800 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
"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
Exact resolution method for general 1D polynomial Schr\"odinger equation
The stationary 1D Schr\"odinger equation with a polynomial potential
of degree N is reduced to a system of exact quantization conditions of
Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations
pairing spectral determinants of (N+2) generically distinct operators, all the
transforms of one quantum Hamiltonian under a cyclic group of complex scalings.
The determinants' zeros define (N+2) semi-infinite chains of points in the
complex spectral plane, and they encode the original quantum problem. Each
chain can now be described by an exact quantization condition which constrains
it in terms of its neighbors, resulting in closed equilibrium conditions for
the global chain system; these are supplemented by the standard
(Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each
chain asymptotically. This reduced problem is then probed numerically for
effective solvability upon test cases (mostly, symmetric quartic oscillators):
we find that the iterative enforcement of all the quantization conditions
generates discrete chain dynamics which appear to converge geometrically
towards the correct eigenvalues/eigenfunctions. We conjecture that the exact
quantization then acts by specifying reduced chain dynamics which can be stable
(contractive) and thus determine the exact quantum data as their fixed point.
(To date, this statement is verified only empirically and in a vicinity of
purely quartic or sextic potentials .)Comment: flatex text.tex, 4 files Submitted to: J. Phys. A: Math. Ge
Zeta functions over zeros of general zeta and -functions
We describe in detail three distinct families of generalized zeta functions
built over the (nontrivial) zeros of a rather general arithmetic zeta or
L-function, extending the scope of two earlier works that treated the Riemann
zeros only. Explicit properties are also displayed more clearly than before.
Several tables of formulae cover the simplest concrete cases: L-functions for
real primitive Dirichlet characters, and Dedekind zeta functions.Comment: latex ell.tex, 1 file, 26 pages, 7 tables; in: Proceedings of The
International Symposium on Zeta Functions, Topology and Quantum Physics (ZTQ
2003) Osaka, Japan March 3-6 200
Exercises in exact quantization
The formalism of exact 1D quantization is reviewed in detail and applied to
the spectral study of three concrete Schr\"odinger Hamiltonians [-\d^2/\d q^2
+ V(q)]^\pm on the half-line , with a Dirichlet (-) or Neumann (+)
condition at q=0. Emphasis is put on the analytical investigation of the
spectral determinants and spectral zeta functions with respect to singular
perturbation parameters. We first discuss the homogeneous potential
as vs its (solvable) limit (an infinite square well):
useful distinctions are established between regular and singular behaviours of
spectral quantities; various identities among the square-well spectral
functions are unraveled as limits of finite-N properties. The second model is
the quartic anharmonic oscillator: its zero-energy spectral determinants
\det(-\d^2/\d q^2 + q^4 + v q^2)^\pm are explicitly analyzed in detail,
revealing many special values, algebraic identities between Taylor
coefficients, and functional equations of a quartic type coupled to asymptotic
properties of Airy type. The third study addresses the
potentials of even degree: their zero-energy spectral
determinants prove computable in closed form, and the generalized eigenvalue
problems with v as spectral variable admit exact quantization formulae which
are perfect extensions of the harmonic oscillator case (corresponding to N=2);
these results probably reflect the presence of supersymmetric potentials in the
family above.Comment: latex txt.tex, 2 files, 34 pages [SPhT-T00/078]; v2: corrections and
updates as indicated by footnote
Anharmonic Oscillators, Spectral Determinant and Short Exact Sequence of affine U_q(sl_2)
We prove one of conjectures, raised by Dorey and Tateo in the connection
among the spectral determinant of anharmonic oscillator and vacuum eigenvalues
of transfer matrices in field theory and statistical mechanics. The exact
sequence of plays a fundamental role in the proof.Comment: 8 pages, Latex 2
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