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<s46>GLYPH<s48> 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<s68>GLYPH<s40>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> (a) GLYPH<s48> GLYPH<s50> GLYPH<s52> GLYPH<s54> GLYPH<s56> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s49> GLYPH<s48>GLYPH<s46>GLYPH<s50> GLYPH<s48>GLYPH<s46>GLYPH<s51> GLYPH<s68>GLYPH<s40>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> (b) GLYPH<s49> GLYPH<s50> GLYPH<s51> GLYPH<s52> GLYPH<s48>GLYPH<s46>GLYPH<s48> GLYPH<s48>GLYPH<s46>GLYPH<s53> GLYPH<s49>GLYPH<s46>GLYPH<s48> GLYPH<s49>GLYPH<s46>GLYPH<s53> GLYPH<s68>GLYPH<s40>GLYPH<s32>GLYPH<s32>GLYPH<s32>GLYPH<s41> (c) FIG. 3: The profile of the density of state function, normalized by the condition ∫ D ( ω ) dω/ω$_{0}$ = 1, for different values of the parameter λ = ω$_{0}$/ω$_{int}$ . a) weak interaction, λ = 6 > λ$_{c}$ ; b) strong interaction, λ = 1 . 5 < λ$_{c}$ ; c) special case λ = λ$_{c}$ , where more strong singularity appears. ( ⃗ k = (1 , 1)). Therefore, near the upper and lower edges of the frequency band, the character of the singularities of the density of states D ( ω ) is universal, D ( ω ) ∝ ( ω − ω$_{0}$ ) near the ω$_{0}$ (see Eq. (13)), and D ( ω ) has a finite jump near the maximal frequency, ω$_{max}$ . At all values of λ , the logarithmic singularity of the form of Δ D ( ω ) = C · ln [ ω$_{c}$/ | ω − ω$_{c}$ | ] is also present (see Fig. 3). For large λ , corresponding to a weak interaction of particles, the frequency grows with | ⃗ k | for all directions of ⃗ k . In this case the situation is standard: saddle points are located at four symmetrical points of the type of X (1,0), and only the three aforementioned singularities are present in the density of state function, see Fig. 3a. However, for small λ , λ < λ$_{c}$ = 3 . 6 the dependence ω ( ⃗ k ) for ⃗ k || (0 , 1) is non-monotonous: a local minimum with a standard parabolic 12