On the essential spectrum of two-dimensional periodic magnetic Schrodingeroperators

For two-dimensional Schrodinger operators with a nonzero constant magnetic field perturbed by an infinite number of periodically disposed, long-range magnetic and electric wells, it is proven that when the inter-well distance (R) grows to infinity, the essential spectrum near the eigenvalues of the 'one well Hamiltonian' is located in mini-bands whose widths shrink faster than any exponential with R. This should be compared with our previous resu lt, which stated that, in the case of compactly supported wells, the mini-b ands shrink Gaussian-like with R.