By Jean-Pierre Aubin

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E. Dt = I Ej(x) - Ej(x ' ) x' I-J. D, then I :S I aD(x) Icol x - x' I. Proof (i) is obvious. To see (ii), let I be the segment joining x and x' and =1 El(x) - E1(X' ) I. Then there is an x" E I such that, let fl IP('\,x)l~ (~)m, ,\=E 1 (x"), where P(,\,x) = I17:1('\ - Ej(x)). g. ) Hence (~)m :S I P(,\,x) - P(>"x") I :S I axP(>" ) Icol x - Since m m j=1 i,j=1 x" I. L. H. Eliasson 48 and I: 1 aD11~ m 1 aD 1 the result follows. To see (iii) let qi(x) be the eigenvector corresponding to Ei(x). Then (D(x) - Ei(x)I)qi(x) = 0, and if we differentiate the relation and take the scalar product with qi (x ) and use that the eigenvectors are orthogonal, then we get an estimate of aEi.

If D is Hermitian, then we also have The constants are independent of m. Proof. 2. By scaling we can assume that (3 = 1 if we replace r by ~. The polynomial P(,\,x) = det(M - D(x)) satisfies Vk 2:: 0, if just 1,\ 1< 2. - Choose a curve ~(x) in I A I~ 1 + piecewise constant in x - keeping a distance 2: ~ to all the roots El(x), ... , Em(x) of P(A, x) and surrounding the first n of these roots. ~(x) may consist of several components so we can choose it to be of length at most mrr. xP(A, ) ICk on I A I~ 1 Consider now the power symmetric functions in the first n roots: Pj(x) = EI(x)j + ...

Lemma 7. 1,', v') -clustering into n' -blocks. 1,', v', p'; n', P') be a perturbation of D I (D' - D)~ b < v'ce-~Ib-alb')k Vk ~ o. c and X, and on s. 3). 1,' many na,s which are separated by a distance at least v. ')4. 3) a fJ 32 L. H. ')2 b')k Vk::::: 0 which gives the result. 4), when the blocks are disjoint, is proven in the same way. 4) when the blocks are equal we consider Va,b(X, y) = Ua,b(X, y) when na(x + y) n nb(x) = 0 and Va,b(X, y) when na(x + y) = nb(x). We define Ve,d and v~,d in the same way as Ue,d and u~,d using v instead of u.