In this note, we continue our analysis of the behavior of self-adjoint Hamiltonians with symmetric double wells given by twin point interactions perturbing various types of “free Hamiltonians” as the distance between the two centers shrinks to zero. In particular, by making the coupling constant to be renormalized and also dependent on the separation distance between the two impurities, we prove that it is possible to rigorously define the unique self-adjoint Hamiltonian that, differently from the one studied in detail by Albeverio and collaborators, behaves smoothly as the separation distance between the impurities shrinks to zero. In fact, we rigorously prove that the Hamiltonian introduced in this note converges in the norm resolvent sense to that of the negative three-dimensional Laplacian perturbed by a single attractive point interaction situated at the origin having double strength, thus making this three-dimensional model more similar to its one-dimensional analog (not requiring the renormalization procedure) as well as to the three-dimensional model involving impurities given by potentials whose range may even be physically very short but non-zero.

We consider an elliptic operator in a planar waveguide with a fast oscillating boundary where we impose Dirichlet, Neumann or Robin boundary conditions assuming that both the period and the amplitude of the oscillations are small. We describe the homogenized operator, establish the norm resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. It is shown that under the homogenization, the type of the boundary condition can change.

Indefinite Sturm–Liouville operators defined on R are often considered as a coupling of two semibounded symmetric operators defined on R ⁺ and R ⁻, respectively. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension.

In this paper, we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm–Liouville problems on R.

In this note, we study the eigenvalue problem for a class of block operator matrix pairs. Our study is motivated by an analysis of abstract differential algebraic equations. Such problems frequently appear in the study of complex systems, e.g. differential equations posed on metric graphs, in mixed variational formulation.

This work describes the development of a model using a zigzag chain of weakly-coupled ball resonators with Neumann boundary conditions. The chain is assumed to be constructed of identical resonators connected through point-like apertures. The connecting points are described by their delta-coupling with a constant intensity. The model is based on the theory of self-adjoint extensions of symmetrical operators. Due to effectively one-dimensional joints, the 3D problem can be solved with assistance from the transfer matrix approach. This allows us to study the spectrum of the physical system. In particular, it is proven that the discrete spectrum of direct zigzag chain is empty while bending deformation leads to the appearance of non-empty discrete spectrum. In addition, the continuous spectrum has band structure. With the help of asymptotic study, we obtain the dependence of the spectrum structure on the geometrical and physical parameters of the system: zigzag angle, bend angle and coupling intensity.

Several explicitly solvable models are constructed for electron tunneling in a system of double two-dimensional periodic arrays of quantum dots with two laterally coupled leads in a homogeneous magnetic field are constructed. The theory of self-adjoint extensions of symmetric operators is used for modelling of electron transport. Dependencies of the transmission coefficient on the magnetic field, the energy of electron and the distance between layers are investigated. The results are compared with those of tunnelling through the corresponding single-layer periodic arrays.

We improve some recent estimates of the rate of convergence for product approximations of solution operators for linear non-autonomous Cauchy problem. The Trotter product formula approximation is proved to converge to the solution operator in the operator-norm. We estimate the rate of convergence of this approximation. The result is applied to diffusion equation perturbed by a time-dependent potential.

For the analysis of the Schrodinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space, quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schrodinger operators with random potentials. For discrete Schr odinger equations on the lattice, only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. For other combinatorial graphs, this is no longer true. Similarly, for quantum graphs the UCP does not hold in general and consequently, the IDS does not need to be continuous.

The electron-impurity scattering coefficient of Bloch waves for one dimensional Dirac comb potential is used for calculating the temperature dependence of conductivity within kinetic theory. We restrict ourselves by scattering on impurities that is also modelled by zero-range potential. The conductivity is obtained by standard averaging in momentum space, it is expressed by integral that is evaluated within temperature expansion.

The sedimentation of nanoparticles in a liquid considering their Brownian diffusion was investigated by using mathematical modeling. The main purpose of this work is investigation of the particles’ behavior in the area adjoining to the bottom of the vessel – the boundary layer.

A quantum random number generator (QRNG) based on the quantum nature of vacuum fluctuations allows one to obtain random bit sequences that can be used in applications that require a high degree of randomness. In that type of quantum random generation system, optical beam splitters with two inputs and two outputs are normally used. A comparison of Y-splitter and spatial beam splitters shows that for two types of optical splitters, the quantum mathematical description of output signals is identical. This allows the use of fiber Y-splitters in practical QRNG schemes. The possibility of generating true random bits was demonstrated experimentally by using quantum random number generator based on homodyne detection.

