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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.

In particular, the spectrum of H is real if π 2 V (x) d x < 1. Proof. 2) defined in the proof of Theorem 9, then tr X= V (x) d x 1 dξ. 2π(|ξ |2 − z) R2 The next result deals with some properties of complex eigenvalues of Schrödinger operators in higher dimensions d ≥ 4. Theorem 12. Let d ≥ 4 and let z ∈ / R+ be an eigenvalue of H = − + i V with V ≥ 0. 3) where ωd−1 is the area of the unit sphere Sd−1 . Proof. 2) and z is an eigenvalue of the operator H , then 1/2 is an eigenvalue of the operator X − 1/2.

It is an elementary application of the Cwikel estimate. Indeed, according to the Birman-Schwinger principle N (E) = n(1, X ), where X is the compact operator defined by the equality √ √ X = V (α(i∇) + E)−1 V . Eigenvalues of Schrödinger Operators with Complex Potentials 39 Let χ be the characteristic function of the ball {|ξ | ∈ Rd : |ξ |2 ≤ µ}. Let us split X such that X = X 1 + X 2 , where √ √ X 2 = V (α(i∇) + E)−1 χ (i∇) V . According to the Ky Fan inequality, n(1, X ) ≤ n(1, 2X 1 ) + n(1, 2X 2 ).