Our journal "Nanosystems: Physics, Chemistry, Mathematics" is devoted to fundamental problems of physics, chemistry and mathematics concerning all aspects of nanosystems science. It considers both theoretical and experimental problems of physics and chemistry of nanosystems, including methods of their design and creation, studies of their structure and properties, behavior under external influences, and the possibility of use. We accept papers directly or conceptually related to the key properties of nanosystems. Nanotechnology has required the creation of new methods of mathematical modeling and mathematical physics, as well as the development of existing methods for their extension to the study of new objects, many of which were previously simply absent. The corresponding mathematical problems will be covered in our journal. The scope of the journal includes all areas of nano-sciences. Papers devoted to basic problems of physics, chemistry and mathematics inspired by nanosystems investigations are welcomed. Both theoretical and experimental works concerning the properties and behavior of nanosystems, problems of their creation and application, mathematical methods of nanosystem studies are considered. The journal publishes scientific reviews (up to 30 journal pages), research papers (up to 15 pages) and letters (up to 5 pages). All manuscripts are peer-reviewed. Authors are informed about the referee opinions and the Editorial decisions.
Current issue
PAPERS, PRESENTED AT THE CONFERENCE
We treat the Nonlinear Schr¨odinger equation (NLSE) on Metric graph. An approach developed earlier for NLSE on interval [14], is extended for star graph. Dirichlet boundary conditions are imposed at the ends of bonds are imposed, while continuity conditions are chosen at the vertex of graph.
In this paper, we study quantum star graphs with time-dependent bond lengths. Quantum dynamics are treated by solving Schrodinger equation with time-dependent boundary conditions given on graphs. The time-dependence of the average kinetic energy is analyzed. The space-time evolution of a Gaussian wave packet is treated for an harmonically breathing star graph.
This paper summarizes the main results of [1] for the spectral asymptotics of the damped wave equation. We define the notion of a high frequency abscissa, a sequence of eigenvalues with imaginary parts going to plus or minus infinity and real parts going to some real number. We give theorems on the number of such high frequency abscissas for particular conditions on the graph. We illustrate this behavior in two particular examples.
In this paper, we treat the inverse scattering problem for the Dirac equation on metric graphs. Using the known scattering data, we recover the potential in the Dirac equation. The Gel’fand-Levitan-Marchenko integral equation is derived and potential is explicitly obtained for the case of a primary star graph.
The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator. Instead of the angle variable operator, we introduce the complex position operator ˆZ = ˆx+iˆy and interpret the order parameter µ = ⟨ ˆZ⟩/⟨ ˆZ† ˆZ⟩ as a measure of certainty of the angle distribution. We prove the relation between the uncertainty of angular momentum and the angle order parameter. We also prove its generalizations and discuss experimental methods for testing these relations.
For the derivation of the dilute Bose–Einstein condensate density and its phase, we have developed the perturbative approach for the solution of the stationary state couple Gross–Pitaevskii hydrodynamic equations. The external disorder potential is considered as a small parameter in this approach. We have derived expressions for the total density, condensate density, condensate density depletion and superfluid velocity of the Bose–Einstein condensate in an infinite length ring with disorder potential having a general form. For the delta correlated disorder, the explicit analytical forms of these quantities (except the superfluid velocity) have been obtained.
We study kicked particle dynamics in a rectangular quantum billiard. The kicking potential is chosen as localized at the center of the billiard. The exact solution for the time-dependent Schr¨odinger equation for a single kicking period is derived. Using this solution, the time-dependence of the average kinetic energy and probability density as a function of spatial coordinates are computed. Different regimes for trapping of the particle in kicking area are analyzed. It is found that depending of the values of kicking parameters, the average kinetic energy can be a periodic or a monotonically growing function of time or can be suppressed. Such behavior is explained in terms of particle trapping regimes. Wave packet dynamics are also studied.
