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Two‐magnon bound states in Heisenberg ferromagnets
1.A. W. Sáenz, W. W. Zachary, and S. Podgor, Bull. Am. Phys. Soc. 15, 1368 (1970).
2.In the case of magnon bound states (resonances), the adjective “low lying” indicates that the corresponding energy eigenvalue (real part of the energy) differs from the lower edge of the pertinent free magnon continuum by an energy which is small in comparison with a typical value of the width of the continuum.
3.In regard to the relevance of the nonexistence of low‐lying magnon bound states to the validity of low‐temperature spin wave calculations, see, e.g., M. Wortis, Phys. Rev. 138, A1126 (1965)
3.and T. Morita and T. Tanaka, J. Math. Phys. 6, 1152 (1965).
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4.R. Silberglitt and A. B. Harris, Phys. Rev. 174, 640 (1968).
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10.In the present paper, we have found it more convenient to refer to specific points, lines, and directions of symmetry than to the corresponding equivalence classes. Naturally, all statements about bound two‐magnon states and allied topics made here for the case when Γ lies along some such point, line, or direction hold at the vectors related to Γ by the pertinent symmetry group operations.
11.R. G. Boyd, Ph.D. Thesis, Univ. of California, Riverside, 1965;
11.R. G. Boyd and J. Callaway, Phys. Rev. 138, A1621 (1965).
12.W. M. Shaw, Bull. Am. Phys. Soc. 13, 501 (1968).
13.The beginnings of a theory of three interacting magnons has been given by C. K. Majumdar, Phys. Rev. B 1, 287 (1970),
13.and C. K. Majumdar and G. Mukhopadyay, Phys. Lett. A 31, 321 (1970).
14.For the sc structure, our definition of agrees with that of the corresponding “reduced zone” of Wortis, Ref. 9.
15.Exception made of one‐dimensional ferromagnets [See Wortis, Ref. 9, Appendix B], only arguments of a heuristic nature appear to have been given in the literature relative to the spacing of consecutive two‐magnon eigenvalues of of given Γ within the corresponding band (2.15) for large N.
16.W. Ledermann, Proc. R. Soc. A 182, 362 (1944).
17.H. Rollnik, Z. Phys. 145, 639 (1956).
18.S. Weinberg, Phys. Rev. 131, 440 (1963).
18.See also R. G. Newton, Scattering Theory of Waves and Particles (McGraw‐Hill, New York, 1966), Chap. 9.
19.The stated differentiability properties of the which were needed in our proof of (2.40), as well as the existence of eigenfunctions of with differentiability attributes strong enough to make meaningful the operations involved in that proof, were derived by combining the relevant facts in pp. 63–65 of T. Kato, Perturbation Theory for Linear Operators (Springer, New York, 1966) with straightforward arguments.
20.One of us (A.W.S.) intends to present this proof in a forthcoming Naval Research Laboratory Report.
21.J. Schwinger, Proc. Natl. Acad. Sci. USA 47, 122 (1961).
21.See also Weinberg, Ref. 18, and G. C. Ghirardi and A. Rimini, J. Math. Phys. 6, 40 (1965).
22.See, for example, G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities (Cambridge U.P., Cambridge, 1958), 2nd ed., p. 28, Theorem 19.
23.See Theorem 18.5, p. 27 of C. C. MacDuffee, The Theory of Matrices (Chelsea, New York, 1946).
24.G. N. Watson, Q. J. Math. 10, 266 (1939).
25.G. S. Joyce, J. Math. Phys. 12, 1390 (1970).
26.Wortis, Ref. 9, proved a nonexistence result, similar to this one of ours, for an analogous class of isotropic sc ferromagnets, for the  direction of Γ. His result can be established very simply by the approach used to prove ours in the text.
27.Compare with the parallel results of Wortis, Ref. 9, for the analogous sc case alluded to in the preceding footnote.
28.L. P. Bouchaert, R. Smoluchowski, and E. P. Wigner, Phys. Rev. 50, 58 (1936). In fact, the maximum dimensionality of an irreducible representation of is two and that of an irreducible representation of is three.
29.W. F. Brinkman and R. J. Elliott, Proc. R. Soc. A 294, 343 (1966);
29.W. F. Brinkman, J. Appl. Phys. 38, 939 (1967).
30.P. A. Fleury and R. Loudon, Phys. Rev. 166, 514 (1968).
30.See also R. J. Elliott, M. F. Thorpe, G. F. Imbusch, R. Loudon, and J. B. Parkinson, Phys. Rev. Lett. 21, 147 (1968) and references contained therein.
30.More recent references can be obtained from P. A. Fleury and H. J. Guggenheim, Phys. Rev. Lett. 24, 1346 (1970).
31.For the NN‐coupled bcc structures under consideration, the are expressable as linear combinations of complete elliptic integrals when Γ is on G or at N, in parallel with what was found by Wortis, Ref. 9, for analogous sc ferromagnets when Γ was on the pertinent zone boundary.
32.One can reduce the to the form (A2) by employing a new variable of integration or and using integration formulas given, e.g., by P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer‐Verlag, Berlin, 1954).
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