Lower Bounds and Isoperimetric Inequalities for Eigenvalues of the Schrödinger Equation

Abstract

The potential which minimizes the lowest eigenvalue of the one-dimensional Schrödinger equation is determined among all potentials V for which the integral of Vn has the prescribed value k. For each value of n and k this potential is found to be a special case of the Epstein-Eckart potentials which were originally introduced because the Schrödinger equation for them could be solved explicitly. The minimum eigenvalue is determined and it provides a lower bound on the lowest eigenvalue of any potential for which ∫Vndx=k. The expression of this fact as an inequality yields an isoperimetric inequality. For an arbitrary potential, each value of n provides one lower bound on the lowest eigenvalue, the largest of which is the best. This best bound is determined for the square well, the exponential, and the inverse power potentials. In the case of the square well, it is compared with the exact value. In the limiting case n = 1 our result reduces to that previously obtained by Larry Spruch, who showed that the delta function has the minimum lowest eigenvalue among all potentials of given ``area.''