We study controllability for a nonhomogeneous string and ring under an axial stretchingtension that varies with time. We consider the boundary control for a string anddistributed control for a ring. For a string, we are looking for a controlf(t) ∈ L 2(0,T) that drives the state solution to rest. We show that for a ring, two forcesare required to achieve controllability. The controllability problem is reduced to amoment problem for the control. We describe the set of initial data which may be driven torest by the control. The proof is based on an auxiliary basis property result.

We study discrete spectrum in spectral gaps of an elliptic periodic second orderdifferential operator in L 2(ℝd )perturbed by a decaying potential. It is assumed that a perturbation is nonnegative andhas a power-like behavior at infinity. We find asymptotics in the large coupling constantlimit for the number of eigenvalues of the perturbed operator that have crossed a givenpoint inside the gap or the edge of the gap. The corresponding asymptotics is power-likeand depends on the observation point.

In this paper, we review some of our recent results in the study of the dynamics ofinteracting Bose gases in the Gross-Pitaevskii (GP) limit. Our investigations focus on thewell-posedness of the associated Cauchy problem for the infinite particle system describedby the GP hierarchy.

We provide a systematic study of boundary data maps, that is, 2 × 2 matrix-valuedDirichlet-to-Neumann and more generally, Robin-to-Robin maps, associated withone-dimensional Schrödinger operators on a compact interval [0, R] withseparated boundary conditions at 0 and R. Most of our results areformulated in the non-self-adjoint context.

Our principal results include explicit representations of these boundary data maps interms of the resolvent of the underlying Schrödinger operator and the associated boundarytrace maps, Krein-type resolvent formulas relating Schrödinger operators corresponding todifferent (separated) boundary conditions, and a derivation of the Herglotz property ofboundary data maps (up to right multiplication by an appropriate diagonal matrix) in thespecial self-adjoint case.

In this paper, we consider one-dimensional wave equation with real-valued square-summablepotential. We establish the long-time asymptotics of solutions by, first, studying thestationary problem and, second, using the spectral representation for the evolutionequation. In particular, we prove that part of the wave travels ballistically ifq ∈ L 2(ℝ+) and this result issharp.

In this short note, we apply the technique developed in [Math. Model. Nat. Phenom., 5 (2010), No. 4, 122-149] to study the long-time evolution for Schrödinger equation with slowly decayingpotential.

We classify the hulls of different limit-periodic potentials and show that the hull of alimit-periodic potential is a procyclic group. We describe how limit-periodic potentialscan be generated from a procyclic group and answer arising questions. As an expositorypaper, we discuss the connection between limit-periodic potentials and profinite groups ascompletely as possible and review some recent results on Schrödinger operators obtained inthis context.

We consider the zigzag half-nanotubes (tight-binding approximation) in a uniform magneticfield which is described by the magnetic Schrödinger operator with a periodic potentialplus a finitely supported perturbation. We describe all eigenvalues and resonances of thisoperator, and theirs dependence on the magnetic field. The proof is reduced to theanalysis of the periodic Jacobi operators on the half-line with finitely supportedperturbations.

We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We focus onphenomena related to unboundedness of the Laplacians. This includes (failure of) essentialselfadjointness, absence of essential spectrum and stochastic incompleteness.

We review some recent results for a class of fluid mechanics equations called activescalars, with fractional dissipation. Our main examples are the surface quasi-geostrophicequation, the Burgers equation, and the Cordoba-Cordoba-Fontelos model. We discussnonlocal maximum principle methods which allow to prove existence of global regularsolutions for the critical dissipation. We also recall what is known about the possibilityof finite time blow up in the supercritical regime.

In this note, I wish to describe the first order semiclassical approximation to thespectrum of one frequency quasi-periodic operators. In the case of a sampling functionwith two critical points, the spectrum exhibits two gaps in the leading orderapproximation. Furthermore, I will give an example of a two frequency quasi-periodicoperator, which has no gaps in the leading order of the semiclassical approximation.

This paper is related to the spectral stability of traveling wave solutions of partialdifferential equations. In the first part of the paper we use the Gohberg-Rouche Theoremto prove equality of the algebraic multiplicity of an isolated eigenvalue of an abstractoperator on a Hilbert space, and the algebraic multiplicity of the eigenvalue of thecorresponding Birman-Schwinger type operator pencil. In the second part of the paper weapply this result to discuss three particular classes of problems: the Schrödingeroperator, the operator obtained by linearizing a degenerate system of reaction diffusionequations about a pulse, and a general high order differential operator. We studyrelations between the algebraic multiplicity of an isolated eigenvalue for the respectiveoperators, and the order of the eigenvalue as the zero of the Evans function for thecorresponding first order system.

We study the number of lattice points in ℝd ,d ≥ 2, lying inside an annulus as a function of the centre of theannulus. The average number of lattice points there equals the volume of the annulus, andwe study the L 1 and L 2 norms ofthe remainder. We say that a dimension is critical, if these norms do not have upper andlower bounds of the same order as the radius goes to infinity. In [Duke Math. J., 107 (2001), No. 2, 209–238], it was proved that in the case of the ball (instead of an annulus)the critical dimensions are d ≡ 1 mod 4. We show that the behaviour ofthe width of an annulus as a function of the radius determines which dimensions arecritical now. In particular, if the width is bounded away from zero and infinity, thecritical dimensions are d ≡ 3 mod 4; if the width goes to infinity, butslower than the radius, then all dimensions are critical, and if the width tends to zeroas a power of the radius, then there are no critical dimensions.

We present a user-friendly version of a double operator integration theory which stillretains a capacity for many useful applications. Using recent results from the lattertheory applied in noncommutative geometry, we derive applications to analogues of theclassical Heinz inequality, a simplified proof of a famous inequality ofBirman-Koplienko-Solomyak and also to the Connes-Moscovici inequality. Our methods aresufficiently strong to treat these inequalities in the setting of symmetric operator normsin general semifinite von Neumann algebras.

A general theorem on the GBDT version of the Bäcklund-Darboux transformation for systemsdepending rationally on the spectral parameter is treated and its applications tononlinear equations are given. Explicit solutions of direct and inverse problems forDirac-type systems, including systems with singularities, and for the system auxiliary tothe N-wave equation are reviewed. New results on explicit construction ofthe wave functions for radial Dirac equation are obtained.

In L 2(ℝd ;ℂn ), we consider a wide class of matrix elliptic secondorder differential operators $\mathcal{A}$ εwith rapidly oscillating coefficients (depending on x/ε).For a fixed τ > 0 and small ε > 0, we findapproximation of the operator exponential exp(− $\mathcal{A}$ ε τ) in the(L 2(ℝd ;ℂn ) →H 1(ℝd ;ℂn ))-operator norm with an error term of orderε. In this approximation, the corrector is taken into account. Theresults are applied to homogenization of a periodic parabolic Cauchy problem.

We prove the instability of threshold resonances and eigenvalues of the linearized NLSoperator. We compute the asymptotic approximations of the eigenvalues appearing from theendpoint singularities in terms of the perturbations applied to the original NLS equation.Our method involves such techniques as the Birman-Schwinger principle and the Feshbachmap.