Distant perturbations of the Laplacian in a multi-dimensional space

We consider the Laplacian in $\mathbb{R}^n$ perturbed by a finite number of distant perturbations those are abstract localized operators. We study the asymptotic behaviour of the discrete spectrum as the distances between perturbations tend to infinity. The main results are the convergence theorem and the asymptotics expansions for the eigenelements. Some examples of the possible distant perturbations are given; they are potential, second order differential operator, magnetic Schrodinger operator, integral operator, and \d-potential

Maximal lengths of exceptional collections of line bundles

In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King's conjecture for toric Fano varieties. More generally, we prove that for any constant $c>\frac34$ there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than c\rk K_0(Y). To obtain varieties without exceptional collections of line bundles, it suffices to put $c=1.$ On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least \frac34 \rk K_0(Y). The constant $\frac34$ is thus maximal with this property.Comment: 27 pages, no figures; misprints and typos corrected, an arithmetic mistake in the proof of Theorem 6.2 corrected, consequently Theorem 6.3 slightly modified, new Lemma 4.4 added, description of the constructed varieties extended, references adde