Refining the comparison theorem of quantum mechanics

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Abstract
If two potentials V1(x) and V2(x) are ordered V12, then the comparison theorem tells the author that the corresponding Schrodinger eigenvalues are ordered E12. He presents some simple conditions which allow him to predict such spectral ordering for the ground state, even when the graphs of the potentials cross over. As illustrations, the truncated quartic oscillator and the Yukawa potential are studied. By allowing Coulomb 'tangents' to cross over the Yukawa in various ways, he is able to improve earlier energy upper bounds which he had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.