Alternatives to Bazley’s special choice for eigenvalue lower bounds

Bazley’s special choice operator is a lesser operator to a positive perturbation of a self-adjoint semi-bounded operator that possesses an exactly soluble base eigenvalue problem. It allows the construction of an exactly soluble intermediate problem that gives eigenvalues not less than the base problem and not greater than the perturbed problem so that lower bounds to the eigenvalues of the perturbed operator are produced. This paper considers alternate derivations of Bazley’s special choice which lead to two alternate methods to determine eigenvalue lower bounds. One is simpler, but gives poorer bounds; the other is more difficult, but sometimes yields superior bounds. Lower bounds to the particle in a box model with a linear perturbation and lower bounds to the helium atom are calculated using the two methods introduced and are compared to those given with Bazley’s special choice.