The authors establish a connection between the one-dimensional Lieb-Thirring inequality for the first moment and an isoperimetric inequality for closed smooth curves with positive curvature and length \(2\pi\). In particular, the problem of minimizing the sum of the negative eigenvalues of Schrödinger operators with two negative eigenvalues is rewritten as geometric problem. An essential input is a change of independent variables from \(x\) to \(\int_{-\infty}^x u_1(t)^2 +u_2(t)^2 \,dt\) (Macke transform) where \(u_1\) and \(u_2\) are the two orthonormal eigenfunctions of the one-dimensional Schrödinger operator under consideration. As second ingredient the dependent variables \(u_1\) and \(u_2\) are rewritten using spherical coordinates. A result is an improved bound on the Lieb-Thirring constant \(L_{1,1}\) for the case of Schrödinger operators with two negative bound states.
For the entire collection see [Zbl 1052.35004].