Calculation of upper and lower bounds for eigenvalues by a boundary condition method: The quartic oscillator, hydrogen molecule ion, and helium atom

A method for computing upper and lower bounds for bound‐state eigenvalues is described. It depends upon inserting trial eigenvalues and observing the nature of the misbehavior of the solutions which are obtained. This misbehavior will change as a sequence of trial eigenvalues crosses the true eigenvalue. Thus in some examples ψ may go to ψ as x → ∞ for E too small, and ψ → — ∞ for x → ∞ for E too large. The advantages of the method are that it gives upper and lower bounds of comparable accuracy, the wave equation need not be separable, and the method can be applied to excited states as well as ground states. So far, we have used series type solutions, so that no integrations or secular equations were required. Three examples are summarized in this paper: the quartic oscillator, the hydrogen molecule ion, and the helium atom.