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Boundary integral method for quantum billiards in a constant magnetic field

We derive a boundary integral equation to compute the eigenvalues of two-dimensional billiards subjected to a magnetic field. The integral requires the Green s function of the boundary-free problem with the magnetic field pointing in the opposite direction. This Green s function is computed for the case of a constant magnetic field perpendicular to the billiard and some applications are discussed. The elliptical billiard is then studied numerically as an example of a nontrivial application.

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