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Quartic anharmonic oscillator with potential V(x)= x(2) + g(2)x4
was the first non-exactly-solvable problem tackled by the
newly-written Schroedinger equation in 1926. Since that time
thousands of articles have been published on the subject, mostly
about the domain of small g(2) (weak coupling regime), although
physics corresponds to g(2) ~ 1, and they were mostly about
energies.This book is focused on studying eigenfunctions as a
primary object for any g(2). Perturbation theory in g(2) for the
logarithm of the wavefunction is matched to the true semiclassical
expansion in powers of : it leads to locally-highly-accurate,
uniform approximation valid for any g(2) [0, ) for eigenfunctions
and even more accurate results for eigenvalues. This method of
matching can be easily extended to the general anharmonic
oscillator as well as to the radial oscillators. Quartic, sextic
and cubic (for radial case) oscillators are considered in detail as
well as quartic double-well potential.
Today Lie group theoretical approach to differential equations has
been extended to new situations and has become applicable to the
majority of equations that frequently occur in applied sciences.
Newly developed theoretical and computational methods are awaiting
application. Students and applied scientists are expected to
understand these methods. Volume 3 and the accompanying software
allow readers to extend their knowledge of computational
algebra.
Written by the world's leading experts in the field, this
up-to-date sourcebook covers topics such as Lie-BAcklund,
conditional and non-classical symmetries, approximate symmetry
groups for equations with a small parameter, group analysis of
differential equations with distributions, integro-differential
equations, recursions, and symbolic software packages. The text
provides an ideal introduction to modern group analysis and
addresses issues to both beginners and experienced researchers in
the application of Lie group methods.
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