to the AFM state with chessboard orientation of the dot’s magnetic moment. 

For all magnetic states of the array investigated here, the dispersion relation is non- analytic as ⃗ k → 0 because of the long-range nature of the dipolar interaction of oscillating magnetic moments of dots and the presence of singularities in dipolar sums. For such modes with finite gap frequency, ω$_{0}$ the magnon spectra have the peculiarity of the type ω ( ⃗ k ) − ω$_{0}$ ∝ | ⃗ k | as ⃗ k → 0. For the AFM structure without a magnetic field, a new sort of singularity, leading to saddle point with four-fold symmetry of the function ω ( ⃗ k ) near the value ⃗ k = 0 is predicted. An important and non-trivial property of most of the modes considered here is the strong anisotropy of the function ω ( ⃗ k ). Note also the decreasing or non-monotonic dependence of the mode frequency ω ( ⃗ k ) on the wave vector ⃗ k , observed for non-small interaction of dots. 

The non-standard dependence ω ( ⃗ k ) for small k produce the change of the density of states D ( ω ). For the spectrum of the FM state, instead of standard two-dimensional Van Hove singularities of a form of finite jump near the gap frequency, ω$_{0}$ , or singularities like ln ( ω$_{c}$/ | ω$_{c}$ − ω | ), weaker singularities of the form D ( ω ) ∼ ( ω − ω$_{0}$ )Θ( ω − ω$_{0}$ ) appear here. On the other hand, for dispersion relations for the FM state (11) the change of the character of its extremum at the point X of the type of (1,0) is predicted at some ratio of the system parameters, ω$_{0}$/ω$_{int}$ ≃ 3 . 6 (see Fig. 2). For this critical value of the parameter ω$_{0}$/ω$_{int}$ the density of states has the singularity D ( ω ) ∝ 1 / | ω − ω$_{c}$ | 1 / $^{4}$, where ω$_{c}$ is the value of frequency at this critical point, which is stronger than the standard finite jump. The value of the parameter ω$_{0}$/ω$_{int}$ depends on the external magnetic field and can be changed continuously during the experiment. 

∗ Electronic address: bivanov@i.com.ua 

1 R. Skomski, J. Phys.: Condens. Matter 15 , R841 (2003); Advanced Magnetic Nanostructures , edited by D. J. Sellmyer and R. Skomski, (Springer, Berlin, 2006). 

2 B. Hillebrands, C. Mathieu, C. Hartmann, M. Bauer, O. Buettner, S. Riedling, B. Roos, S. O. Demokritov, B. Bartenlian, C. Chappert, D. Decanini, F. Rosseaux, E. Cambril, A. Muller, B. Hoffmann, and U. Hartmann, J. Magn. Magn. Mater. 75, 10 (1997). 

3 C. Miramond, C. Fermon, F. Rousseaux, D. Decanini and F. Carcenac, J. Magn. Magn. Mat. 

