Stability estimates for the lowest eigenvalue of a Schrödinger operator

There is a family of potentials that minimize the lowest eigenvalue of a Schrödinger eigenvalue under the constraint of a given L^p norm of the potential. We give effective estimates for the amount by which the eigenvalue increases when the potential is not one of these optimal potentials. Our results are analogous to those for the isoperimetric problem and the Sobolev inequality. We also prove a stability estimate for Hölder's inequality, which we believe to be new.