1,993 research outputs found
An exact solution method for 1D polynomial Schr\"odinger equations
Stationary 1D Schr\"odinger equations with polynomial potentials are reduced
to explicit countable closed systems of exact quantization conditions, which
are selfconsistent constraints upon the zeros of zeta-regularized spectral
determinants, complementing the usual asymptotic (Bohr--Sommerfeld)
constraints. (This reduction is currently completed under a certain vanishing
condition.) In particular, the symmetric quartic oscillators are admissible
systems, and the formalism is tested upon them. Enforcing the exact and
asymptotic constraints by suitable iterative schemes, we numerically observe
geometric convergence to the correct eigenvalues/functions in some test cases,
suggesting that the output of the reduction should define a contractive
fixed-point problem (at least in some vicinity of the pure case).Comment: flatex text.tex, 4 file
From exact-WKB towards singular quantum perturbation theory
We use exact WKB analysis to derive some concrete formulae in singular
quantum perturbation theory, for Schr\"odinger eigenvalue problems on the real
line with polynomial potentials of the form , where
even, and . Mainly, we establish the limiting forms of global
spectral functions such as the zeta-regularized determinants and some spectral
zeta functions.Comment: latex text.tex, 3 files, 2 figures, 14 pages
http://www-spht.cea.fr/articles/T03/192 [SPhT-T03/192], submitted to Publ.
RIMS, Kyoto Univ. (special 40th anniversary issue
Simplifications of the Keiper/Li approach to the Riemann Hypothesis
The Keiper/Li constants are asymptotically () sensitive to the Riemann Hypothesis, but highly elusive
analytically and difficult to compute numerically. We present quite explicit
variant sequences that stay within the abstract Keiper--Li frame, and appear
simpler to analyze and compute.Comment: 21 pages, 6 figure
"Exact WKB integration'' of polynomial 1D Schr\"odinger (or Sturm-Liouville) problem
We review an "exact semiclassical" resolution method for the general
stationary 1D Schr\"odinger equation with a polynomial potential. This method
avoids having to compute any Stokes phenomena directly; instead, it basically
relies on an elementary Wronskian identity, and on a fully exact form of
Bohr--Sommerfeld quantization conditions which can also be viewed as a
Bethe-Ansatz system of equations that will "solve" the general polynomial 1D
Schr\"odinger problem.Comment: latex txt12.tex, 4 files, 3 figures, 18 pages Differential equations
and Stokes phenomenon Groningen, The Netherlands May 28-30 2001
[SPhT-T01/146
Zeta-regularisation for exact-WKB resolution of a general 1D Schr\"odinger equation
We review an exact analytical resolution method for general one-dimensional
(1D) quantal anharmonic oscillators: stationary Schr\"odinger equations with
polynomial potentials. It is an exact form of WKB treatment involving spectral
(usual) vs "classical" (newer) zeta-regularisations in parallel. The central
results are a set of Bohr--Sommerfeld-like but exact quantisation conditions,
directly drawn from Wronskian identities, and appearing to extend Bethe-Ansatz
formulae of integrable systems. Such exact quantisation conditions do not just
select the eigenvalues; some evaluate the spectral determinants, and others the
wavefunctions, for the spectral parameter in general position.Comment: 17 pages, 2 figures. V2: minor amendments throughout, with one typo
corrected in eq.(19
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