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We consider a general class of two-dimensional lattice sums consisting of complex powers s of inverse quadratic functions. We consider two cases, one where the quadratic function is negative definite and another more restricted case where it is positive definite. In the former, we use a representation due to H. Kober, and consider the limit u →∞, where the lattice becomes ever more elongated along one period direction (the one-dimensional limit). In the latter, we use an explicit evaluation of the sum due to Zucker and Robertson. In either case, we show that the one-dimensional limit of the sum is given in terms of ζ (2 s ) if Re( s )>1/2 and either ζ (2 s −1) or ζ (2−2 s ) if Re( s )s )=1/2. We also comment on a class of sums that involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle. We show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it; others are identically zero on it, while still others have no zeros on it.

A description of the properties of \L with complex characters is given. By using these, together with the more familiar \L with real characters, it is shown how certain two dimensional lattice sums, which previously could not be put into closed form, may now be expressed in this way., Comment: 21 pages, 1 table

We discuss general properties of doubly periodic sums over the square lattice, linking phased, Bloch-type sums in the direct lattice with displaced sums in the reciprocal lattice using the Poisson summation formula. We discuss cardinal points, where the sums reduce to a single product of Dirichlet L functions, and exhibit all cardinal points for the square lattice. We introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2,3,4,5. For the last case, the analytic expressions for the sums involve complementary L functions, or alternatively L functions with complex characters. © 2007 American Institute of Physics.

Series of the form Σk = 1∞(2k2k)−1 k−n may be expressed as log sin integrals and are shown to be summable exactly in terms of Dirichlets L-series for values of n up to and including 5. Other related series are also discussed and several exact results are given.

The two-pion exchange contributions to the Λ-N and Λ-Λ potentials areobtained on the basis of the Mandelstam representation of the scattering amplitude. Formulae for V 0 , V s and V T in the expression V = V 0 + V s ( σ 1 · σ 2 )+ V T S 12 +… are given in terms of the parameters related to the π-N and π-Λ scattering amplitudes. The quantity V 0 involves the S-wave π-λ scattering amplitude which is as yet experimentally unknown; V S and V T depend only on the P-wave gp-N and π-λ interactions, which are assumed to be dominated by the 3-3 resonance N ∗ and Y 1 ∗ resonance (1385 MeV), respectively. Higher partial waves are ignored. Only V S and V T are evaluated numerically. It is found that V S and V T are extremely small . This implies that the strongspin-independence of the phenomenological Λ-N force is characteristic of the short-range part of the potential.

Experimental results showing the effect of anode current on the frequency of crossed-field oscillators are presented. The results for M-type backward-wave oscillators are shown to be qualitatively explained simply by taking into account the potential deresion due to the presence of the electron beam.

Nogami has developed a method for treating the pairing interaction in nuclei by using a model Hamiltonian Hλ1Nλ2N2. Here H is theoriginal Hamiltonian and N the nucleon number operator. This method is applied here to several examples and its accuracy is discussed.