SPECTRAL PROPERTIES OF A CHARGED-PARTICLE IN ANTIDOT ARRAY - A LIMITING CASE OF QUANTUM BILLIARD

A model of the periodic array of quantum antidots in the presence of a uniform magnetic field is suggested. The model can be conceived as a periodic lattice of resonators (curvilinear triangles) connected throu gh ''infinitely small'' openings at the vertices of the triangles. The model Hamiltonian is obtained by means of operator extension theory i n indefinite metric spaces. In the case of rational magnetic flux thro ugh an elementary cell of the lattice, the dispersion equation is foun d in an explicit form with the help of harmonic analysis on the magnet ic translation group. It is proved, at least in the case of integer fl ux, that the spectrum of the model Hamiltonian consists of three parts : (1) Landau levels (they correspond to the classical orbits lying bet ween antidots); (2) extended states that correspond to the classical p ropagation trajectories; and (3) bound states satisfying the dispersio n equation; they correspond to the classical chaotic orbits rotating a round single antidots. Among other things, methods of finding the Gree n's function for some planar domains with curvilinear boundaries are d erived. (C) 1996 American Institute of Physics.