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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Almost Mathieu operator ： ウィキペディア英語版 Almost Mathieu operator In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by : $(u)(n)\; =\; u(n+1)\; +\; u(n-1)\; +\; 2\; \backslash lambda\; \backslash cos(2\backslash pi\; (\backslash omega\; +\; n\backslash alpha))\; u(n),\; \backslash ,$ acting as a self-adjoint operator on the Hilbert space $\backslash ell^2(\backslash mathbb)$. Here $\backslash alpha,\backslash omega\; \backslash in\backslash mathbb,\; \backslash lambda\; >\; 0$ 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. For $\backslash lambda\; =\; 1$, the almost Mathieu operator is sometimes called Harper's equation. == The spectral type == If $\backslash alpha$ is a rational number, then $H^\_\backslash omega$ is a periodic operator and by Floquet theory its spectrum is purely absolutely continuous. Now to the case when $\backslash alpha$ is irrational. Since the transformation $\backslash omega\; \backslash mapsto\; \backslash omega\; +\; \backslash alpha$ is minimal, it follows that the spectrum of $H^\_\backslash omega$ does not depend on $\backslash omega$. 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 $\backslash omega$. It is now known, that *For , $H^\_\backslash omega$ has surely purely absolutely continuous spectrum. (This was one of Simon's problems.) *For $\backslash lambda=\; 1$, $H^\_\backslash omega$ has almost surely purely singular continuous spectrum. (It is not known whether eigenvalues can exist for exceptional parameters.) *For $\backslash lambda\; >\; 1$, $H^\_\backslash omega$ has almost surely pure point spectrum and exhibits Anderson localization. (It is known that almost surely can not be replaced by surely.) That the spectral measures are singular when $\backslash lambda\; \backslash geq\; 1$ follows (through the work of Last and Simon) from the lower bound on the Lyapunov exponent $\backslash gamma(E)$ given by : $\backslash gamma(E)\; \backslash geq\; \backslash max\; \backslash .\; \backslash ,$ This lower bound was proved independently by Avron, Simon and Michael Herman, after an earlier almost rigorous argument of Aubry and André. In fact, when $E$ belongs to the spectrum, the inequality becomes an equality (the Aubry-André formula), proved by Jean Bourgain and Svetlana Jitomirskaya. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Almost Mathieu operator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.