Derivation and reflection properties of a transmission-free absorbing potential

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Abstract
The criterion for the validity of the JWKB approximation is used to derive a first order differential equation for a negative imaginary absorbing potential, −iε(r). The resulting absorbing potential switches on as rapidly as possible with increasing r without causing too much reflection, and it has a second order pole at the end of the absorbing region which eliminates all transmission. It is completely specified by a single physical parameter: the minimum energy Emin at which absorption is required; the width of the absorbing region can be chosen to correspond to the de Broglie wavelength at this minimum energy. With this choice, the exact quantum mechanical reflection probability of the potential is <1% at E=Emin, <0.1% at E=2Emin, and <0.01% for all E⩾3Emin. Because of its lack of parameters and its convenient reflection properties, we anticipate that the new potential (and other potentials of a similar form) will find useful application in both time-dependent and time-independent quantum scattering calculations.