On the Lieb-Thirring constants L_gamma,1 for gamma geq 1/2

Let $E_i(H)$ denote the negative eigenvalues of the one-dimensional Schr\"odinger operator $Hu:=-u^{\prime\prime}-Vu,\ V\geq 0,$ on $L_2({\Bbb R})$. We prove the inequality \sum_i|E_i(H)|^\gamma\leq L_{\gamma,1}\int_{\Bbb R} V^{\gamma+1/2}(x)dx, (1) for the "limit" case $\gamma=1/2.$ This will imply improved estimates for the best constants $L_{\gamma,1}$ in (1), as $1/2<\gamma<3/2.Comment: AMS-LATEX, 15 page

Quasi-regular asymptotic behavior of the solution of a singularly perturbed Cauchy problem for linear systems of differential matrix equations

The author constructs an asymptotic expansion in powers of epsilon of the solution on a finite interval of t of the singularly perturbed matrix Cauchy problem epsilon dot{Z}=A(t)Z+ZB(t),quad Z(0,epsilon)=Z_0, where A(t) and B(t) are real square matrices depending smoothly on t. It is assumed that the eigenvalues of the matrices are simple and have non-positive real parts. A norm estimate of the error term is given. An expansion is also given for the solution of the non-homogeneous Cauchy problem

Remarks on the Cwikel-Lieb-Rozenblum and Lieb-Thirring Estimates for Schrödinger Operators on Riemannian Manifolds

Let M be a general complete Riemannian manifold and consider a Schrödinger operator − + V on L2(M). We prove Cwikel-Lieb-Rozenblum as well as Lieb-Thirring type estimates for − + V . These estimates are given in terms of the potential and the heat kernel of the Laplacian on the manifold. Some of our results hold also for Schrödinger operators with complex-valued potentials