An exact solution method for 1D polynomial Schrödinger equations

Details An exact solution method for 1D polynomial Schrödinger equations An exact solution method for 1D polynomial Schrödinger equations

1999-02-10

arxiv.org/abs/math-ph/9902016v1

Physics

Mathematical Physics

flatex text.tex, 4 files

Scientific paper

10.1088/0305-4470/32/32/311

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 $q^4$ case).

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