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Using typical experimental techniques it is difficult to separate the effects of carrier density and disorder on the superconducting transition in two dimensions. Using a simple fabrication procedure based on metal layer dewetting, we have produced graphene sheets decorated with a non-percolating network of nanoscale tin clusters. These metal clusters both efficiently dope the graphene substrate and induce long-range superconducting correlations. This allows us to study the superconducting transition at fixed disorder and variable carrier concentration. We find that despite structural inhomogeneity on mesoscopic length scales (10-100 nm), this material behaves electronically as a homogenous dirty superconductor. Our simple self-assembly method establishes graphene as an ideal tunable substrate for studying induced two-dimensional electronic systems at fixed disorder and our technique can readily be extended to other order parameters such as magnetism.

The use of orthonormal wavelet basis functions for solving singular integral scattering equations is investigated. It is shown that these basis functions lead to sparse matrix equations which can be solved by iterative techniques. The scaling properties of wavelets are used to derive an efficient method for evaluating the singular integrals. The accuracy and efficiency of the wavelet transforms is demonstrated by solving the two-body T-matrix equation without partial wave projection. The resulting matrix equation which is characteristic of multiparticle integral scattering equations is found to provide an efficient method for obtaining accurate approximate solutions to the integral equation. These results indicate that wavelet transforms may provide a useful tool for studying few-body systems., Comment: 11 pages, 4 figures

We take advantage of the fact that in lambda phi ^4 problems a large field cutoff phi_max makes perturbative series converge toward values exponentially close to the exact values, to make optimal choices of phi_max. For perturbative series terminated at even order, it is in principle possible to adjust phi_max in order to obtain the exact result. For perturbative series terminated at odd order, the error can only be minimized. It is however possible to introduce a mass shift in order to obtain the exact result. We discuss weak and strong coupling methods to determine the unknown parameters. The numerical calculations in this article have been performed with a simple integral with one variable. We give arguments indicating that the qualitative features observed should extend to quantum mechanics and quantum field theory. We found that optimization at even order is more efficient that at odd order. We compare our methods with the linear delta-expansion (LDE) (combined with the principle of minimal sensitivity) which provides an upper envelope of for the accuracy curves of various Pade and Pade-Borel approximants. Our optimization method performs better than the LDE at strong and intermediate coupling, but not at weak coupling where it appears less robust and subject to further improvements. We also show that it is possible to fix the arbitrary parameter appearing in the LDE using the strong coupling expansion, in order to get accuracies comparable to ours., Comment: 10 pages, 16 figures, uses revtex; minor typos corrected, refs. added

We demonstrate that wavelet bases have features that make them advantageous for solving momentum-space scattering integral equations. Using the example of two nucleons interacting with the Malfliet-Tjon V interaction, we show it is possible to reduce the size of the matrix representation of the Lippmann-Scwhinger equation by 96% with a loss of accuracy of only a few parts in a million. We also demonstrate that wavelet methods provide an accurate means for treating the singularities that appear in the scattering integral equations., Comment: 3 pages, 2 figures

Wavelets are a useful basis for constructing solutions of the integral and differential equations of scattering theory. Wavelet bases efficiently represent functions with smooth structures on different scales, and the matrix representation of operators in a wavelet basis are well-approximated by sparse matrices. The basis functions are related to solutions of a linear renormalization group equation, and the basis functions have structure on all scales. Numerical methods based on this renormalization group equation are discussed. These methods lead to accurate and efficient numerical approximations to the scattering equations. These notes provide a detailed introduction to the subject that focuses on numerical methods. We plan to provide periodic updates to these notes., Comment: 80 pages, 4 figures

We show that the use of wavelet bases for solving the momentum-space scattering integral equation leads to sparse matrices which can simplify the solution. Wavelet bases are applied to calculate the K-matrix for nucleon-nucleon scattering with the s-wave Malfliet-Tjon V potential. We introduce a new method, which uses special properties of the wavelets, for evaluating the singular part of the integral. Analysis of this test problem indicates that a significant reduction in computational size can be achieved for realistic few-body scattering problems., Comment: 26 pages, Latex, 6 eps figures