where ω is the probing parameter with which we "test" the given Hamiltonian Ĥ. In functional analysis, usually, it is introduced to research the spectrum (i.e., the set of eigenvalues) of a given operator simply by checking the analyticity of the resolvent operator (in this case Ĝ). The key idea here is that whenever ω, colloquially speaking, hits the eigenvalue λ, then the Green's function is supposed to diverge in such a way, that the numerator in the Laurent expansion 121 tells us right away about the eigenwavefunctions associated with the given eigenvalue λ. The advantages of this approach and its mathematical details will become clear in the next sections. If one has a task of understanding the indirect interactions happening through the surface, it seems obvious that taking into account the CDW or FOs is an essential step towards the successful solution. Derivation of a Friedel Oscillation problem, such as mentioned in has been known for decades, it should be not too hard to apply the results of it to the exact problem. The reality, though, is much tougher than at first glance, and the introduction of any change into the mathematical model of this system meets huge analytical complications. It is relatively easy (well, possible) to deliver the solution for a system with a spherical symmetry even in d-dimensional case (section 3.4), although any symmetry breakage like a crystal structure (sections 3.5, 3.6) complicates life drastically even if we consider only the electrostatic interactions only(we work in the green function approach and explain the relation between the interaction and its energy or Hamiltonian in section 3.3. If one wants to encounter more sophisticated interactions like RKKY interaction(the unprepared reader might think of it as simply the magnetic interactions), see section 3.8 or two-body interaction section 3.7. The abovementioned theory works for electron gas defined in a pretty abstract manner, thus the derived formalism should work as well for all systems with electron gas including the graphene (section 3.10). Section 3.11 is related to the dynamic CDWs or time-dependent FOs. We apologize in advance to anyone who might consider this part trivial.