An isoperimetric type inequality for the principal eigenvalue of Schrödinger operators depending on the curvature of a loop

Let κ be the curvature of a smooth closed curve of length 2π. The 2π  -periodic eigenvalue problems u″−gκ2u+λu=0u″−gκ2u+λu=0, where g is a real parameter, are considered and upper and lower bounds for their principal eigenvalues are obtained. The bounds are of isoperimetric type and characterize uniquely the curve in the case where equality occurs.