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How could we formulate the Bloch theorem for a semi-infinite crystal?

For simplicity I suggest assuming that the crystal boundary is along one of its crystallographic planes. One could also assume a mirror symmetry in respect to this plane in an infinite crystal.

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  • $\begingroup$ What are you trying to get at with the formulation? There will be additional states associated with the truncation of the crystal. $\endgroup$
    – Jon Custer
    Mar 29, 2020 at 16:30
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    $\begingroup$ @JonCuster I think the question has merits on its own. However in practice it is an offshoot of another question that I was thinking about, see here physics.stackexchange.com/a/540000/247642 $\endgroup$
    – Roger V.
    Mar 29, 2020 at 16:52
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    $\begingroup$ I wasn’t questioning the question, but asking for clarification. For example, crystal truncation x-ray scattering is a very surface sensitive technique - are you looking for something like that? $\endgroup$
    – Jon Custer
    Mar 29, 2020 at 17:03
  • $\begingroup$ @JonCuster Specifically here I am asking forr mathematical result. But to the question that I cited in the previous comment x-ray scattering may be relevant. I am mainly interested in theoretical description. $\endgroup$
    – Roger V.
    Mar 29, 2020 at 17:06

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It has something to do with the Wiener-Hopf method.

Basically, if you recall quantum mechanics and how the Green's function for a Schrodinger equation is found there, it is a Fourier transform of an evolution operator. However, the evolution operator has the initial condition, and, typically, you do not know pre-history, which prevents you from making the usual Fourier transform straightforwardly (similar to your half-infinite problem), Nevertheless, by splitting the Green's tensor (and the evolution operator, correspondingly), into an advanced and retarded parts, one can actually perform the Fourier transform.

Even though all of the above sounds abstract, but it is just an example of how this can be done. I can refer to the following article, where this has been used to solve a similar problem: https://journals.aps.org/prb/abstract/10.1103/PhysRevB.84.125402

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