Physics - Mathematics Department - California Institute of Technology

More documents

Recommendations

Info

$MATH 5b; Solutions to Homework Set 1 January 2009 1. As K is a ...$

640 E. H. Lieb, I. M. Sigal, B. Simon and W. Thirring where G α is given by (3.3). The symmetry condition (1) is obvious, and (3) holds since φ(y)= 1 if y^l. F a Φ0 implies that 1*1^1-2^ \x a \ ^(1 -ε^lxU, and Σ bfa since φ(y) Φ 0 implies \y| ^ 1 — 2ε. Since \x — y\~* ^ G α (x,j/), we have proven (4). That leaves the key conditions (5') and (2'). For (2'), we use Theorem 3.2. This guarantees us that there is an a where the last two factors in F a are 1, and so there is an a with F a(x) =
Approximate Neutrality of Large-Z Ions have that Thus bfa X \Vbηa\ 2 ^N-HlnNf\xa\- 2 \φ'(N-'\xa\ X Gα^(xfl,x&))T. (4.6) As a final gradient estimate, since |VJx α | | = 1 and X bfa (x a ,x b ) 2 ], J and α ^ 1. Thus, since |xj X G α ^(x fl ,x b ) ^ N when φ'( ) ^0, 1 f a — 1 i i V /^ / xJ > G^ (x fl ,x ι a\ JLJ a N \ ai Since |x a | ^(1 — 2ε) 5|x|oo on suppF α , we see that, for α#0. by using (4.3)-(4.7). Thus Σ a,b Since Xi 72 ^ i, we can take γ = N~ 1/2 and obtain: or X bfa 641 (4-7) as required. The J's constructed in this way are continuous but are only piecewise C 1 . By convoluting with a smooth, totally symmetric function of very small support, we can arrange for C 00 J's which still obey the required properties. • Remark. By using φ = ψ m in the above construction for m suitable, we can reduce N 1/2 to any desired positive power of N. 5. The Main Theorem Here we will prove Theorem 1.1. Given the construction in the last section, this follows Sigal [12] fairly closely. Pick ε > 0. We shall prove ϊϊm N(Z)/Z ^ (1 - 3ε)~ K
Page 1 and 2: Communications in Commun. Math. Phy
Page 3 and 4: Approximate Neutrality of Large-Z I
Page 5: Approximate Neutrality of Large-Z I
Page 9 and 10: Approximate Neutrality of Large-Z I

Physics - Mathematics Department - California Institute of Technology

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?