135 research outputs found
Using Spectral Method as an Approximation for Solving Hyperbolic PDEs
We demonstrate an application of the spectral method as a numerical
approximation for solving Hyperbolic PDEs. In this method a finite basis is
used for approximating the solutions. In particular, we demonstrate a set of
such solutions for cases which would be otherwise almost impossible to solve by
the more routine methods such as the Finite Difference Method. Eigenvalue
problems are included in the class of PDEs that are solvable by this method.
Although any complete orthonormal basis can be used, we discuss two
particularly interesting bases: the Fourier basis and the quantum oscillator
eigenfunction basis. We compare and discuss the relative advantages of each of
these two bases.Comment: 19 pages, 14 figures. to appear in Computer Physics Communicatio
Massive Jackiw-Rebbi Model
In this paper we analyze a generalized Jackiw-Rebbi (J-R) model in which a
massive fermion is coupled to the kink of the model as a
prescribed background field. We solve this massive J-R model exactly and
analytically and obtain the whole 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 () and its slope at (). The
graph of the energies as a function of , 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 larger than a
critical value . This is in sharp contrast to the
usual (massless) J-R model where the zero mode and hence the fermion number
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 , which is zero in the massless J-R
model.Comment: 10 pages, 3 figure
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