By M. Aizenman (Chief Editor)

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The memoryless compound channel associated with I is given by the family {N ⊗l : S(H⊗l ) → S(K⊗l )}ł∈N,N ∈I. In the rest of the paper we will simply write I for that family. Each compound channel can be used in three different scenarios: 1. the informed decoder 2. the informed encoder 3. the case of uninformed users. In the following three subsections we will give definitions of codes and capacity for these cases. 1. The Informed Decoder. An (l, kl )-code for I with informed decoder is a pair (P l , {RlN : N ∈ I}) where: 1.

The use of complex potentials in nuclear physics. Ann. Phys. 45(1), 113–131 (1967) 3. : On the spectrum of singular boundary value problems. Matem. Sb. 55, 125–174 (1961) 4. : Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). Dordrecht: D. , 1987 5. : Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. Math. 106, 93–100 (1977) 6. : Linear operators and their spectra, Cambridge Studies in Advanced Mathematics 106, Cambridge: Cambridge University Press, 2007 7.

Thus, ( λ j − s)+ ≤ tr (T2 + 2s)− . 3) µ Let τ j be negative eigenvalues of T2 . 4) with p > 1/2 and d ≥ 2. If now q > p > 1/2 then ∞ |τ j |q = q E q−1 N (E)d E 0 j ≤C Rd W d/4+ p d x + µd/2−1 Rd W 1/2+ p d x |λ1 |q− p . 4) it follows that the lowest eigenvalue τ1 satisfies the inequality |τ1 |r −1/2 ≤ C Rd W d/4+r −1/2 d x + µd/2−1 Rd Wrdx =C µ (W ). Eigenvalues of Schrödinger Operators with Complex Potentials 45 Hence for q > p > 1/2 and r > 1 we arrive at |τ j |q ≤ C Rd j W d/4+ p d x + µd/2−1 Rd W 1/2+ p d x | µ (W )| 2(q− p)/(2r −1) .

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

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