Scattering of a relativistic scalar particle by a cusp potential

Abstract

We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.PACS numbers: 03.65. Pm, 03.65.Nk, 03.65.Ge The study of low-momentum scattering of nonrelativistic particles by one-dimensional potentials is a well studied and understood problem [1]. Here we have that, as momentum goes to zero, the reflection coefficient goes to unity unless the potential V (x) supports a zero energy resonance. In this case the transmission coefficient goes to unity, becoming a transmission resonance [2]. Recently, this result has been generalized to the Dirac equation [3], showing that transmission resonances at k = 0 in the Dirac equation take place for a potential barrier V = V (x) when the corresponding potential well V = −V (x) supports a supercritical state. Kennedy [4] has shown that this result is also valid for a Woods-Saxon potential. More recently, transmission resonances and half-bound states have been discussed for a Dirac particle scattered by a cusp potential [5,6] as well as for a class of short range potentials [7]. The bound states for scalar relativistic particles satisfying the Klein-Gordon equation are qualitatively different from the previous case. Here, for short-range attractive potentials the Schiff-Snyder effect [8,9,10,11,12,13,14] takes place, i.e for a given potential strength two bound states appear, one with positive norm and another with negative norm. Such states can be associated with a particle-antiparticle creation process. No antiresonant states appear [11,12].The absence of resonant overcritical states for the Klein-Gordon equation in the presence of short-range potential interactions does not prevent the existence of transmission resonances for given values of the potential.Quantum effects associated with scalar particles in the presence of external potentials have been extensively discussed in the literature [10,14]. Among quantum effects, we have that transmission resonance is one of the most interesting phenomena. For given values of the energy and the proper choice of the shape of the effective barrier, the probability of transmission reaches a maximum such as that obtained in the study of superradiance [14], where the amplitude of the scattered solutions by a rotating Kerr black hole is even larger than the amplitude of the incident wave. Analogous phenomena can also be obtained due to the presence of strong electromagnetic potentials [15].Recently, transmission resonances for the Klein-Gordon equation in the presence of a Woods-Saxon potential barrier have been computed [16]. The transmission coefficient as a function of the energy and the potential am...

Transmission resonance for a Dirac particle in a one-dimensional Hulthén potential

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