A Simple proof of the generalized Lieb-Thirring inequalities in one-space dimension

Abstract

The Lieb-Thirring inequalities give a sharp upper bound for the L p -norm of a function which is the pointwise sum of the squares of a finite orthonormal sequence of functions that are elements of a suitable Sobolev space [LT]. Originally proven for the functions defined on the whole n-dimensional Euclidean space, they were later extended to bounded domains and to suborthogonal sequences of functions [GMT]. Here, we present a simple proof of these inequalities for bounded intervals in one space dimension utilizing simple Sobolev inequalities and standard results from Hilbert space theory.