Grover’s algorithm is a quantum algorithm for searching specified elements in an unsorted list. It has many valuable applications. The utilization of Grover’s algorithm, to adapt it to accelerate the works of well-known classical algorithms, is very promising, and it is one of the fastest algorithms to solve such problems like global optimization and graph coloring. In this regard, it is very important to study the stability of the Grover’s algorithm, to know how distortion of the circuit’s elements affects on it results. This work presents the results of the simulation of Grover’s algorithm, research of its stability with respect to perturbations of quantum logic circuit elements and its dependencies from the number of qubits, used in quantum circuit. Another part of this research was realized on IBM quantum processor and shows the stability of the 2-qubit Grover’s algorithm.

The problem of a nanowires conductivity is studied from a kinetic point of view for quasiclassical Bloch electrons in an electric field. Few statements of problems with cylindrical symmetry for the integro-differential Kolmogorov equation are formulated: the dynamic Cauchy problem and two stationary boundary regime ones. The first is for an empty cylinder with scattering of the conduction electrons on walls, the second takes into account scattering on defects inside the wire. The integro-differential equations are transformed to integral ones and solved iteratively. There are two types of expansions with the leading terms in the right and left sides. The iteration series is constructed and its convergence studied.

Ultrasonic near-field levitation allows suspension of a moderately large object at a height of tens of microns above sound actuator. We developed an asymptotic approach to describe the air dynamics in the gap between an acoustic source and the levitating object. The suggested method allows computation of the lifting force. Due to resolving of both viscous and inertial effects, it remains applicable across a wide range of levitation distances. The paper explains theoretical background of the model and presents a numerical solution of the obtained equations. The results are compared to published numerical and experimental data showing very good agreement.

The correlation between the structure, resistance and UV-irradiation impact on conductivity of polystyrene-based composites with multilayered graphene oxide flakes was observed. It is established that composites structure and conducting properties depend on concentration, surface modification and the methods by which graphene oxide was incorporated into the polystyrene matrix.

There are many quantum computing systems, some of which are still being developed today. To develop quantum calculation systems, IBM provides access to the 5-qubit quantum computer ‘IBM Quantum Experience’. Quantum computers must deal with the loss of information due to environmental disturbances. Quantum systems cannot be completely isolated. Noise can be a cause of different errors in the quantum circuits. In this work, we observe distortions in quantum circuits and investigate the noise stability of different quantum gates. We investigate a method for calculating the quantum state of the superconducting qubit, used in ‘IBM Quantum Experience’, after an interaction with a quantum operator.

The present study examines the electric field distribution in the structure made of two eccentric dielectric cylinders. In oder to find the potential of electric field, we employ a bipolar coordinate system. The obtained results allow one to quantify the impact of the eccentricity on effective characteristics of a periodic set of eccentric cylinders. Special attention is paid to the analysis of the structure polarizability.

Modern methods of cancer treatment include chemotherapy and radiotherapy, but they are often characterized by low efficacy and high toxicity. The effectiveness of cancer therapy is often limited by a lack of effective systems for drug delivery to the tumor site. Cerium oxide nanoparticles are able to act as radioprotectors and as radiosensitizers exhibiting selective toxicity in the tumor microenvironment, providing for their tremendous potential in treating cancer. However, methods for controlled delivery of CeO₂ nanoparticles to the tumor have not been investigated nor described yet. In this article, we consider different approaches to the development of new ceria nanoparticle-based theranostic agents. Modification of polyelectrolyte microcapsules with nano-ceria appears to be the most promising method. Our design proposals are based on the synergistic pharmacological action of ceria-based nanomaterials and anticancer pharmaceuticals with the ability to control and visualize their sites of localization.

Gd_0.1Ti_0.1Zr_0.1Ce_0.7O₂ solid solution with crystallite size of 10 nm, specific surface area of 85 m² /g and pore size of 2–6 nm has been prepared by a simple co-precipitation method with sonication and characterized by several methods. Among the characterization methods was laser desorption ionization-time of flight mass spectrometry (LDI-TOF) which was used to characterize the surface of the catalyst (fresh and used in CO oxidation) and thereby determine the catalytic sites (active sites of oxidation).