Spectral problem for plane periodical structures
P.G. Akishin, S.I. Vinitsky
Joint Institute for Nuclear Research,
141980 Dubna, Russia
For the plane Helmholtz problem of the periodical disc structures
with the nonzero phase shifts conditions
and the Dirichlet conditions on the disc boundaries the approach
based on the boundary integral equations for the simple layer potentials
of the lattice Green function is applied [1].
The mathematical aspects of the problem, such as the question of reality of
the spectrum , the problem of discretization of the continue boundary
integral equations, the question of solving arising discrete nonlinear
problem are considered.
The computer code for solving the plane Helmholtz problem has been created.
The comparison of numerical results for orthogonal basis lattice vectors
and zero phase shifts conditions with known ones [2] has been performed.
Some applications of the proposed approach for the
quantum billiards of a general type are given.