404 research outputs found
Weakly bound states in heterogeneous waveguides: a calculation to fourth order
We have extended a previous calculation of the energy of a weakly
heterogeneous waveguide to fourth order in the density perturbation, deriving
its general expression. For particular configurations where the second and
third orders both vanish, we discover that the fourth order contribution lowers
in general the energy of the state, below the threshold of the continuum. In
these cases the waveguide possesses a localized state. We have applied our
general formula to a solvable model with vanishing second and third orders
reproducing the exact expression for the fourth order.Comment: 20 pages, one table, two figures; minor changes with respect to the
first versio
Convergence acceleration of series through a variational approach
By means of a variational approach we find new series representations both
for well known mathematical constants, such as and the Catalan constant,
and for mathematical functions, such as the Riemann zeta function. The series
that we have found are all exponentially convergent and provide quite useful
analytical approximations. With limited effort our method can be applied to
obtain similar exponentially convergent series for a large class of
mathematical functions.Comment: 19 pages, 9 figures, 1 table Accepted for publication on the Journal
of Mathematical Analysis and Application
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