Abstract:
We prove sharp L^2 boundary decay estimates for the eigenfunctions of certain second order elliptic operators acting in a bounded region, and of their first order space derivatives, using only the Hardy inequality. We then deduce bounds on the change of the eigenvalues when the region is reduced slightly in size, subject to DBCs.

Abstract:
A number of classical results reflect the fact that if a holomorphic function maps the unit disk into itself taking the origin into the origin, and if some boundary point $b$ maps to the boundary, then the map is a magnification at $b$. We prove a sharp quantitative version of this result which also sharpens a classical result of Loewner, and which implies that the map is a strict magnification at $b$ unless it is a rotation.

Abstract:
Let $f$ be a function on the unit circle and $D_n(f)$ be the determinant of the $(n+1)\times (n+1)$ matrix with elements $\{c_{j-i}\}_{0\leq i,j\leq n}$ where $c_m =\hat f_m\equiv \int e^{-im\theta} f(\theta) \f{d\theta}{2\pi}$. The sharp form of the strong Szeg\H{o} theorem says that for any real-valued $L$ on the unit circle with $L,e^L$ in $L^1 (\f{d\theta}{2\pi})$, we have \[ \lim_{n\to\infty} D_n(e^L) e^{-(n+1)\hat L_0} = \exp \biggl(\sum_{k=1}^\infty k\abs{\hat L_k}^2\biggr) \] where the right side may be finite or infinite. We focus on two issues here: a new proof when $e^{i\theta}\to L(\theta)$ is analytic and known simple arguments that go from the analytic case to the general case. We add background material to make this article self-contained.

Abstract:
We use chiral perturbation theory to investigate hadronic properties in strong electric and magnetic fields. A strong-field power counting is employed, and results for pions and nucleons are obtained using Schwinger's proper-time method. In the limit of weak fields, we accordingly recover the well known one-loop chiral perturbation theory results for the electric and magnetic polarizabilities of pions and nucleons. In strong constant fields, we extend the Gell-Mann-Oakes-Renner relation. For the case of electric fields, we find that non-perturbative effects result in hadron decay. For sufficiently strong magnetic fields, the chiral analysis confirms that the nucleon hierarchy becomes inverted giving rise to proton beta-decay. Properties of asymptotic expansions are explored by considering weak field limits. In the regime where the perturbative expansion breaks down, the first-order term gives the best agreement with the non perturbative result.

Abstract:
Electrokinetic boundary conditions are derived for AC electrokinetic (ACEK) phenomena over leaky dielectric (i.e., semiconducting) surfaces. Such boundary conditions correlate the electric potentials across the semiconductor-electrolyte interface (consisting of the electric double layer (EDL) inside the electrolyte solutions and the space charge layer (SCL) inside the semiconductors) under AC electric fields with arbitrary wave forms. The present electrokinetic boundary conditions allow for evaluation of induced zeta potential contributed by both bond charges (due to electric polarization) and free charges (due to electric conduction) from the leaky dielectric materials. Subsequently, we demonstrate the applications of these boundary conditions in analyzing the ACEK phenomena around a semiconducting cylinder. It is concluded that the flow circulations exist around the semiconducting cylinder and are shown to be stronger under an AC field with lower frequency and around a cylinder with higher conductivity.

Abstract:
In this paper, instability at an interface between two miscible liquids with identical mechanical properties but different electrical conductivities is analyzed in the presence of an electric field that is perpendicular to the interface. A parallel electric field case was considered in a previous work [1]. A sharp Eulerian interface is considered between the two miscible liquids. Linear stability analysis leads to an analytic solution for the critical condition of instability. The mechanism of instability is analyzed. Key differences between the perpendicular and parallel electric field cases are discussed. The effect of a microchannel geometry is studied and the relevant non-dimensional parameters are identified.

Abstract:
Recent experiments on ultracold atomic gases in an optical lattice potential have produced a Mott insulating state of Rb atoms. This state is stable to a small applied potential gradient (an `electric' field), but a resonant response was observed when the potential energy drop per lattice spacing (E), was close to the repulsive interaction energy (U) between two atoms in the same lattice potential well. We identify all states which are resonantly coupled to the Mott insulator for E close to U via an infinitesimal tunneling amplitude between neighboring potential wells. The strong correlation between these states is described by an effective Hamiltonian for the resonant subspace. This Hamiltonian exhibits quantum phase transitions associated with an Ising density wave order, and with the appearance of superfluidity in the directions transverse to the electric field. We suggest that the observed resonant response is related to these transitions, and propose experiments to directly detect the order parameters. The generalizations to electric fields applied in different directions, and to a variety of lattices, should allow study of numerous other correlated quantum phases.

Abstract:
In most electronic devices, electric current of both types (electrons and holes) flows through a junction. Usually the boundary conditions have been formulated exclusively for open circuit. The boundary conditions proposed here bypass this limitation by the first time, as far as we are aware. Besides, these new boundary conditions correctly describe current flow in a circuit, i.e., closed circuit conditions, which are the usual operation conditions for electronic devices and for the measurement of many transport properties. We also have generalized the case (as much as it is possible in a classical treatment), so self-consistent boundary conditions to describe current-flow through a contact between two arbitrary conducting media are developed in the present work. These boundary conditions take into account a recently developed theory: influence of temperature space inhomogeneity due to the interfaces and quasi-particles temperature-mismatch on thermo-generation and recombination. They also take into account surface resistance, surface recombination rates and possible temperature discontinuities at the interface due to finite surface thermo-conductivity. The temperature difference between current-carriers and phonon subsystems is also included in this approach.

Abstract:
We find theoretically energy spectrum of a graphene monolayer in a strong constant electric field using a tight-binding model. Within a single band, we find quantized equidistant energy levels (Wannier-Stark ladder), separated by the Bloch frequency. Singular interband coupling results in mixing of the states of different bands and anticrossing of corresponding levels, which is described analytically near Dirac points and is related to the Pancharatnam-Berry phase. The rate of interband tunneling, which is proportional to the anticrossing gaps in the spectrum, is only inversely proportional to the tunneling distance, in a sharp contrast to conventional solids where this dependence is exponential. This singularity will have major consequences for graphene behavior in strong ultrafast optical fields, in particular, leading to non-adiabaticity of electron excitation dynamics.

Abstract:
In the GaN-based heterostructures, this paper reports that the strong electric fields induced by polarization effects at the structure boundaries complicate the electric--static equilibrium and the boundary conditions. The basic requirements of electric--static equilibrium for the heterostructure systems are discussed first, and it is deduced that in the application of the coupled Schr\"{o}dinger--Poisson model to the heterostructures of electric--static equilibrium state, zero external electric field guarantees the overall electric neutrality, and there is no need to introduce the charge balance equation. Then the relation between the screening of the polar charges in GaN-based heterostructures and the possible boundary conditions of the Poisson equation is analysed, it is shown that the various boundary conditions are equivalent to each other, and the surface charge, which can be used in studying the screening of the polar charges, can be precisely solved even if only the conduction band energy is correctly known at the surface. Finally, through the calculations on an AlGaN/GaN heterostructure with typical structure parameters by the coupled Schr\"{o}dinger--Poisson model under the various boundary conditions, the correctness of the above analyses are validated.