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In [[mathematical physics]], the '''almost Mathieu operator'''<!--, named after [[?????? Mathieu]], --> arises in the study of the [[quantum Hall effect]]. It is given by
: <math> [H^{\lambda,\alpha}_\omega u](n) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, </math>


acting as a [[self-adjoint operator]] on the Hilbert space <math>\ell^2(\mathbb{Z})</math>. Here <math>\alpha,\omega \in\mathbb{T}, \lambda > 0</math> are parameters. In [[pure mathematics]], its importance comes from the fact of being one of the best-understood examples of an [[ergodic]] [[Schrödinger operator]].  For example, three problems (now all solved) of [[Barry Simon]]'s fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator.<ref>{{cite book |first=Barry |last=Simon |chapter=Schrödinger operators in the twenty-first century |title=Mathematical Physics 2000 |pages=283–288 |publisher=Imp. Coll. Press |location=London |year=2000 |isbn=186094230X }}</ref>


For <math>\lambda = 1</math>, the almost Mathieu operator is sometimes called '''Harper's equation'''.
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== The spectral type ==
 
If <math>\alpha</math> is a [[rational number]], then <math>H^{\lambda,\alpha}_\omega</math>
is a periodic operator and by [[Floquet theory]] its [[spectrum (functional analysis)|spectrum]] is purely [[absolutely continuous]].
 
Now to the case when <math>\alpha</math> is [[Irrational number|irrational]].
Since the transformation <math> \omega \mapsto \omega + \alpha </math> is minimal, it follows that the spectrum of <math>H^{\lambda,\alpha}_\omega</math> does not depend on <math> \omega </math>. On the other hand, by ergodicity, the supports of absolutely continuous, singular continuous, and pure point parts of the spectrum are almost surely independent of <math> \omega </math>.
It is now known, that
*For <math>0 < \lambda < 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has surely purely absolutely continuous spectrum. <ref>{{cite journal |first=A. |last=Avila |year=2008 |title=The absolutely continuous spectrum of the almost Mathieu operator |work=Preprint |arxiv=0810.2965 }}</ref> (This was one of Simon's problems.)
*For <math>\lambda= 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has almost surely purely singular continuous spectrum.<ref>{{cite journal |last=Gordon |first=A. Y. |last2=Jitomirskaya |first2=S. |last3=Last |first3=Y. |last4=Simon |first4=B. |title=Duality and singular continuous spectrum in the almost Mathieu equation |journal=[[Acta Mathematica|Acta Math.]] |volume=178 |year=1997 |issue=2 |pages=169–183 |doi=10.1007/BF02392693 }}</ref> (It is not known whether eigenvalues can exist for exceptional parameters.)
*For <math>\lambda > 1</math>, <math>H^{\lambda,\alpha}_\omega</math> has almost surely pure point spectrum and exhibits [[Anderson localization]].<ref>{{cite journal |last=Jitomirskaya |first=Svetlana Ya. |title=Metal-insulator transition for the almost Mathieu operator |journal=[[Annals of Mathematics|Ann. of Math.]] |volume=150 |year=1999 |issue=3 |pages=1159–1175 |doi= |jstor=121066 }}</ref> (It is known that almost surely can not be replaced by surely.)<ref>{{cite journal |first=J. |last=Avron |first2=B. |last2=Simon |title=Singular continuous spectrum for a class of almost periodic Jacobi matrices |journal=[[Bulletin of the American Mathematical Society|Bull. Amer. Math. Soc.]] |volume=6 |year=1982 |issue=1 |pages=81–85 |doi= |zbl=0491.47014 }}</ref><ref>{{cite journal |first=S. |last=Jitomirskaya |first2=B. |last2=Simon |title=Operators with singular continuous spectrum, III. Almost periodic Schrödinger operators |journal=[[Communications in Mathematical Physics|Comm. Math. Phys.]] |volume=165 |year=1994 |issue=1 |pages=201–205 |zbl=0830.34074 }}</ref>
 
That the spectral measures are singular when <math> \lambda \geq 1 </math> follows (through the work of Last and Simon)
<ref>{{cite journal |first=Y. |last=Last |first2=B. |last2=Simon |title=Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators |journal=[[Inventiones Mathematicae|Invent. Math.]] |volume=135 |year=1999 |issue=2 |pages=329–367 |doi=10.1007/s002220050288 }}</ref>
from the lower bound on the [[Lyapunov exponent]] <math>\gamma(E)</math> given by
: <math> \gamma(E) \geq \max \{0,\log(\lambda)\}. \, </math>
 
