# Saxon-Hutner theorem

A theorem concerning energy gaps in one-dimensional random alloy models described by the Schrödinger equation or the Dirac equation. It states that forbidden energies that are common to a pure crystal $A$ and a pure crystal $B$( with the same lattice constant) will always be forbidden energies in any arrangement of the atoms of $A$ and $B$ in a substitutional solid solution.

Making use of the one-to-one correspondence between the real localized atomic potentials and $( 2 \times 2 )$- transfer matrices belonging to any of the isomorphic three-dimensional Lie groups ${ \mathop{\rm SU} } ( 1,1 )$, ${ \mathop{\rm SL} } ( 2, \mathbf R )$ or ${ \mathop{\rm Sp} } ( 2, \mathbf R )$, this can be described in mathematical terms as follows. If $A ^ {r _ {1} } B ^ {s _ {1} } \dots A ^ {r _ {k} } B ^ {s _ {k} }$ is an arbitrary linear chain consisting of two types of atoms $A$ and $B$, each having representatives $r _ {i} , s _ {i} \in \mathbf Z ^ {+}$ in the $i$ th period. Then the group nature of the individual transfer matrices $M _ {A}$ and $M _ {B}$ makes it possible to express the total transfer matrix $M$ of the elementary cell as the product $M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {1} } M _ {A} ^ {r _ {1} }$, and the forbidden energies for electrons propagating there are given by the condition $| { { \mathop{\rm tr} } ( M ) } | > 2$.

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

$$\left | { { \mathop{\rm tr} } ( M _ {A} ) } \right | , \left | { { \mathop{\rm tr} } ( M _ {B} ) } \right | > 2,$$

is it true that for any arrangement $A ^ {r _ {_ 1 } } B ^ {s _ {_ 1 } } \dots A ^ {r _ {k} } B ^ {s _ {k} }$ of atoms of type $A$ and $B$ one has

$$\left | { { \mathop{\rm tr} } ( M _ {B} ^ {s _ {k} } M _ {A} ^ {r _ {k} } \dots M _ {B} ^ {s _ {_ 1 } } M _ {A} ^ {r _ {_ 1 } } ) } \right | > 2 ?$$

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

How to Cite This Entry:
Saxon-Hutner theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Saxon-Hutner_theorem&oldid=48612
This article was adapted from an original article by I.M. Mladenov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article