Abstract:
We solve the Klein-Gordon equation in the presence of a spatially one-dimensional cusp potential. The scattering solutions are obtained in terms of Whittaker functions and the condition for the existence of transmission resonances is derived. We show the dependence of the zero-reflection condition on the shape of the potential. In the low momentum limit, transmission resonances are associated with half-bound states. We express the condition for transmission resonances in terms of the phase shifts.

Abstract:
We have solved exactly the two-component Dirac equation in the presence of a spatially one-dimensional Hulth\'en potential, and presented the Dirac spinors of scattering states in terms of hypergeometric functions. We have calculated the reflection and transmission coefficients by the matching conditions on the wavefunctions, and investigated the condition for the existence of transmission resonances. Furthermore, we have demonstrated how the transmission-resonance condition depends on the shape of the potential.

Abstract:
We study a family of chaotic maps with limit cases the tent map and the cusp map (the cusp family). We discuss the spectral properties of the corresponding Frobenius--Perron operator in different function spaces including spaces of analytic functions. A numerical study of the eigenvalues and eigenfunctions is performed.

Abstract:
Quantum above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential is analytically considered within the mean field Gross-Pitaevskii approximation. Reformulating the problem of reflectionless transmission as a quasi-linear eigenvalue problem for the potential depth, an approximation for the specific height of the potential that supports reflectionless transmission of the incoming matter wave is derived via modification of the Rayleigh-Schroedinger time-independent perturbation theory. The approximation provides highly accurate description of the resonance position for all the resonance orders if the nonlinearity parameter is small compared with the incoming particles chemical potential. Notably, the result for the first transmission resonance turns out to be exact, i.e., the derived formula for the resonant potential height gives the exact value of the first nonlinear resonances position for all the allowed variation range of the involved parameters, the nonlinearity parameter and chemical potential. This has been shown by constructing the exact solution of the problem for the first resonance. Furthermore, the presented approximation reveals that, in contrast to the linear case, in the nonlinear case reflectionless transmission may occur not only for potential wells but also for potential barriers with positive potential height. It also shows that the nonlinear shift of the resonance position from the position of the corresponding linear resonance is approximately described as a linear function of the resonance order. Finally, a compact (yet, highly accurate) analytic formula for the n-th order resonance position is constructed via combination of analytical and numerical methods.

Abstract:
We calculate the tunneling process of a Dirac particle across two square barriers separated a distance $d$, as well as the scattering by a double cusp barrier where the centers of the cusps are separated a distance larger than their screening lengths. Using the scattering matrix formalism, we obtain the transmission and reflection amplitudes for the scattering processes of both configurations. We show that, the presence of transmission resonances modifies the Lorentizian shape of the energy resonances and induces the appearance of additional maxima in the transmission coefficient in the range of energies where transmission resonances occur. We calculate the Wigner time-delay and show how their maxima depend on the position of the transmission resonance.

Abstract:
This resonance is similar in character to other resonances in atomic physics. It is parameterized by its energy and its lifetime. A numerical discretization technique, the mapped Fourier grid method (MFG) is extended to the Dirac equation and is used to solve for the resonance parameters of a quasimolecular supercritical 1S$\sigma$ state which arises, e.g., in a uranium-uranium collision. Direct methods using only the MFG method are shown to give reasonable estimates for the resonance parameters. Analytic continuation methods such as complex scaling (CS) of the coordinate or adding a complex absorbing potential (CAP) are then applied. They allow for more accurate determinations of the supercritical resonance parameters. The (extrapolated) augmented analytic continuation methods are used to investigate the effects of higher-order couplings, beyond the monopole approximation. For the nearly charge symmetric system of U$^{92+}$-Cf$^{98+}$, it is shown that the S-D quadrupole coupling is the next dominant interaction after the monopole interaction. Collisions near the Coulomb barrier, in which supercritical resonance states occur, are also calculated using the MFG. Results are presented for the collision of U$^{92+}$-U$^{92+}$ for a zero-impact-parameter Coulomb trajectory at a center-of-mass energy of 740MeV. The enhancement effects to the positron spectrum due to nuclear sticking at closest approach is examined. Results are given for sticking times of 2,5,10$\times10^{-21}$s. Nuclear sticking is shown to offer the possibility of demonstrating experimentally the existence of supercritical resonance states.

Abstract:
We investigate the enhanced microwave transmission through the array of metallic coaxial annular apertures (MCAAs) experimentally and theoretically. The even-mode and the odd-mode surface resonances are clarified from the spatial field distributions and the dispersion diagram. The impact of local resonance is thoroughly embodied in the even-mode surface resonant states, while the odd-mode surface resonances are scaled by periodicity, invariant to different local geometry of the unit cell, and invisible in measurements. The enhanced transmission is the collective selections on the interplay between the local resonances and the evanescent Bloch wave channels on the surface. Transmission measurements for different inner diameter of the apertures show that the transmissivity extrema with respect to the specific angles precisely correspond to the degenerate points in the dispersion diagram of surface resonances.

Abstract:
We present here a detailed multifractal scaling study for the electronic transmission resonances with the system size for an infinitely large one dimensional perfect and imperfect quasiperiodic system represented by a sequence of $\delta$-function potentials. The electronic transmission resonances in the energy minibands manifest more and more fragmented nature of the transmittance with the change of system sizes. We claim that when a small perturbation is randomly present at a few number of sites, the nature of electronic states will change and this can be understood by studying the electronic transmittance with the change of system size. We report the different critical states manifested in the size variation of the transmittance corresponding to the resonant energies for both perfect and imperfect cases through multifractal scaling study for few of these resonances.

Abstract:
In this paper we show how transmission metallic gratings with very narrow and deep enough slits can exhibit transmission resonances for wavelengths larger than the period of the grating. By using a transfer matrix formalism and a quasi-analytical model based on a modal expansion, we show that there are two possible ways of transferring light from the upper surface to the lower one: by the excitation of coupled surface plasmon polaritons on both surfaces of the metallic grating or by the coupling of incident plane waves with waveguide resonances located in the slits. Both mechanisms can lead to almost perfect transmittance for those particular resonances.

Abstract:
A theory of image-potential states is presented for the general case where these surface electronic states are resonant with a bulk continuum. The theory extends the multiple scattering approach of Echenique and Pendry into the strong coupling regime while retaining independence from specific forms of surface and bulk potentials. The theory predicts the existence of a well-resolved series of resonances for arbitrary coupling strengths. Surprisingly, distinct image-potential resonances are thus expected to exist on almost any metal surface, even in the limiting case of jellium.