9 research outputs found
Variational Perturbation Theory for Summing Divergent Non-Borel-Summable Tunneling Amplitudes
We present a method for evaluating divergent non-Borel-summable series by an
analytic continuation of variational perturbation theory. We demonstrate the
power of the method by an application to the exactly known partition function
of the anharmonic oscillator in zero spacetime dimensions. In one spacetime
dimension we derive the imaginary part of the ground state energy of the
anharmonic oscillator for {\em all negative values of the coupling constant
, including the nonanalytic tunneling regime at small-. As a highlight of
the theory we retrieve from the divergent perturbation expansion the action of
the critical bubble and the contribution of the higher loop fluctuations around
the bubble.Comment: Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of
paper (including all PS fonts) at
http://www.physik.fu-berlin.de/~kleinert/34
End-To-End Distribution Function Function of Stiff Polymers for all Persistence Lengths
We set up recursion relations for calculating all even moments of the
end-to-end distance of a Porod-Kratky wormlike chains in dimensions. From
these moments we derive a simple analytic expression for the end-to-end
distribution in three dimensions valid for all peristence lengths. It is in
excellent agreement with Monte Carlo data for stiff chains and goes properly
over into the Gaussian random-walk distributions for low stiffness.Comment: Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of
paper (including all PS fonts) at
http://www.physik.fu-berlin.de/~kleinert/345 Mathematica programs at
http://www.physik.fu-berlin.de/~kleinert/b5/pgm1
Perturbation Theory for Particle in a Box
Recently developed strong-coupling theory open up the possibility of treating
quantum-mechanical systems with hard-wall potentials via perturbation theory.
To test the power of this theory we study here the exactly solvable quantum
mechanics of a point particle in a one-dimensional box. Introducing an
auxiliary harmonic mass term , the ground-state energy E^{(0) can be
expanded perturbatively in powers of , where is the box size. The
removal of the infrared cutoff requires the resummation of the series at
infinitely strong coupling. We show that strong-coupling theory yields a
fast-convergent sequence of approximations to the well-known quantum-mechanical
energy E^{(0)= \pi ^2/2d^2.Comment: Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of
paper also at http://www.physik.fu-berlin.de/~kleinert/28
Perturbation Theory for Path Integrals of Stiff Polymers
The wormlike chain model of stiff polymers is a nonlinear -model in
one spacetime dimension in which the ends are fluctuating freely. This causes
important differences with respect to the presently available theory which
exists only for periodic and Dirichlet boundary conditions. We modify this
theory appropriately and show how to perform a systematic large-stiffness
expansions for all physically interesting quantities in powers of ,
where is the length and the persistence length of the polymer. This
requires special procedures for regularizing highly divergent Feynman integrals
which we have developed in previous work. We show that by adding to the
unperturbed action a correction term , we can calculate
all Feynman diagrams with Green functions satisfying Neumann boundary
conditions. Our expansions yield, order by order, properly normalized
end-to-end distribution function in arbitrary dimensions , its even and odd
moments, and the two-point correlation function
Dependence of Variational Perturbation Expansions on Strong-Coupling Behavior. Inapplicability of delta-Expansion to Field Theory
We show that in applications of variational theory to quantum field theory it
is essential to account for the correct Wegner exponent omega governing the
approach to the strong-coupling, or scaling limit. Otherwise the procedure
either does not converge at all or to the wrong limit. This invalidates all
papers applying the so-called delta-expansion to quantum field theory.Comment: Author Information under
http://www.physik.fu-berlin.de/~kleinert/institution.html . Latest update of
paper (including all PS fonts) at
http://www.physik.fu-berlin.de/~kleinert/34
New Optimization Methods for Converging Perturbative Series with a Field Cutoff
We take advantage of the fact that in lambda phi ^4 problems a large field
cutoff phi_max makes perturbative series converge toward values exponentially
close to the exact values, to make optimal choices of phi_max. For perturbative
series terminated at even order, it is in principle possible to adjust phi_max
in order to obtain the exact result. For perturbative series terminated at odd
order, the error can only be minimized. It is however possible to introduce a
mass shift in order to obtain the exact result. We discuss weak and strong
coupling methods to determine the unknown parameters. The numerical
calculations in this article have been performed with a simple integral with
one variable. We give arguments indicating that the qualitative features
observed should extend to quantum mechanics and quantum field theory. We found
that optimization at even order is more efficient that at odd order. We compare
our methods with the linear delta-expansion (LDE) (combined with the principle
of minimal sensitivity) which provides an upper envelope of for the accuracy
curves of various Pade and Pade-Borel approximants. Our optimization method
performs better than the LDE at strong and intermediate coupling, but not at
weak coupling where it appears less robust and subject to further improvements.
We also show that it is possible to fix the arbitrary parameter appearing in
the LDE using the strong coupling expansion, in order to get accuracies
comparable to ours.Comment: 10 pages, 16 figures, uses revtex; minor typos corrected, refs. adde
September 1999 Z. Physics Letters A 260 1999 182--189 www.elsevier.nlrlocaterphysleta Perturbation theory for particle in a box
Recently developed strong-coupling theory opens up the possibility of treating quantum-mechanical systems with hard-wall potentials via perturbation theory. To test the power of this theory we study here the exactly solvable quantum mechanics of a point particle in a one-dimensional box. Introducing an auxiliary harmonic frequency term v, the ground-state energy E Z0