3,328 research outputs found

    Semi-spectral Chebyshev method in Quantum Mechanics

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    Traditionally, finite differences and finite element methods have been by many regarded as the basic tools for obtaining numerical solutions in a variety of quantum mechanical problems emerging in atomic, nuclear and particle physics, astrophysics, quantum chemistry, etc. In recent years, however, an alternative technique based on the semi-spectral methods has focused considerable attention. The purpose of this work is first to provide the necessary tools and subsequently examine the efficiency of this method in quantum mechanical applications. Restricting our interest to time independent two-body problems, we obtained the continuous and discrete spectrum solutions of the underlying Schroedinger or Lippmann-Schwinger equations in both, the coordinate and momentum space. In all of the numerically studied examples we had no difficulty in achieving the machine accuracy and the semi-spectral method showed exponential convergence combined with excellent numerical stability.Comment: RevTeX, 12 EPS figure

    An integral equation method for solving neumann problems on simply and multiply connected regions with smooth boundaries

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    This research presents several new boundary integral equations for the solution of Laplace’s equation with the Neumann boundary condition on both bounded and unbounded multiply connected regions. The integral equations are uniquely solvable Fredholm integral equations of the second kind with the generalized Neumann kernel. The complete discussion of the solvability of the integral equations is also presented. Numerical results obtained show the efficiency of the proposed method when the boundaries of the regions are sufficiently smooth

    Fast integral equation methods for the modified Helmholtz equation

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    We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or O(Nlog⁥N)O(N\log N) operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

    Wiener-Hopf solution for impenetrable wedges at skew incidence

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    A new Wiener-Hopf approach for the solution of impenetrable wedges at skew incidence is presented. Mathematical aspects are described in a unified and consistent theory for angular region problems. Solutions are obtained using analytical and numerical-analytical approaches. Several numerical tests from the scientific literature validate the new technique, and new solutions for anisotropic surface impedance wedges are solved at skew incidence. The solutions are presented considering the geometrical and uniform theory of diffraction coefficients, total fields, and possible surface wave contribution

    Emergent bubbling geometries in gauge theories with SU(2|4) symmetry

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    We study the gauge/gravity duality between bubbling geometries in type IIA supergravity and gauge theories with SU(2|4) symmetry, which consist of N=4 super Yang-Mills on R×S3/ZkR\times S^3/Z_k, N=8 super Yang-Mills on R×S2R\times S^2 and the plane wave matrix model. We show that the geometries are realized as field configurations in the strong coupling region of the gauge theories. On the gravity side, the bubbling geometries can be mapped to electrostatic systems with conducting disks. We derive integral equations which determine the charge densities on the disks. On the gauge theory side, we obtain a matrix integral by applying the localization to a 1/4-BPS sector of the gauge theories. The eigenvalue densities of the matrix integral turn out to satisfy the same integral equations as the charge densities on the gravity side. Thus we find that these two objects are equivalent.Comment: 29 pages, 3 figures; v2: typos corrected and a reference adde
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