In this presentation, we wish to provide an overview of the spectral features for the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-interaction centered at the origin or by a pair of identical attractive δ-interactions symmetrically situated with respect to the origin. Given that such Hamiltonians represent the mathematical model for quantum dots with sharply localized impurities, we cannot help having the renowned article by Bruning, Geyler and Lobanov [1] as our key reference. We shall also compare the spectral features of the aforementioned three-dimensional models with those of the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ′-interaction in one dimension, fully investigated in [2, 3], given the existence in both models of the remarkable spectral phenomenon called ”level crossing”. The rigorous definition of the self-adjoint Hamiltonian for the singular double well model will be provided through the explicit formula for its resolvent (Green’s function). Furthermore, by studying in detail the equation determining the ground state energy for the double well model, it will be shown that the concept of “positional disorder”, introduced in [1] in the case of a quantum dot with a single δ-impurity, can also be extended to the model with the twin impurities in the sense that the greater the distance between the two impurities is, the less localized the ground state will be. Another noteworthy spectral phenomenon will also be determined; for each value of the distance between the two centers below a certain threshold value, there exists a range of values of the strength of the twin point interactions for which the first excited symmetric bound state is more tightly bound than the lowest antisymmetric bound state. Furthermore, it will be shown that, as the distance between the two impurities shrinks to zero, the 3D-Hamiltonian with the singular double well (requiring renormalization to be defined) does not converge to the one with a single δ-interaction centered at the origin having twice the strength, in contrast to its one-dimensional analog for which no renormalization is required. It is worth stressing that this phenomenon has also been recently observed in the case of another model requiring the renormalization of the coupling constant, namely the one-dimensional Salpeter Hamiltonian perturbed by two twin attractive δ-interactions symmetrically situated at the same distance from the origin.

The self-adjoint Schr¨odinger operator A_δ,α with a δ-interaction of constant strength α supported on a compact smooth hypersurface C is viewed as a self-adjoint extension of a natural underlying symmetric operator S in L²(Rⁿ). The aim of this note is to construct a boundary triple for S^∗ and a self-adjoint parameter Θ_δ,α in the boundary space L²(C) such that A_δ,α corresponds to the boundary condition induced by Θ_δ,α. As a consequence, the well-developed theory of boundary triples and their Weyl functions can be applied. This leads, in particular, to a Krein-type resolvent formula and a description of the spectrum of A_δ,α in terms of the Weyl function and Θ_δ,α.

We solve the Cauchy problem for the Schrӧdinger equation corresponding to the family of Hamiltonians H_γ(t) in L²(R) which describes a δ′-interaction with time-dependent strength 1/γ(t). We prove that the strong solution of such a Cauchy problem exists whenever the map t → γ(t) belongs to the fractional Sobolev space H^3/4(R), thus weakening the hypotheses which would be required by the known general abstract results. The solution is expressed in terms of the free evolution and the solution of a Volterra integral equation.

We review the main results of our recent work on singular perturbations supported on bounded hypersurfaces. Our approach consists in using the theory of self-adjoint extensions of restrictions to build self-adjoint realizations of the n-dimensional Laplacian with linear boundary conditions on (a relatively open part of) a compact hypersurface. This allows one to obtain Kreın-like resolvent formulae where the reference operator coincides with the free selfadjoint Laplacian in Rⁿ, providing in this way with an useful tool for the scattering problem from a hypersurface. As examples of this construction, we consider the cases of Dirichlet and Neumann boundary conditions assigned on an unclosed hypersurface.

Analytical solutions for the Stokes equations in a cavity bounded by two confocal semi-ellipses and two line segments are derived here. The exact solution for the stream function, in the form of a Fourier series, is obtained. Eddy structure is described for different boundary conditions.

We consider a model of point-like interaction between electrons and bosons in a cavity. The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization. The model fits into the extension theory of symmetric operators. Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators. The approach allows explicit computation of the Weyl function.

The most intensive X-ray diffraction peaks for three types of carbon allotropes are analyzed: i) temperature-annealed nanodiamond powder (carbon “onions”), ii) multi-walled carbon nanotubes, iii) layers of epitaxial graphene. A reconstruction of the X-ray diffraction pattern using an intershell distribution, obtained by high resolution transmission electron microscopy, was compared to the XRD data. For a qualitative analysis of the diffraction profiles, the method of convolution of Lorentzians (size broadening profile), together with a statistical consideration of interlayer spacings (lattice strain broadening profile) were used. For the case of iii) the statistical distribution reduces to a Gaussian and the method itself transforms to a best fit procedure of the classical Voigt function to the experimental data. For cases i) and ii) and the high-resolution electron microscopy-reconstructed data, the method fits the experiment better using either negatively or positively-skewed statistical distributions, correspondingly. A model of particles with a spiral internal structure and with radius-dependent spacings between the successive turns may explain experimental data for these cases. The data for epitaxial graphene allows different interpretations, including fluctuations of lattice spacings caused by distortions of the valence bands and angles in the graphene planes or by the formation of scrolls.

We apply the method of far-field matching to remodel laser beams and study light scattering from spherical particles illuminated by a Laguerre-Gaussian (LG) light beam. The optical field in the near-field region is analyzed for purely azimuthal LG beams characterized by a nonzero azimuthal mode number m_LG. The morphology of photonic nanojets is shown to significantly vary, depending the mode number and the scatterer’s characteristics. The cases of negative index metamaterial and metallic Mie scatterers are discussed. We also discuss the symmetry properties of laser beams and related results for the optical forces. The near-field structure of optical vortices associated with the components of the electric field, being highly sensitive to the mode number, is found to be determined by the twofold rotational symmetry.

A common approach to establishing long-distance synchronization links in quantum communication (QC) systems is based on using optical signals transmitted in cables, where they decay and are distorted. It is necessary to evaluate the transformation of the signal parameters during propagation and their influence on the QC systems. We investigate the temperature dependence of the synchronization signal phase of a subcarrier wave quantum communication system (SCWQC) in optical fiber cables. A temperature model was created in order to determine the signal phase delay in the cable. We estimate the influence of daily temperature fluctuations on the phase delay in ground- and air-based cables. For systems operating with ground-based cables, they do not have any significant impact on the synchronization of the signal phase. However, for systems operating through air-based cables, phase adjustment is required every 158 ms for stable operation. This allowed us to optimize the parameters for a calibration procedure of a previously-developed SCWQC system, increasing the overall sifted key generation rate.

Quantum random number generators based on vacuum fluctuations produce truly random numbers that can be used for applications are requiring a high degree of randomness. A beam splitter with two inputs and two outputs is normally used in these systems. In this paper, mathematical descriptions were obtained for the use of such beam splitter and fiber Y-splitter in quantum random number generation systems with homodyne detection. We derived equations which allowed us to estimate the impact of the scheme parameters’ imperfection upon measurement results. We also obtained mathematical expressions, demonstrating the equivalence of quantum descriptions for a Y-splitter and a beam splitter with two inputs.

By using the example of transfer between fine structure levels in the helium atom, the possibility of laser generation without inversion has been studied.