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Non-relativistic rates for the decay of 2s hydrogen atoms to the ground state by single-photon and two-photon emission in the presence of a homogeneous magnetic field of arbitrary strength (0 4.7 x lo6 T) are calculated by variational procedures. Over the whole range of B, two-photon emission is the dominant process. As the magnetic field grows, the two-photon decay rate increases. It is found that the Markov approximation can be applied to the two-photon decay for magnetic fields of strength B 3 4.7 x lo3 T. B

Abstract. Let ℓ > 0 be arbitrary. We introduce the extremal quantitieswhere the supremum is taken over all not identically zero non-negative positive definite functions. We are interested in the question how large can the above extremal quantities be? This problem was originally posed by Yu. Shteinikov and S. Konyagin for the case ℓ = 2. In this note we obtain exact values for the right limits lim ε→0+ G(k + ε) and lim ε→0+ C(k + ε) (k ∈ N), and sufficiently close bounds for other values of ℓ. We point out that the problem provides an extension of the classical problem of Wiener.MSC 2000 Subject Classification. Primary 42A82, 42A38, 26D15.

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