In this paper we analyze a generalized Jackiw-Rebbi (J-R) model in which a massive fermion is coupled to the kink of the $lambdaphi^4$ model as a prescribed background field. We solve this massive J-R model exactly and analytically and obtain the who
le spectrum of the fermion, including the bound and continuum states. The mass term of the fermion makes the potential of the decoupled second order Schrodinger-like equations asymmetric in a way that their asymptotic values at two spatial infinities are different. Therefore, we encounter the unusual problem in which two kinds of continuum states are possible for the fermion: reflecting and scattering states. We then show the energies of all the states as a function of the parameters of the kink, i.e. its value at spatial infinity ($theta_0$) and its slope at $x=0$ ($mu$). The graph of the energies as a function of $theta_0$, where the bound state energies and the two kinds of continuum states are depicted, shows peculiar features including an energy gap in the form of a triangle where no bound states exist. That is the zero mode exists only for $theta_0$ larger than a critical value $(theta_0^{textrm{c}})$. This is in sharp contrast to the usual (massless) J-R model where the zero mode and hence the fermion number $pm1/2$ for the ground state is ever present. This also makes the origin of the zero mode very clear: It is formed from the union of the two threshold bound states at $theta_0^{textrm{c}}$, which is zero in the massless J-R model.
In this paper we present a complete and exact spectral analysis of the $(1+1)$-dimensional model that Jackiw and Rebbi considered to show that the half-integral fermion numbers are possible due to the presence of an isolated self charge conjugate zer
o mode. The model possesses the charge and particle conjugation symmetries. These symmetries mandate the reflection symmetry of the spectrum about the line $E=0$. We obtain the bound state energies and wave functions of the fermion in this model using two different methods, analytically and exactly, for every arbitrary choice of the parameters of the kink, i.e. its value at spatial infinity ($theta_0$) and its scale of variations ($mu$). Then, we plot the bound state energies of the fermion as a function of $theta_0$. This graph enables us to consider a process of building up the kink from the trivial vacuum. We can then determine the origin and evolution of the bound state energy levels during this process. We see that the model has a dynamical mass generation process at the first quantized level and the zero-energy fermionic mode responsible for the fractional fermion number, is always present during the construction of the kink and its origin is very peculiar, indeed. We also observe that, as expected, none of the energy levels crosses each other. Moreover, we obtain analytically the continuum scattering wave functions of the fermion and then calculate the phase shifts of these wave functions. Using the information contained in the graphs of the phase shifts and the bound states, we show that our phase shifts are consistent with the weak and strong forms of the Levinson theorem. Finally, using the weak form of the Levinson theorem, we confirm that the number of the zero-energy fermionic modes is exactly one.
