Communications In Mathematical Physics - Volume 267 by M. Aizenman (Chief Editor)

By M. Aizenman (Chief Editor)

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Part 2. Let V1 and V2 denote the respective eigenspaces for the Laplacian on P that contain f 1 and f 2 . The representation theory for the diagonal action of P on ×2 P can be used to decompose the space V1 × V2 as a finite direct sum of spaces, ⊕v Wv , where the restriction to the diagonal in P × P identifies Wv with a direct sum of copies of a single eigenspace of the Laplacian on P. This notation is such that pairs Wv and Wv with v = v map via this restriction homorphism to distinct eigenspaces of the Laplacian on P.

23). 25) 1≤a≤d σ 2σ where υ(τ ) = Vol(Y ) τ or 2π ln(τ ), and where bi j (τ ) is bounded as τ → ∞. Here, the case with υ (τ ) linear in τ arises when Y is compact and L annihilates the constants; and the other case arises when Y = R and m = 0. 26) 1≤i, j≤N 1≤a≤d where fa ≡ 1≤i≤N (∂ai + ∂ai ). 24). 24) is valid with Aˆ (∗τ ·z,τ ·z) replaced by Aˆ ≡ j ac (z i , z j ) 1≤i, j≤N ∂ai ∂a + 1≤a≤d 1≤i, j≤N 1≤i, j≤N j bi j (τ ) + ∂ai ∂a 1≤a≤d j ac (z i , z j ) 1 +υ(τ ) 2 ∂ai ∂a 1≤a≤d j j (∂ai ∂a +∂ai ∂a ).

4. The existence of a τ -independent bound can be deduced by first writing = α bα α , where bα ∈ C and where each α is either constant or has the form (z,F) . Writing in this way exhibits the fact that it is sufficient to find a τ independent bound for any given (z, F) version of Q(Rτ (z,F) , Rτ (z,F) ). The existence of the latter bound is argued next. Consider first the cases where Y is compact and L has no kernel, where Y = Rn>1 , and where Y = R and m > 0. To start, fix some very large number r , and use R ∈ ×2N R to denote the diagonal point (r, .

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