GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s53> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s53> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s53> GLYPH<s48> GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s97> GLYPH<s32> GLYPH<s68>GLYPH<s40> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> GLYPH<s68>GLYPH<s40> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> GLYPH<s32> GLYPH<s32> (a) h=0. GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s50> GLYPH<s46>GLYPH<s53> GLYPH<s51>GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s97> GLYPH<s68>GLYPH<s40> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> GLYPH<s32> (b) h=2.5. GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s46>GLYPH<s52> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s50>GLYPH<s46>GLYPH<s50> GLYPH<s50>GLYPH<s46>GLYPH<s52> GLYPH<s50> GLYPH<s46>GLYPH<s54> GLYPH<s50> GLYPH<s46>GLYPH<s56> GLYPH<s51>GLYPH<s46>GLYPH<s48> GLYPH<s51> GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s53> GLYPH<s49>GLYPH<s48> GLYPH<s68>GLYPH<s40> GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> GLYPH<s97> GLYPH<s32> GLYPH<s32> (c) h=5. FIG. 7: The density of state for two branches of magnons for the AFM dot array with β = 5, normalized by the condition ∫ D ( ω ) dω/ω$_{a}$ = 1, ω$_{a}$ = γH$_{a}$ , for each band, for different values of the magnetic field. a) the field equals zero, pay attention that these two branches are overlapped. b) h = 2 . 5, at moderate field values a linear peculiarity of the density of states is present inside of the lower band of collective modes; c) h = 5 presents the case of large fields, where a linear peculiarity of the density of states corresponds to lower frequency, and the finite jump is present inside the low energy band. The characteristic frequencies are mentioned by vertical arrows. one expects the standard peculiarity of D ( ω ) like a finite discontinuity near the maximal frequency. For the upper band a nonstandard behavior manifests itself only at small fields h < 1 . 2, when at points like X(1,0) minima appear, while saddle points move along directions like (1,0) into the Brillouin zone. The situation here resembles that described for the FM case and we do not discuss it here. The scheme of Van Hove singularities for the lower branch are much less standard. First of all, for the low band at all values of field the global maximum is located at the points like X(1,0). A standard peculiarity of the density of states like a finite jump corresponds to this frequency. At h = 0 (the zero field case is special and will be considered separately) both at the edges of the Brillouin zone [points like M(1,1)] and in its center the minima are present, 21