The motion of a quantum particle in a time-dependent circular billiard is studied on the basis of the Schr¨odinger equation with time-dependent boundary conditions. The cases of monotonically expanding (contracting), non-harmonically, harmonically breathing circles the case when billiard wall suddenly disappears are explored in detail. The exact analytical solutions for monotonically expanding and contracting circles are obtained. For all cases, the time-dependence of the quantum average energy is calculated. It is found that for an harmonically breathing circle, the average energy is time-periodic in the adiabatic regime with the same period as that of the oscillation. For intermediate frequencies which are comparable with the initial frequency of the particle in unperturbed billiard, such periodicity is broken. However, for very high frequencies, the average energy once again becomes periodic. A qualitative analysis of the border between adiabatic and non-adiabatic regimes is provided.
Non-stationary second harmonic generation by femtosecond pulses, taking into account both group velocity mismatch and dispersion in nonlinear photonic crystals (quasi-phase matched crystals) with domains of arbitrary sizes has been studied numerically. A simulated-annealing algorithm, working on the basis of numerical calculation, is developed to design quasi-phase matching gratings which can yield the desired amplitude and phase profile for second-harmonic pulses in the presence of pump depletion.
In this article, the problem of nanocatalysis is considered when the catalysts are gold nanoparticles. The main experimental facts are presented and basic qualitative dependences are highlighted. The hypothesis considers the role of Tamm states of gold nanoparticles, with the modification of these states to reduce nanoparticle sizes. A semi-quantitative quantum-chemical reaction scheme of oxygen dissociation with gold nanocatalysis is shown. A theoretical answer to the basic experimental test has been obtained.
The particle dynamics in one side corrugated rectangular billiard system is investigated with the help of numerical analysis. The dependence of chaotic behavior in particle dynamics on the corrugation height h is shown. The focusing mechanism of the corrugated billiard is investigated by analyzing the dependence of the total path on particle incident angle.
REGULAR PAPERS
We consider the problem of reconstructing the time-dependent history of electromagnetic fields from Maxwell’s system of equations for an homogeneous anisotropic medium. As additional information, the Fourier image of electric and magnetic field intensity vectors for values ν = 0 of transformation parameter are given. It is shown that if the given functions satisfy some conditions of agreement and smoothness, the solution of the posed problem is uniquely defined in a class of continuously differentiable functions.
The distribution function for the perimeter of a simply connected cluster containing undoped lattice sites is based on percolation theory and the hypothesis of scale invariance. The Renyi entropy for doped graphene at low temperatures was calculated on the basis of this distribution function.
In the present paper, we consider the Hamiltonian H(K), K ∈ T3 := (−π; π]3 of a system of three arbitrary quantum mechanical particles moving on the three-dimensional lattice and interacting via zero range potentials. We find a finite set Λ ⊂ T3 such that for all values of the total quasi-momentum K ∈ Λ the operator H(K) has infinitely many negative eigenvalues accumulating at zero. It is found that for every K ∈ Λ, the number N(K; z) of eigenvalues of H(K) lying on the left of z, z < 0, satisfies the asymptotic relation lim z→−0 N(K; z)| log |z||−1 = U0 with 0 < U0 < ∞, independently on the cardinality of Λ.
Investigation of the concentration dependence of the size and type C60[=C(COOH)2]3 aggregation in aqueous solutions at 25 ◦C was accomplished with the help of a dynamic light scattering method. It was determined that three types of aggregates are realized in the solutions. The average number of C60[=C(COOH)2]3 molecules in smaller aggregates and all types of aggregates were calculated. One can see that over the whole concentration range, from 0.01 to 10 g/dm3, aqueous solutions of C60[=C(COOH)2]3 are characterized by sub-micro-heterogeneous behavior (because second-type aggregates with the linear dimensions – hundreds of nm are formed in all solutions). Additionally, the most concentrated solution (C = 10 g/dm3) is characterized by micro-heterogeneous or colloid behavior (because third-type aggregates with the linear dimensions on the order of µm – are formed). In order to describe or explain such behavior, a stepwise aggregation model was invoked.
ISSN 2305-7971 (Online)
