassical quantization, scat- E-mail address: zahmed@apsara.barc.ernet.in (Z. Ahmed). Fig. 1. Depiction of three cases of Ginocchio potential, where x - and y -axes have been scaled. This potential is expressed as V(x) = (-λ 2 ν(ν + 1) + ((1 - λ 2 )/4)[2 - (7 - λ 2 )y 2 + 5(1 - λ 2 )y 4 ]) × (1 - y 2 ), with λ 2 x = tanh -1 (y) -  1 - λ 2 tanh -1 [y  1 - λ 2 ], -1  y  1 [6]. tering and tunneling, closely lying doublets in dou- ble wells, bound states embedded in positive energy continuum, Dirac equation etc. have all been revis- ited very extensively. An excellent review of these in- 0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII:S0375-9601(01)00084-6