content_image/373750.jpg content_image/373749.jpg content_image/373748.jpg content_image/373747.jpg content_image/373746.jpg
For analysis of the sum σ$_{c}$ ( ⃗ k ) near the point ⃗ k = ⃗ k$_{0}$ we note that because of the evident relations 

<img src='content_image/373750.jpg'>

the value of σ$_{c}$ ( ⃗ k$_{0}$ ) = 0. The next terms of the expansion σ$_{c}$ ( ⃗ k ) over the vector components ⃗ q = ⃗ k − ⃗ k$_{0}$ can be written easily, 

<img src='content_image/373749.jpg'>

They comprise both real terms proportional to q x − q y and imaginary terms like q$_{x}$q$_{y}$ . After simple algebra, these terms are presented via the sum D introduced above and one more convergent sum of similar structure 

<img src='content_image/373748.jpg'>

Finally, in the quadratic approximation over ⃗ q = ⃗ k − ⃗ k$_{0}$ near the point ⃗ k$_{0}$ of the type (1,1) one can present the sum as 

<img src='content_image/373747.jpg'>

This function, in contrast with σ ( ⃗ k ) near the same point ⃗ k$_{0}$ , is anisotropic. However, expressions for the frequencies contain σ ( ⃗ k ) and σ$_{c}$ ( ⃗ k ) only in the combination σ $^{2}$( ⃗ k ) − 9 | σ$_{c}$ ( ⃗ k ) | $^{2}$, therefore for presentation of the spectrum of collective modes accurate to | ⃗ q | 2 it is not essential to take σ$_{c}$ ( ⃗ k ) into account, and to the accuracy of q 2 the spectrum near the point ⃗ k$_{0}$ of the type (1,1) is radially symmetric over components of the vector ⃗ q . 

The point with ⃗ k$_{0}$ of the type (1,0) including the points like ⃗ k$_{0}$ = ± π⃗ e$_{x}$/a or ⃗ k$_{0}$ = ± π⃗ e$_{y}$/a , possess lower (biaxial) symmetry than the points ⃗ k = 0 or points of the type (1,1). The rather complicated study in the vicinity of this point ( ⃗ q = ⃗ k − ⃗ k$_{0}$ is small) shows the analytical dependence for the components of ⃗ q , in this case for the components of ⃗ q parallel and perpendicular to ⃗ k$_{0}$ 

<img src='content_image/373746.jpg'>