This lower bound was proved independently by Avron, Simon and [[Michael Herman (mathematician)|Michael Herman]], after an earlier almost rigorous argument of Aubry and André. In fact, when <math> E </math> belongs to the spectrum, the inequality becomes an equality (the Aubry-André formula), proved by [[Jean Bourgain]] and Svetlana Jitomirskaya.<ref>{{cite journal |first=J. |last=Bourgain |first2=S. |last2=Jitomirskaya |title=Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential |journal=[[Journal of Statistical Physics]] |volume=108 |year=2002 |issue=5–6 |pages=1203–1218 |doi=10.1023/A:1019751801035 }}</ref>
 
== The structure of the spectrum ==
 
[[Image:Hofstadter's_butterfly.png|thumb|Hofstadter's Butterfly]]
 
Another striking characteristic of the almost Mathieu operator is that its spectrum is a [[Cantor set]] for all irrational <math>\alpha</math> and <math>\lambda > 0</math>. This was shown by [[Artur Avila|Avila]] and [[Svetlana Jitomirskaya|Jitomirskaya]] solving the by-then famous "Ten Martini Problem"<ref>{{cite journal |first=A. |last=Avila |first2=S. |last2=Jitomirskaya |title=The Ten Martini problem |work=Preprint |year=2005 |id={{ArXiv|math|0503363}} }}</ref> (also one of Simon's problems) after several earlier results (including generically<ref>{{cite journal |first=J. |last=Bellissard |first2=B. |last2=Simon |title=Cantor spectrum for the almost Mathieu equation |journal=[[Journal of Functional Analysis|J. Funct. Anal.]] |volume=48 |year=1982 |issue=3 |pages=408–419 |doi=10.1016/0022-1236(82)90094-5 }}</ref> and almost surely<ref>{{cite journal |last=Puig |first=Joaquim |title=Cantor spectrum for the almost Mathieu operator |journal=Comm. Math. Phys. |volume=244 |year=2004 |issue=2 |pages=297–309 |doi=10.1007/s00220-003-0977-3 }}</ref> with respect to the parameters).
 
Furthermore, the measure of the spectrum of the almost Mathieu operator is known to be
: <math>Leb(\sigma(H^{\lambda,\alpha}_\omega)) = |4 - 4 \lambda| \, </math>
 
for all <math>\lambda > 0</math>. For <math> \lambda = 1 </math> this means that the spectrum has zero measure (this was first proposed by [[Douglas Hofstadter]] and later became one of Simon's problems<ref>{{cite journal |first=A. |last=Avila |first2=R. |last2=Krikorian |title=Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles |journal=[[Annals of Mathematics]] |volume=164 |year=2006 |issue=3 |pages=911–940 |doi=10.4007/annals.2006.164.911 }}</ref>). For <math> \lambda \neq 1 </math>, the formula was discovered numerically by Aubry and André and proved by Jitomirskaya and Krasovsky.
 
The study of the spectrum for <math> \lambda =1 </math> leads to the [[Hofstadter's butterfly]], where the spectrum is shown as a set.
 
== References ==
 
{{reflist|2}}
 
[[Category:Spectral theory]]
[[Category:Mathematical physics]]

Latest revision as of 23:18, 27 February 2014


Before heading to the celebration it's a good concept to have your concerns prepared in your thoughts. Pace courting is very fast-fire as you want to know as a lot as feasible after your 3 minutes is up.



There is absolutely nothing that will flip off a lady faster than a guy who seems to be full of himself. That is accurate for nerd dating as well as for dating in the real world. Girls don't like to be around guys who think they are "all that and a bag of chips." They do like a man who is confident but not arrogant, they like a guy who is interesting, but not just intrigued in himself or intercourse.

The first thing to keep in mind on a baby boomers dating website is that no matter how a lot people say they like people for "who they are", the bottom line is, they decide you by your look initial.

According to the US Division of justice a woman gets raped each two minutes in The united states. And forty seven %25 of these women had been on a date. Yes many have discovered true love off the Internet and have even start dating or even married.

Pursuing a woman or a man to a day will require a great deal of research as 1 requirements to know well about the person. Getting to know about their interests and skills is extremely vital. It is essential to know what you need from your cherished 1 as it will be useful for you to tell a reason so as to day with him or her. Sustaining a great physique language is a basic requirement for you to day as good body language produces a good image and identification.

In good courting things transfer on quickly because there is no worry of rejection or misuse. What happens to many failing partnership is, two dating individuals move in different paces. 1 may want to head for marriage after sometime of courting whilst the other wants to go sluggish before stating "i do". This lack of reading from the exact same script may untrue the partners to go different ways. geek dating guarantees that a ideal match is found where two individuals have the exact same courting agendas and share a common see about courting as nicely as the world at big. A pace day reveals as a lot about him/herself in a time span of 3-five minutes. You may believe 5 minutes is small time to discover a perfect match but the environment of the http://GeekDates.net occasion guarantees that all the expectations are satisfied.

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