Boundary integral method for quantum billiards in a constant magnetic field

Abstract

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. ͓S1063-651X͑96͒05612-7͔ PACS number͑s͒: 05.45.ϩb, 03.65.Ϫw ប A•"ϩk 2 ϪA 2 /ប 2 ͬ ͑r͒ϭ0, ͑3͒ *Present address: School of Mathematics, University Walk, Bristol BS8 1TW, United Kingdom.

Magnetic edge states

“…This is a spectral equation which was derived by Tiago et al [33] (except for a misprint in their paper).…”

“…We proceed to extend these ideas to magnetic billiards. A step in this direction was taken by Tiago et al [33], who essentially propose a null-field method 8 [104] for (interior) magnetic billiards. It involves the irregular Green function (A.14) in angular momentum decomposition.…”

“…The null field method is an alternative scheme to quantize magnetic billiards in the interior [33]. We include it for completeness although its practical use is limited.…”

The boundary integral method for magnetic billiards

“…An independent solution to (7) exists which grows exponentially beyond this classically allowed region. It may be called irregular free Green function and was used in the null-field method approach [17] for reasons to be explained below. In the following only the regular Green function will be used.…”

“…It seems natural to extend these ideas to the magnetic problem. A step in this direction was taken by Tiago et al [17] who essentially propose a null-field method [18] which involves the irregular Green function in the angular momentum decomposition. A drawback of their approach is that the latter function must be known for large angular momenta which is practically inaccessible numerically.…”