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On demontre la borne superieure optimale λ 2 /λ 1 ≤4 pour le rapport des deux premieres valeurs propres des operateurs de Schrodinger a 1 dimension a potentiels non negatifs. L'egalite est vraie si et seulement si le potentiel s'annule identiquement

Best constant for the ratio of the first two eigenvalues of one-dimensional Schrödinger operators with positive potentials