This paper based on our previous works which dealth with ultrathin crystalline films. The quantum wires are quasi $1$D crystalline systems bounded in two perpendicular directions ($y$ and z) while infinite in the $x$ direction. The phonon excitation energies are defined by the poles of Green's functions, which be found from the condition that the $3D$ determinant of the set of equation of motion vanishes. Using Chebyshev's polynomial representation of this block matrix, we obtain the solutions as roots of the characteristic matrix equation. By numerical analysis (for simplest cases: $N_{y/z}\in$ [2,5]) we conjectured that the solution of this equation is of the same form as for thin films, but with the discretization along the $y$ direction too.