A theoretical study of a waveguide with a uniformly curved section is presented within the envelope function approximation. Utilizing analytical solutions in each part of the waveguide, exact expression of the scattering matrix of the system is derived. Based on it, a conductance of the waveguide is calculated for the wide range of the bend angle and radius. It is shown that a quasibound state formed as a result of the bend, at some critical parameters of the curve becomes a true bound state with infinite lifetime. It has its degenerate continuum counterpart, but does not interact with it. As a result of a constructive resonant interference in the bend, dip in the conductance, which is an essential property for the noncritical values, vanishes with full transmission being observable instead. Mathematical and physical interpretation of these results is given, and characteristic features of the critical parameters are discussed. Comparison with quantum waveguides with other types of nonuniformity is performed.

Analytical solutions of the Schrödinger equation for the one‐dimensional quantum well with all possible permutations of the Dirichlet and Neumann boundary conditions (BCs) in perpendicular to the interfaces uniform electric field are used for the comparative investigation of their interaction and its influence on the properties of the system. Limiting cases of the weak and strong voltages allow an easy mathematical treatment and its clear physical explanation; in particular, for the small , the perturbation theory derives for all geometries a linear dependence of the polarization on the field with the BC‐dependent proportionality coefficient being positive (negative) for the ground (excited) states. Simple two‐level approximation elementary explains the negative polarizations as a result of the field‐induced destructive interference of the unperturbed modes and shows that in this case the admixture of only the neighboring states plays a dominant role. Different magnitudes of the polarization for different BCs in this regime are explained physically and confirmed numerically. Hellmann‐Feynman theorem reveals a fundamental relation between the polarization and the speed of the energy change with the field. It is proved that zero‐voltage position entropies are BC independent and for all states but the ground Neumann level (which has ) are equal to while the momentum entropies depend on the edge requirements and the level. Varying electric field changes position and momentum entropies in the opposite directions such that the entropic uncertainty relation is satisfied. Other physical quantities such as the BC‐dependent zero‐energy and zero‐polarization fields are also studied both numerically and analytically. Applications to different branches of physics, such as ocean fluid dynamics and atmospheric and metallic waveguide electrodynamics, are discussed. image

A theoretical study of a waveguide with a uniformly curved section and an embedded quantum dot is presented within the envelope function approximation. For the quantum dot being extremely localized in the direction of the electron propagation, exact form of the scattering matrix of the system is derived, and a conductance of the waveguide is calculated. It is shown that if, for the straight wire with dot, in the fundamental mode conductance is a monotonically increasing function of the Fermi energy, then, after bending the waveguide, one has Fano resonances on the conductance-Fermi energy dependence. The resonances appear as a result of mixing by the bend of the longitudinal and transverse electron motion in the straight parts of the waveguide and concomitant intersubband interaction. For the small bend angle (large bend radius) the width of the resonance grows narrower with decreasing the angle (increasing the radius), until in the limit of the zero angle (infinite radius) one recovers true bound state in the continuum and the corresponding monotonic conductance of the straight channel. A dependence of the resonance width on the parameters of the bend and the dot is investigated; in particular, it is shown that, as a result of coherent resonant phenomena in the superposition of the bend and the dot, true bound states in the continuum can be formed also for the nonzero bend angle and the finite bend radius. Miscellaneous cases of bending the waveguide which, if uncurved, already exhibits the Fano resonance, are also investigated; for example, it is shown that in this situation, as a result of the bend, resonances can collapse too, again producing true bound states in the continuum. The case of inversion of the Fano resonances, i.e., a change of their minimum and maximum mutual location, is also analyzed. Mathematical and physical interpretation of the obtained results is given, and characteristic features of the critical parameters at which the Fano resonances collapse, are discussed. It is demonstrated that currents flowing in the waveguide, near the Fano resonances drastically change their behavior from laminar to vortical structure, and an evolution of the vortices is described. Parallels are drawn to the other types of the guiding structures, such as electromagnetic waveguides and acoustic ducts.

A comparative study of 1D quantum structures which allows analytic expressions for the position and momentum Rényi R ( α ) and Tsallis T ( α ) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms the special status of the Gaussian distribution: at any parameter α it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2 ≤ α ≤ 1 that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi-infinite range of the dimensionless parameter α where momentum components exist strongly depends on the position potential and/or boundary condition for the position wave function. Asymptotic limits reveal that in either space the entropies R ( α ) and T ( α ) approach their Shannon counterpart, α = 1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are also indicated.