The perturbation of some exactly soluble problems in wave mechanics by the method of potential envelopes

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Abstract
Suppose the function epsilon k( nu ) which represents the kth bound-state eigenvalue of the Hamiltonian h=(- Delta + nu phi (r)+U(r)) is known exactly for all allowed values of nu >or=0. The authors present the corresponding eigenvalue Ek( nu ) of the Hamiltonian H=(- Delta + nu f( phi (r))+U(r)), where f( phi ) is a smooth, increasing, and either convex or concave transformation of the potential phi (r). An application of the method of potential envelopes yields a simple formula for an upper or lower bound to Ek( nu ) according to whether the transformation f( phi ) is concave or convex. The example phi (r)=(-r-1+ omega r), U(r)= omega 2r2, and f( phi )= lambda -1(elambda phi -1) for lambda >0 is discussed in detail.

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