# 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.

#### References

[a1] | D. Saxon, R. Hutner, Philips Res. Rep. , 4 (1949) pp. 81 |

[a2] | H. Matsuda, Progr. Theor. Phys. , 27 (1962) pp. 81 |

[a3] | J. Hori, Progr. Theor. Phys. , 32 (1964) pp. 371 |

[a4] | L. Dworin, Phys. Rev. , 138 A (1965) pp. 1121 |

[a5] | B. Tong, S. Tong, Phys. Rev. , 180 (1969) pp. 739 |

[a6] | I. Mladenov, C.R. Acad. Sci. (Sofia) , 38 (1985) pp. 993 |

[a7] | I. Mladenov, Phys. Lett. A , 137 (1989) pp. 313 |

**How to Cite This Entry:**

Saxon-Hutner theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Saxon-Hutner_theorem&oldid=48612