Abstract:
We study high energy resonances for the operator $-\Delta_{V,\partial\Omega}:=-\Delta+\delta_{\partial\Omega}\otimes V $ when $V$ has strong frequency dependence. The operator $-\Delta_{V,\partial\Omega}$ is a Hamiltonian used to model both quantum corrals and leaky quantum graphs. Since highly frequency dependent delta potentials are out of reach of the more general techniques in previous work, we study the special case where $\Omega=B(0,1)\subset \mathbb{R}^2$ and $V\equiv h^{-\alpha }V_0>0$ with $\alpha\leq 1$. Here $h^{-1}\sim \Re \lambda$ is the frequency. We give sharp bounds on the size of resonance free regions for $\alpha\leq 1$ and the location of bands of resonances when $5/6\leq \alpha\leq 1$. Finally, we give a lower bound on the number of resonances in logarithmic size strips: $-M\log \Re \lambda\leq \Im \lambda \leq 0$.

Abstract:
A self-consistent model is proposed here to study the nonlinear resonance and the hysteresis phenomena in the vertical oscillations of a charged micro-particle in a RF sheath. In this model, the charging process of the micro-particle and the sheath dynamics is considered self-consistently. And also, various forces acting on the particle are fully taken into account in the Newton's equations of the micro-particle. By solving the equation, we simulate the motions of the micro-particle in the sheath, under the excitations of the probe. Numerical results reproduce well the recent experimental observations; at the same time, we find that these nonlinearities are not only due to the structure of the sheath, but also due to the charging process of the micro-particle, ion drag force, neutral gas friction and the excitation of the probe.

Abstract:
We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.

Abstract:
We study high energy resonances for the operators $-\Delta +\delta_{\partial\Omega}\otimes V$ and $-\Delta+\delta_{\partial\Omega}'\otimes V\partial_\nu$ where $\Omega$ is strictly convex with smooth boundary, $V:L^2(\partial\Omega)\to L^2(\partial\Omega)$ may depend on frequency, and $\delta_{\partial\Omega}$ is the surface measure on $\partial\Omega$. These operators are model Hamiltonians for quantum corrals and leaky quantum graphs. We give a quantum version of the Sabine Law from the study of acoustics for both the $\delta$ and $\delta'$ interactions. It characterizes the decay rates (imaginary parts of resonances) in terms of the system's ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For the $\delta$ interaction we show that generically there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of the $\delta'$ interaction, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate and is optimal in the case of the unit disk in the plane. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose--Taylor parametrix for complex energies. We use these constructions to give a complete microlocal description of boundary layer operators and to prove sharp high energy estimates on the boundary layer operators in the case that $\partial\Omega$ is smooth and strictly convex.

Abstract:
We demonstrate an efficient double-layer light absorber by exciting plasmonic phase resonances. We show that the addition of grooves can cause mode splitting of the plasmonic waveguide cavity modes and all the new resonant modes exhibit large absorptivity greater than 90%. Some of the generated absorption peaks have wide-angle characteristics. Furthermore, we find that the proposed structure is fairly insensitive to the alignment error between different layers. The proposed plasmonic nano-structure designs may have exciting potential applications in thin film solar cells, thermal emitters, novel infrared detectors, and highly sensitive bio-sensors.

Abstract:
Nonlinear magneto-optical resonances have been measured in an extremely thin cell (ETC) for the D1 transition of rubidium in an atomic vapor of natural isotopic composition. All hyperfine transitions of both isotopes have been studied for a wide range of laser power densities, laser detunings, and ETC wall separations. Dark resonances in the laser induced fluorescence (LIF) were observed as expected when the ground state total angular momentum F_g was greater than or equal to the excited state total angular momentum F_e. Unlike the case of ordinary cells, the width and contrast of dark resonances formed in the ETC dramatically depended on the detuning of the laser from the exact atomic transition. A theoretical model based on the optical Bloch equations was applied to calculate the shapes of the resonance curves. The model averaged over the contributions from different atomic velocity groups, considered all neighboring hyperfine transitions, took into account the splitting and mixing of magnetic sublevels in an external magnetic field, and included a detailed treatment of the coherence properties of the laser radiation. Such a theoretical approach had successfully described nonlinear magneto-optical resonances in ordinary vapor cells. Although the values of certain model parameters in the ETC differed significantly from the case of ordinary cells, the same physical processes were used to model both cases. However, to describe the resonances in the ETC, key parameters such as the transit relaxation rate and Doppler width had to be modified in accordance with the ETC's unique features. Agreement between the measured and calculated resonance curves was satisfactory for the ETC, though not as good as in the case of ordinary cells.

Abstract:
We prove an abstract criterion stating resolvent convergence in the case of operators acting in different Hilbert spaces. This result is then applied to the case of Laplacians on a family $X_\eps$ of branched quantum waveguides. Combining it with an exterior complex scaling we show, in particular, that the resonances on $X_\eps$ approximate those of the Laplacian with ``free'' boundary conditions on $X_0$, the skeleton graph of $X_\eps$.

Abstract:
We study analytically the evolution of superconductivity in clean quasi-two-dimensional multiband supercon- ductors as the film thickness enters the nanoscale region by mean-field and semiclassical techniques. Tunneling into the substrate and finite lateral size effects, which are important in experiments, are also considered in our model. As a result, it is possible to investigate the interplay between quantum coherence effects, such as shape resonances and shell effects, with the potential to enhance superconductivity, and the multiband structure and the coupling to the substrate that tend to suppress it. The case of magnesium diboride, which is the conventional superconductor with the highest critical temperature, is discussed in detail. Once the effect of the substrate is considered, we still observe quantum size effects such as the oscillation of the critical temperature with the thickness but without a significant enhancement of superconductivity. In thin films with a sufficiently longer superconducting coherence length, it is, however, possible to increase the critical temperature above the bulk limit by tuning the film thickness or lateral size.

Abstract:
The usefulness of a theoretical metric for civilization is that it can help to identify the kinds of progress which society can make that is universalized for all humanity. Societal systems perform the functions which provide the values and performance of the society, and wherein societal problems occur. In the concept of the level of “civilization” of a society, four kinds of measures can assess the progress of a society in attaining universalized values: Truth, Good, Beautiful, and Wealth. The value of Truth in our civilization is methodologically investigated by science. The value of Good in our civilization is politically pursued through democracy. The value of Beautiful in our civilization is seen in the preservation of the environment of the Earth. The value of Wealth in our civilization is generated through industrialization of societal production. We apply the theory to the historical case of the International Court of Justice and Yugoslav War Crimes to examine empirical evidence about the validity of a theoretical metric.

Abstract:
The efficiency of thin disk accretion onto black holes depends on the inner boundary condition, specifically the torque applied to the disk at the last stable orbit. This is usually assumed to vanish. I estimate the torque on a magnetized disk using a steady magnetohydrodynamic inflow model originally developed by Takahashi et al., 1990. I find that the efficiency epsilon can depart significantly from the classical thin disk value. In some cases epsilon > 1, i.e. energy is extracted from the black hole.