Connection between the Lieb-Thirring conjecture for Schrödinger operators and an isoperimetric problem for ovals on the plane. (English) Zbl 1087.81042
Conca, Carlos (ed.) et al., Partial differential equations and inverse problems. Proceedings of the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems, Santiago, Chile, January 6–18, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3448-7/pbk). Contemporary Mathematics 362, 53-61 (2004).
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].
For the entire collection see [Zbl 1052.35004].
Reviewer: Heinz Siedentop (München)
MSC:
81T10 | Model quantum field theories |
53A04 | Curves in Euclidean and related spaces |
35J10 | Schrödinger operator, Schrödinger equation |
34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |