Editing Bloch's theorem (section)

== History and related equations ==

The concept of the Bloch state was developed by Felix Bloch in 1928<ref>{{cite journal|author=Felix Bloch|author-link=Felix Bloch|title=Über die Quantenmechanik der Elektronen in Kristallgittern|journal=Zeitschrift für Physik| volume=52 | issue=7–8| pages=555–600 |year=1928|doi=10.1007/BF01339455|bibcode = 1929ZPhy...52..555B |s2cid=120668259|language=de}}</ref> to describe the conduction of electrons in crystalline solids. The same underlying mathematics, however, was also discovered independently several times: by [[George William Hill]] (1877),<ref>{{cite journal|doi=10.1007/BF02417081| author=George William Hill|author-link=George William Hill|title=On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon|journal=Acta Math.|volume=8|pages=1–36 |year=1886|url=https://zenodo.org/record/1691491|doi-access=free}} This work was initially published and distributed privately in 1877.</ref> [[Gaston Floquet]] (1883),<ref>{{cite journal|author=Gaston Floquet|author-link=Gaston Floquet | title=Sur les équations différentielles linéaires à coefficients périodiques|journal= Annales Scientifiques de l'École Normale Supérieure|volume=12|pages=47–88 |year=1883|doi=10.24033/asens.220|doi-access=free}}</ref> and [[Alexander Lyapunov]] (1892).<ref>{{cite book|author=Alexander Mihailovich Lyapunov|author-link=Aleksandr Lyapunov|title=The General Problem of the Stability of Motion|location=London|publisher= Taylor and Francis|year= 1992}} Translated by A. T. Fuller from Edouard Davaux's French translation (1907) of the original Russian dissertation (1892).</ref> As a result, a variety of nomenclatures are common: applied to [[ordinary differential equations]], it is called [[Floquet theory]] (or occasionally the ''Lyapunov–Floquet theorem''). The general form of a one-dimensional periodic potential equation is [[Hill differential equation|Hill's equation]]:<ref name=Magnus_Winkler>
{{cite book 
|first1=W|last1= Magnus |author-link=Wilhelm Magnus|first2=S|last2= Winkler
|title=Hill's Equation
|year= 2004 
|page=11 
|publisher=Courier Dover
|isbn=0-486-49565-5
|url=https://books.google.com/books?id=ML5wm-T4RVQC&q=%22hill's+equation%22}}
</ref>
<math display="block">\frac {d^2y}{dt^2}+f(t) y=0, </math>
where {{math|''f''(''t'')}} is a periodic potential. Specific periodic one-dimensional equations include the [[Kronig–Penney model]] and [[Mathieu function|Mathieu's equation]].

Mathematically Bloch's theorem is interpreted in terms of unitary characters of a lattice group, and is applied to [[spectral geometry]].<ref>Kuchment, P.(1982), ''Floquet theory for partial differential equations'', RUSS MATH SURV., 37, 1–60</ref><ref>{{cite journal |author=Katsuda, A. |author2=Sunada, T |author2-link=Toshikazu Sunada |year=1987 |title=Homology and closed geodesics in a compact Riemann surface |journal=Amer. J. Math. |volume=110 |issue=1 |pages=145–156 |doi=10.2307/2374542| jstor=2374542 }}</ref><ref>{{cite journal |author=Kotani M |author2=Sunada T. |year=2000 |title=Albanese maps and an off diagonal long time asymptotic for the heat kernel |journal=Comm. Math. Phys. |volume=209 |issue=3 |pages=633–670 |doi=10.1007/s002200050033 | bibcode = 2000CMaPh.209..633K |s2cid=121065949 }}</ref>