View source for Saxon-Hutner theorem

A theorem concerning energy gaps in one-dimensional random alloy models described by the [[Schrödinger equation|Schrödinger equation]] or the [[Dirac equation|Dirac equation]]. It states that forbidden energies that are common to a pure crystal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100501.png" /> and a pure crystal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100502.png" /> (with the same lattice constant) will always be forbidden energies in any arrangement of the atoms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100503.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100504.png" /> in a substitutional solid solution.

Making use of the one-to-one correspondence between the real localized atomic potentials and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100505.png" />-transfer matrices belonging to any of the isomorphic three-dimensional Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100507.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100508.png" />, this can be described in mathematical terms as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s1100509.png" /> is an arbitrary linear chain consisting of two types of atoms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005011.png" />, each having representatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005012.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005013.png" />th period. Then the group nature of the individual transfer matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005015.png" /> makes it possible to express the total transfer matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005016.png" /> of the elementary cell as the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005017.png" />, and the forbidden energies for electrons propagating there are given by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005018.png" />.

In the transfer-matrix approach, the theorem takes the form of the following question: Given that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005019.png" /></td> </tr></table>

is it true that for any arrangement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005020.png" /> of atoms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005022.png" /> one has

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s110/s110050/s11005023.png" /></td> </tr></table>

Relying on quite different techniques, several non-equivalent sufficient conditions guaranteeing its validity have been derived.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D. Saxon,   R. Hutner,   ''Philips Res. Rep.'' , '''4'''  (1949)  pp. 81</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Matsuda,   ''Progr. Theor. Phys.'' , '''27'''  (1962)  pp. 81</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Hori,   ''Progr. Theor. Phys.'' , '''32'''  (1964)  pp. 371</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Dworin,   ''Phys. Rev.'' , '''138 A'''  (1965)  pp. 1121</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B. Tong,   S. Tong,   ''Phys. Rev.'' , '''180'''  (1969)  pp. 739</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I. Mladenov,   ''C.R. Acad. Sci. (Sofia)'' , '''38'''  (1985)  pp. 993</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I. Mladenov,   ''Phys. Lett. A'' , '''137'''  (1989)  pp. 313</TD></TR></table>