Semiclassical asymptotic behavior of the lower spectral bands of the Schrödinger operator with a trigonal-symmetric periodic potential

We study the semiclassical approximation of the lower bands of the Schrödinger operator with a periodic two-dimensional potential with a trigonal symmetry and consider the cases where the potential has one or two wells in the elementary cell. We obtain the exponentially small asymptotic behavior of the band width and find the dispersion relations. We investigate the form of the Bloch functions. Solving this problem is the first step in studying the more complicated (and more physically interesting) problem of tunnel effects in rotating dimers.