Spectral theory and L∞-time decay estimates for Klein–Gordon equations on two half axes with transmission: The tunnel effect

Consider two copies N₁, N₂ of the interval [0, ∞]. Consider Klein-Gordon equations with (different) constant coefficients on ? × N_j ( = time × space). Assume the coincidence of the values of the solution at the boundary points of the N_j for all times and a transmission condition relating its first (one-sided) space derivatives at these points. Under a symmetry condition, we extend the spatial part of the equation and the transmission conditions to a self-adjoint operator (by Friedrichs extension) and reformulate our problem in terms of an abstract wave equation in a suitable Hilbert space. We derive an expansion of the solution in generalized eigenfunctions of this self-adjoint extension and show, that the L^∞-norms (in space) of the solution and its first k space derivatives at the time t decay for t → ∞ at least as const. t^¼, if the initial conditions satisfy a compatibility condition of order k derived in this paper. The loss of decay rate in comparison with the full line case (const. t^?½, cf. [28]) is caused by the tunnel effect. Further we show that an abstract wave equation in a Hilbert space with a Friedrichs extension as spatial part can always be derived from a stationarity principle for an associated action-type functional. This yields a physical legitimation of our model by the principle of stationary action and moreover a criterion for the physical interpretability of all models created by the linear interaction concept [4, 6, 8, 10], in particular for the coupling of media of different dimension (alternative to [13, 16] for similar models).     

Simple,Rapid and Sensitive Electrochemical Method for the Determination of the Triketone Herbicide Sulcotrione in River Water Using a Glassy Carbon Electrode

We report a novel, simple, rapid and sensitive electrochemical method for the determination of sulcotrione, a member of the relatively new class of triketone herbicides, using differential pulse voltammetry on a glassy carbon electrode. Its electrochemical behavior including influences of electrolyte composition, pH and scan rate was studied to select optimal experimental parameters for its determination. In Britton? Robinson buffer at pH 3 sulcotrione provided a well‐defined reduction peak at ?0.84 V (vs. Ag/AgCl electrode), with good repeatability (relative standard deviation of 2.3 % for 8 measurements at 10 µM concentration level). With optimized parameters differential pulse voltammetry rendered two linear concentration ranges from 0.2 to 2 µM and from 2 to 50 µM with a detection limit of 0.05 µM. The proposed procedure was successfully applied to the determination of sulcotrione in spiked river water samples with satisfactory recoveries (93–109 %). The developed method may represent a simple, rapid and sensitive alternative to highly toxic mercury electrodes and chromatographic methods.     

Non-Autonomous Evolution Equations on Nonsmooth Domair

The solution of a stationary boundary value problem on a domain with conical points has singularities near these points. Here we first consider existence results in appropriate weighted Sobolev spaces in order to incorporate the singularities. We secondly use these results to prove existence, uniqueness and regularity of solutions of non-autonomous second order evolution equations on such domains.     

Radiological hazard assessment around two lignite-fired power plants in Kosovo

Journal of Radioanalytical and Nuclear Chemistry - The environmental radioactivity (40K, 238U, 232Th and 137Cs) in soils around Kosova-A and Kosova-B power plants is determined by high-resolution...     

Electrochemical sensors for the simultaneous determination of zinc,cadmium and lead using a Nafion/ionic liquid/graphene composite modified screen-printed carbon electrode

A simple, low cost, and highly sensitive electrochemical sensor, based on a Nafion/ionic liquid/graphene composite modified screen-printed carbon electrode (N/IL/G/SPCE) was developed to determine zinc (Zn(II)), cadmium (Cd(II)), and lead (Pb(II)) simultaneously. This disposable electrode shows excellent conductivity and fast electron transfer kinetics. By in situ plating with a bismuth film (BiF), the developed electrode exhibited well-defined and separate peaks for Zn(II), Cd(II), and Pb(II) by square wave anodic stripping voltammetry (SWASV). Analytical characteristics of the BiF/N/IL/G/SPCE were explored with calibration curves which were found to be linear for Zn(II), Cd(II), and Pb(II) concentrations over the range from 0.1 to 100.0 ng L⁻¹. With an accumulation period of 120 s detection limits of 0.09 ng mL⁻¹, 0.06 ng L⁻¹ and 0.08 ng L⁻¹ were obtained for Zn(II), Cd(II) and Pb(II), respectively using the BiF/N/IL/G/SPCE sensor, calculated as 3σ value of the blank. In addition, the developed electrode displayed a good repeatability and reproducibility. The interference from other common ions associated with Zn(II), Cd(II) and Pb(II) detection could be effectively avoided. Finally, the proposed analytical procedure was applied to detect the trace metal ions in drinking water samples with satisfactory results which demonstrates the suitability of the BiF/N/IL/G/SPCE to detect heavy metals in water samples and the results agreed well with those obtained by inductively coupled plasma mass spectrometry.     

Delayed reflection of the energy flow at a potential step for dispersive wave packets

We study Klein–Gordon equations with constant coefficients and different dispersion relations on two one‐dimensional semi‐infinite media coupled with transmission conditions. We obtain lower and upper bounds of the reflected part of the energy flow at the connecting point when the frequency band involved in the initial signal is sufficiently narrow. We detect a phenomenon of delayed reflection for low frequency wave packets, which is in accordance with the recent experiments of Haibel and Nimtz. The result is then generalized for a star‐shaped network of n semi‐infinite branches connected at one point. Copyright © 2004 John Wiley & Sons, Ltd.     

Lossless error estimates for the stationary phase method with applications to propagation features for the Schrödinger equation

We consider a version of the stationary phase method in one dimension of A. Erdélyi, allowing the phase to have stationary points of non‐integer order and the amplitude to have integrable singularities. After having completed the original proof and improved the error estimate in the case of regular amplitude, we consider a modification of the method by replacing the smooth cut‐off function employed in the source by a characteristic function, leading to more precise remainder estimates. We exploit this refinement to study the time‐asymptotic behaviour of the solution of the free Schrödinger equation on the line, where the Fourier transform of the initial data is compactly supported and has a singularity. We obtain asymptotic expansions with respect to time in certain space‐time cones as well as uniform and optimal estimates in curved regions, which are asymptotically larger than any space‐time cone. These results show the influence of the frequency band and of the singularity on the propagation and on the decay of the wave packets. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple tunnel effect for dispersive waves on a star-shaped network: an explicit formula for the spectral representation

We consider the Klein?CGordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential that is constant but different on each branch. The corresponding spatial operator is self-adjoint, and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier-type inversion formula in terms of an expansion in generalized eigenfunctions. Further, we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore, the approach via the Sturm?CLiouville theory for systems is not well-suited. The considerable effort to construct explicit formulas involving the generalized eigenfunctions that incorporate the tunnel effect is justified for example by the perspective to study the influence of this effect on the L ^??-time decay of solutions.