# Download Communications In Mathematical Physics - Volume 292 by M. Aizenman (Chief Editor) PDF By M. Aizenman (Chief Editor)

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2) Then the form v is compact on d[a]. This means that the form v is continuous on Hγ [a] and the corresponding operator Q (determined by the relations aγ [Qx, y] = v[x, y] for x, y ∈ d[a]) is compact on Hγ [a]. 3) on the domain D(H ) = (I +i Q)−1 D(A). It is clear that the operator H can be interpreted as the sum H = A + i V. Proposition 2. 3) is densely defined and closed. Proof. Let us first prove that H is densely defined. Assume the opposite, that there is a non-zero vector h ∈ d[a] such that aγ [(I + i Q)−1 u, h] = 0 for all vectors u ∈ D(A).

M of the operator H and has no intersection with the spectrum of H . Then the projection onto the span of the corresponding root vectors is given by the formula P= i 2π (H − z)−1 dz. 5) 34 A. Laptev, O. 4) with Hn instead of H , will converge to P. That means that the norm ||P − Pn || will be small for sufficiently large n. Now, in order to draw a conclusion about eigenvalues of Hn , we can apply the following statement. Lemma 1 (see, for example, ). If P and P0 are two projections such that rank P = rank P0 , then ||P − P0 || ≥ 1.

Moreover, suppose that λ j ∈ Cle f t are eigenvalues of the operator H , and τ j are negative eigenvalues of A enumerated in the order of increasing real parts. Then n n λj ≤ |τ j | 1 1 for all n. Indeed, let P be the orthogonal projection onto the span of eigenvectors x j corresponding to λ j , 1 ≤ j ≤ n. Then n tr H P = λj. 7) where the minimum is taken over all orthogonal projections P of rank n with the property RanP ⊂ d[a]. 8). Corollary 1. Let γ > 0. Then n n ( λ j + γ )− ≤ 1 (τ j + γ )− .