Abstract
We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate the existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits Gaussian Unitary Ensemble (GUE) like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunnelling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.
Highlights
The boundary integral method, has a similar demand on matrix size as the scaling method and can in principle be applied to arbitrary shapes as it is based on the Green’s theorem that shows how the wavefunction inside the billiard can be represented with its normal derivative on the boundary
We mean that the system itself is timereversal invariant but different disjoint chaotic components are not mapped onto themselves upon time reversal operation
Such situation is possible in the so-called Monza billiard due to unidirectionality of the classical motion [16]
Summary
The scaling method as proposed by Vergini and Saraceno [11] allows one to compute a number of states of a quantum billiard, or any other system described by the Helmholtz equation with the Dirichlet boundary condition, lying close to a chosen reference value of the wavenumber k without losing any state in a chosen interval. The latter problem has been resolved by Bäcker [9] by calculating not the determinant directly but the singular value decomposition (SVD) of the matrix A and following the behaviour of individual singular values At this point, we introduce a similar procedure to the one presented for the scaling method, namely trying to determine the solution close to a chosen reference value of the wavenumber k0 by varying k = k0 + δk. In terms of the order of the formal perturbation parameter , whose powers give the order of the terms with respect to the small parameter δk0 and should be set to one in the final result This shows that the procedure can only be applied to those eigenvectors u whose wavenumbers lie close to the reference value k0. We may eliminate the spurious solutions by solving the Neumann problem for each of the holes by performing the same computation for each hole, where the boundary taken is just the boundary of the hole in question, and discard the levels obtained for each hole from the spectrum of the total problem
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