Sharp Lieb-Thirring inequalities in high dimensions

. We show how a matrix version of the Buslaev-FaddeevZakharov trace formulae for a one-dimensional Schrodinger operator leads to Lieb-Thirring inequalities with sharp constants L cl fl;d with fl 3=2 and arbitrary d 1. 0. INTRODUCTION Let us consider a Schrodinger operator in L 2 (R d ) -\\Delta + V ; (0.1) where V is a real-valued function. In [22] Lieb and Thirring proved that if fl ? max(0; 1 - d=2), then there exist universal constants L fl;d satisfying 1 tr (-\\Delta + V) fl - L fl;d Z R d V fl+ d 2 - (x) dx : (0.2) In the critical case d 3 and fl = 0 the bound (0.2) is known as the CwikelLieb -Rozenblum (CLR) inequality, see [7, 19, 24] and also [6, 18]. For the remaining case d = 1, fl = 1=2 the estimate (0.2) has been verified in [26], see also [13]. On the other hand it is known that (0.2) fails for fl = 0 if d = 2 and for 0 fl ! 1=2 if d = 1. If V 2 L fl+ d 2 (R d ), then the inequalities (0.2) are accompanied by the Weyl type asymptotic formula lim f...

Hali tarjima qilinmagan