then Ïƒac(âˆ’âˆ† + V ) = R. It was proved for the one-dimensional case by Deift and Killip [5] (see also [11, 18, 6]). For some Dirac operators, this conjecture was shown to be true for d = 1 by M. Kreinâ€¦ (More)

Let pn(x) be the orthonormal polynomials associated to a measure dÎ¼ of compact support in R . If E / âˆˆ supp(dÎ¼), we show there is a Î´ > 0 so that for all n, either pn or pn+1 has no zeros in (E âˆ’ Î´,â€¦ (More)

For two-dimensional Euler equation on the torus, we prove that the Lâˆž norm of the gradient can grow superlinearly for some infinitely smooth initial data. We also show the exponential growth of theâ€¦ (More)

We prove Rakhmanovâ€™s theorem for Jacobi matrices without the additional assumption that the number of bound states is finite. This result solves one of Nevaiâ€™s open problems. Consider a measure dÎ¼â€¦ (More)

We present general principles for the preservation of a.c. spectrum under weak perturbations. The SchrÃ¶dinger operators on the strip and on the Caley tree (Bethe lattice) are considered. In thisâ€¦ (More)

We prove that the massless Dirac operator in R with long-range potential has an a.c. spectrum which fills the whole real line. The Dirac operators with matrix-valued potentials are considered as well.

For the two-dimensional Euler equation on the torus, we prove that the Lâˆžâ€“norm of the vorticity gradient can grow as double exponential over arbitrary long but finite time provided that at time zeroâ€¦ (More)

Preface In the recent years, the theory of orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC) enjoyed the considerable development. In these lecture notes, we will explainâ€¦ (More)

Conjecture 0.1. Do the following conditions: vn â†’ 0 and vn+q âˆ’ vn âˆˆ l (q âˆˆ Zâ€“ fixed) guarantee that Ïƒac(J) = [âˆ’2, 2]? The symbol Ïƒac(J) conventionally denotes the absolutely continuous (a.c.)â€¦ (More)