REFINING THE COMPARISON THEOREM OF QUANTUM-MECHANICS

If two potentials V1 (x) and V2(x) are ordered V1 < V2, then the comparison theorem tells us that the corresponding Schrodinger eigenvalues are ordered E1 < E2. We present some simple conditions which allow us 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, we are able to improve earlier energy upper bounds which we had obtained by representing the Yukawa potential as a smooth transformation of the Coulomb potential, and also to augment these results with lower bounds.