Metric deformation and eigenvalue problem in 2D for an irregular boundary

Details Metric deformation and eigenvalue problem in 2D for an irregular boundary Metric deformation and eigenvalue problem in 2D for an irregular boundary

2011-06-28

arxiv.org/abs/1106.5707v1

Physics

Quantum Physics

4 pages, 1 table

Scientific paper

We prescribe a new formulation for solving the Helmholtz equation in 2D with irregular boundary. A suitable diffeomorphism is used to annul the asymmetries in the boundary by mapping it into an equivalent circle. This results in a modification of the metric in the interior of the region and manifests itself in the appearance of new gauge dependent source terms in the original homogeneous equation. The modified equation is then solved perturbatively. This method allows us to retain the simple form of the boundary condition at the cost of complicating the form of the original equation. It is seen to work reasonably well even for boundaries with large deviations from a circle by comparing our results with the exactly/numerically obtained ones. The Fourier representation of the boundary ensures the convergence of the perturbation series.

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