On the Maximum and Minimum of Some Functionals for the Eigenvalue Problem of Sturm-Liouville Type

Let λn(q) be the nth eigenvalue of the Sturm-Liouville equation y′′+(λ−q(x))y=0, y(−l/2) = y(l/2) = 0. With certain restrictions on the class of functions q we determine the shapes of the solutions of the extremal problems for the functionals λn(q) and the minimum problem for the functional λ2(q) − λ1(q).