Level comparison theorems

Abstract

Two theorems are established which state that in certain circumstances the nth eigenvalue of one Schrödinger operator is higher or lower than the ( n − 1)th eigenvalue of a second Schrödinger operator. One condition is the positivity of the difference of the two potentials and a certain behavior of this difference at the boundary. The other condition is in each theorem a differential inequality of the second order on one of the potentials. The proof involves a comparsion of the second logarithmic derivative of wave functions. By specifying the difference of the two potentials, specific forms of the theorems give new theorems on level spacings as well as old and new theorems on level ordering. A third theorem deals with perturbations of factorizable systems. Here explicit formulas connect the level splittings again with second order differential expressions of the perturbing potential.