On the essential spectrum of two-dimensional periodic magnetic Schrodinger operators

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 result, which stated that, in the case of compactly supported wells, the mini-bands shrink Gaussian-like with R.