翻訳と辞書 Words near each other ・ Lieboch ・ Liebowitz ・ Liebowitz social anxiety scale ・ Liebschützberg ・ Liebsdorf ・ Liebshausen ・ Liebstadt ・ Liebstedt ・ Liebster Gott, wenn werd ich sterben? BWV 8 ・ Liebster Immanuel, Herzog der Frommen, BWV 123 ・ Liebster Jesu, mein Verlangen, BWV 32 ・ Liebt sie dich so wie ich? ・ Liebvillers ・ Lieb–Liniger model ・ Lieb–Oxford inequality ・ Lieb–Thirring inequality ・ Liechelkopf ・ Liechtenstein ・ Liechtenstein alcohol tax referendum, 1929 ・ Liechtenstein Alps referendum, 1967 ・ Liechtenstein at the 1936 Summer Olympics ・ Liechtenstein at the 1936 Winter Olympics ・ Liechtenstein at the 1948 Summer Olympics ・ Liechtenstein at the 1948 Winter Olympics ・ Liechtenstein at the 1952 Summer Olympics ・ Liechtenstein at the 1956 Winter Olympics ・ Liechtenstein at the 1960 Summer Olympics ・ Liechtenstein at the 1960 Winter Olympics ・ Liechtenstein at the 1964 Summer Olympics ・ Liechtenstein at the 1964 Winter Olympics Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Lieb–Thirring inequality ： ウィキペディア英語版 Lieb–Thirring inequality In mathematics and physics, Lieb–Thirring inequalities provide an upper bound on the sums of powers of the negative eigenvalues of a Schrödinger operator in terms of integrals of the potential. They are named after E. H. Lieb and W. E. Thirring. The inequalities are useful in studies of quantum mechanics and differential equations and imply, as a corollary, a lower bound on the kinetic energy of $N$ quantum mechanical particles that plays an important role in the proof of stability of matter.〔E. H. Lieb, W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger hamiltonian and their relation to Sobolev inequalities, Studies in Mathematical Physics, Princeton University Press (1976), 269–303〕 ==Statement of the inequalities== For the Schrödinger operator $-\backslash Delta+V(x)=-\backslash nabla^2+V(x)$ on $\backslash mathbb^n$ with real-valued potential $V(x):\backslash mathbb^n\backslash to\backslash mathbb$, the numbers $\backslash lambda\_1\backslash le\backslash lambda\_2\backslash le\backslash dots\backslash le0$ denote the (not necessarily finite) sequence of negative eigenvalues. Then, for $\backslash gamma$ and $n$ satisfying one of the conditions $\backslash begin$ \gamma\ge\frac12&,\,n=1,\\ \gamma>0&,\,n=2,\\ \gamma\ge0&,\,n\ge3, \end there exists a constant $L\_$, which only depends on $\backslash gamma$ and $n$, such that where $V(x)\_-:=\backslash max(-V(x),0)$ is the negative part of the potential $V$. The cases $\backslash gamma>1/2,n=1$ as well as $\backslash gamma>0,n\backslash ge2$ were proven by E. H. Lieb and W. E. Thirring in 1976 〔 and used in their proof of stability of matter. In the case $\backslash gamma=0,\; n\backslash ge3$ the left-hand side is simply the number of negative eigenvalues, and proofs were given independently by M. Cwikel.,〔M. Cwikel, Weak type estimates for singular values and the number of bound states of Schrödinger operators, Ann. of Math. (2) 106 (1977), no. 1, 93–100〕 E. H. Lieb 〔E. H. Lieb, Bounds on the eigenvalues of the Laplace and Schroedinger operators, Bull. Amer. Math. Soc. 82 (1976), no. 5, 751–753〕 and G. V. Rozenbljum.〔G. V. Rozenbljum, Distribution of the discrete spectrum of singular differential operators, Izv. Vysš. Učebn. Zaved. Matematika (1976), no. 1(164), 75–86〕 The resulting $\backslash gamma=0$ inequality is thus also called the Cwikel–Lieb–Rosenbljum bound. The remaining critical case $\backslash gamma=1/2,\; n=1$ was proven to hold by T. Weidl 〔T. Weidl, On the Lieb–Thirring constants $L\_$ for $\backslash gamma\; \backslash ge\; 1/2$, Comm. Math. Phys. 178 (1996), no. 1, 135–146〕 The conditions on $\backslash gamma$ and $n$ are necessary and cannot be relaxed. 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「Lieb–Thirring inequality」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.