Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues

We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain \Omega\subset\RR^n with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.Comment: 18 pages, 3 figure

Perturbative Analysis of the Method of Particular Solutions for Improved Inclusion of High-Lying Dirichlet Eigenvalues

The Dirichlet eigenvalue or “drum” problem in a domain $\Omega\subset\mathbb{R}^2$ becomes numerically challenging at high eigenvalue (frequency) E. In this regime the method of particular solutions (MPS) gives spectral accuracy for many domain shapes. It requires a number of degrees of freedom scaling as $\sqrt{E}$, the number of wavelengths on the boundary, in contrast to direct discretization for which this scaling is E. Our main result is an inclusion bound on eigenvalues that is a factor $O(\sqrt{E})$ tighter than the classical bound of Moler–Payne and that is optimal in that it reflects the true slopes of curves appearing in the MPS. We also present an MPS variant that cures a normalization problem in the original method, while evaluating basis functions only on the boundary. This method is efficient at high frequencies, where we show that, in practice, our inclusion bound can give three extra digits of eigenvalue accuracy with no extra effort

Difficulty of distinguishing product states locally

Non-locality without entanglement is a rather counter-intuitive phenomenon in which information may be encoded entirely in product (unentangled) states of composite quantum systems in such a way that local measurement of the subsystems is not enough for optimal decoding. For simple examples of pure product states, the gap in performance is known to be rather small when arbitrary local strategies are allowed. Here we restrict to local strategies readily achievable with current technology; those requiring neither a quantum memory nor joint operations. We show that, even for measurements on pure product states there can be a large gap between such strategies and theoretically optimal performance. Thus even in the absence of entanglement physically realizable local strategies can be far from optimal for extracting quantum information.Comment: 5 pages, 1 figur

GaAsP on GaP top solar cells

GaAsP on GaP top solar cells as an attachment to silicon bottom solar cells are being developed. The GaAsP on GaP system offers several advantages for this top solar cell. The most important is that the gallium phosphide substrate provides a rugged, transparent mechanical substrate which does not have to be removed or thinned during processing. Additional advantages are that: (1) gallium phosphide is more oxidation resistant than the III-V aluminum compounds, (2) a range of energy band gaps higher than 1.75 eV is readily available for system efficiency optimization, (3) reliable ohmic contact technology is available from the light-emitting diode industry, and (4) the system readily lends itself to graded band gap structures for additional increases in efficiency

N/P GaAs concentrator solar cells with an improved grid and bushbar contact design

The major requirements for a solar cell used in space applications are high efficiency at AMO irradiance and resistance to high energy radiation. Gallium arsenide, with a band gap of 1.43 eV, is one of the most efficient sunlight to electricity converters (25%) when the the simple diode model is used to calculate efficiencies at AMO irradiance, GaAs solar cells are more radiation resistant than silicon solar cells and the N/P GaAs device has been reported to be more radiation resistant than similar P/N solar cells. This higher resistance is probably due to the fact that only 37% of the current is generated in the top N layer of the N/P cell compared to 69% in the top layer of a P/N solar cell. This top layer of the cell is most affected by radiation. It has also been theoretically calculated that the optimized N/P device will prove to have a higher efficiency than a similar P/N device. The use of a GaP window layer on a GaAs solar cell will avoid many of the inherent problems normally associated with a GaAlAs window while still proving good passivation of the GaAs surface. An optimized circular grid design for solar cell concentrators has been shown which incorporates a multi-layer metallization scheme. This multi-layer design allows for a greater current carrying capacity for a unit area of shading, which results in a better output efficiency