No abstract uploaded!

Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant$L$_{0,3}which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension$d$≥3) for the quotient$L$_{0,d}/L^{cl}_{0,d}is the so-called classical constant. This gives some improvement in large dimensions.

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold of$T$^{*}$R$^{$n$+1}-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.

No abstract uploaded!

We give a concrete and surprisingly simple characterization of compact sets $ K \subset \mathbb{R}^{{2 \times 2}} $ for which families of approximate solutions to the inclusion problem$Du$∈$K$are compact. In particular our condition is algebraic and can be tested algorithmically. We also prove that the quasiconvex hull of compact sets of 2 × 2 matrices can be localized. This is false for compact sets in higher dimensions in general.