613 research outputs found
Approximate closed-form formulas for the zeros of the Bessel Polynomials
We find approximate expressions x(k,n) and y(k,n) for the real and imaginary
parts of the kth zero z_k=x_k+i y_k of the Bessel polynomial y_n(x). To obtain
these closed-form formulas we use the fact that the points of well-defined
curves in the complex plane are limit points of the zeros of the normalized
Bessel polynomials. Thus, these zeros are first computed numerically through an
implementation of the electrostatic interpretation formulas and then, a fit to
the real and imaginary parts as functions of k and n is obtained. It is shown
that the resulting complex number x(k,n)+i y(k,n) is O(1/n^2)-convergent to z_k
for fixed kComment: 9 pages, 2 figure
Finite-size effects on a lattice calculation
We study in this paper the finite-size effects of a non-periodic lattice on a
lattice calculation. To this end we use a finite lattice equipped with a
central difference derivative with homogeneous boundary conditions to calculate
the bosonic mass associated to the Schwinger model. We found that the
homogeneous boundary conditions produce absence of fermion doubling and chiral
invariance, but we also found that in the continuum limit this lattice model
does not yield the correct value of the boson mass as other models do. We
discuss the reasons for this and, as a result, the matrix which cause the
fermion doubling problem is identified.Comment: 8 pages, no figures, extended version, five references adde
Aliasing modes in the lattice Schwinger model
We study the Schwinger model on a lattice consisting of zeros of the Hermite
polynomials that incorporates a lattice derivative and a discrete Fourier
transform with many properties. Such a lattice produces a Klein-Gordon equation
for the boson field and the exact value of the mass in the asymptotic limit if
the boundaries are not taken into account. On the contrary, if the lattice is
considered with boundaries new modes appear due to aliasing effects. In the
continuum limit, however, this lattice yields also a Klein-Gordon equation with
a reduced mass.Comment: Enlarged version, 1 figure added, 11 page
Lattice calculations on the spectrum of Dirac and Dirac-K\"ahler operators
We present a matrix technique to obtain the spectrum and the analytical index
of some elliptic operators defined on compact Riemannian manifolds. The method
uses matrix representations of the derivative which yield exact values for the
derivative of a trigonometric polynomial. These matrices can be used to find
the exact spectrum of an elliptic operator in particular cases and in general,
to give insight into the properties of the solution of the spectral problem. As
examples, the analytical index and the eigenvalues of the Dirac operator on the
torus and on the sphere are obtained and as an application of this technique,
the spectrum of the Dirac-Kahler operator on the sphere is explored.Comment: 11 page
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