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Let λ j be the eigenvalues of the operator H = −d 2 /d x 2 + V lying inside the semi-infinite strip b = {z : 0 < z < b, z > 0}. Then for any γ > 7/4, r ∈ (γ − 21 , γ ) and V ∈ L ∞ (Rd ) such that W = (|V |2 + 4 V )+ ∈ L r −1/4 we have γ −r | λ j |γ ≤ C||W ||∞ (b + W λj∈ 1/2 ∞ ) Rd b |W |r −1/4 d x . Proof. 6) where W ∞ appears when we estimate the lowest eigenvalue λ1 of the operator T2 = 21 (−d 2 /d x 2 − µ)2 − W . Now consider the part of the strip b = {z : 0 < z < b, z > 0} satisfying z > s > 0.

8). 4. Let T be a bounded operator in a Hilbert space, whose spectrum outside the unit circle {z : |z| > 1} is discrete. Suppose also that the essential spectrum of the operator (T ∗ T )1/2 is contained in [0, 1]. Let λ j be the eigenvalues of the operator T lying outside of the unit circle, and let s j > 1 be the eigenvalues of (T ∗ T )1/2 . 10) 1 for all values of n. One should mention also that, if one of the sequences ends at j = j0 , we extend it by setting it equal to 1 for j > j0 . This inequality was discovered for compact operators by H.

A. Appendix . . . . . . . . . . . . . . . . . . 55 60 62 74 79 80 87 91 92 94 94 1. Introduction The determination of capacities of quantum channels in various settings has been a field of intense work over the last decade. In contrast to classical information theory, to any 56 I. Bjelakovi´c, H. Boche, J. Nötzel quantum channel we can associate in a natural way different notions of capacity depending on what is to be transmitted over the channel and which figure of merit is chosen as the criterion for the success of the particular quantum communication task.