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In this paper:

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.74.125402

In the appendix the author attempts to arrive at the spectral density of states of the surface of a half-space. To do this, he arrives at the Green's function of the surface atom of 1D atomic chain and then tries to extend it to the 3D problem with the following argument:

'In the case of a real 3D surface, we can split the problem

into multiple independent problems, each one corresponding

to a different wave vector parallel to the surface, q . Because

of parallel momentum conservation, each of these problems

is completely decoupled from the rest, and can be described

by an effective one-dimensional system. Now, instead of

having −2t for the diagonal elements, we must take into account

the dispersion parallel to the surface, so the diagonal

elements are kd=−2t+c^2 q^2 at the inner layers'

Here he is talking about the dynamical matrix of the 1D chain. On moving to 3D, he mentions that the diagonal elements should be added by an energy term c^2 q^2.

Is this obvious? I am not able to intuitively understand why the diagonal elements of the dynamic matrix will be shifted by this energy term.. Can anybody help me out with this? For e.g., Why just the diagonal terms? I tried to discuss this with the author, he mentioned - it is akin to the tight binding Hamiltonian for electrons, your diagonal element needs to be shifted by the energy corresponding to the momentum parallel to the surface.

I am afraid it is still not clear to me.

Thanks!

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# Dynamical matrix and dispersion

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