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

By M. Aizenman (Chief Editor)

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The condition that the sesquilinear form v is generated by a self-adjoint operator V is excessive. 3), as soon as we know that v[u, u] = aγ [Qu, u], where Q is compact in the space Hγ [a]. This remark allows one to consider the case when the elliptic operator A = 2 is perturbed by a differential operator of first order. Under the above assumptions, the difference between the resolvents of the operators A and H is compact. Hence, the spectrum σ (H ) of the operator H is discrete in C\σ (A). 2. Let H be as described above.

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.

That implies that non-real eigenvalues of − + it V escape any compact subset of C, as t → 0. It does not necessary imply that the eigenvalues tend to infinity as t → 0, because they might simply reach the positive real semi-axis for some t > 0 (see Theorem 15). Proof of Theorem 9. 2) has an eigenvalue 1. Eigenvalues of Schrödinger Operators with Complex Potentials 49 Suppose that z > 0. Then the real part of the operator X is positive and, consequently, the spectrum of this operator lies in the right half plane.

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