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Stationary 1D Schr\"odinger equations with polynomial potentials are reducedto explicit countable closed systems of exact quantization conditions, whichare selfconsistent constraints upon the zeros of zeta-regularized spectraldeterminants, complementing the usual asymptotic (Bohr--Sommerfeld)constraints. (This reduction is currently completed under a certain vanishingcondition.) In particular, the symmetric quartic oscillators are admissiblesystems, and the formalism is tested upon them. Enforcing the exact andasymptotic constraints by suitable iterative schemes, we numerically observegeometric convergence to the correct eigenvalues/functions in some test cases,suggesting that the output of the reduction should define a contractivefixed-point problem (at least in some vicinity of the pure $q^4$ case).Comment: flatex text.tex, 4 file

Exact resolution method for general 1D polynomial Schrödinger equation