×

2-magnon scattering in the Heisenberg model. (English) Zbl 0879.47043

Summary: We prove the asymptotic completeness for 2-magnon scattering in the Heisenberg model. The proof is based on a Mourre-estimate. The results equally apply to the scattering of two interacting particles on a lattice.

MSC:

47N50 Applications of operator theory in the physical sciences
81U05 \(2\)-body potential quantum scattering theory

References:

[1] H.A. Bethe , Zur Theorie der Metalle , Z. Physik , Vol. 71 , 1931 , p. 205 . Zbl 0002.37205 | JFM 57.1587.01 · Zbl 0002.37205
[2] F. Bloch , Zur Theorie des Ferromagnetismus , Z. Physik , Vol. 61 , 1930 , p. 206 . JFM 56.1319.02 · JFM 56.1319.02
[3] H.L. Cycon , R.G. Froese , W. Kirsch and B. Simon , Schrödinger Operators , Springer , 1987 . Zbl 0619.47005 · Zbl 0619.47005
[4] J. Dixmier , Les algèbres d’opérateurs dans l’espace hilbertien ( Algèbres de von Neumann ), Gauthier-Villars , 1969 . Zbl 0175.43801 · Zbl 0175.43801
[5] F.J. Dyson , General Theory of Spin Wave Interactions , Phys. Rev. , Vol. 102 , 1956 , p. 1217 . MR 78231 | Zbl 0072.23601 · Zbl 0072.23601 · doi:10.1103/PhysRev.102.1217
[6] J. Hanus , Bound States in the Heisenberg Ferromagnet , Phys. Rev. Letters , Vol. 11 , 1963 , p. 336 . MR 155645
[7] K. Hepp , Scattering Theory in the Heisenberg Ferromagnet , Phys. Rev. B , Vol. 5 , 1972 , p. 95 . · Zbl 0231.31006
[8] B. Helffer and J. Sjöstrand , Equation de Schrödinger avec champs magnétique et équation de Harper in Schrödinger Operators , H. HOLDEN, A. JENSEN, eds., Springer , 1989 . MR 1037319 | Zbl 0699.35189 · Zbl 0699.35189
[9] W. Hunziker and I.M. Sigal , The general theory of N-body quantum systems, to appear . MR 1332036
[10] S. Lojasiewicz , Introduction to complex analytic geometry , Birkhäuser , 1991 . MR 1131081 | Zbl 0747.32001 · Zbl 0747.32001
[11] J.F. Perez , Bound and Scattering States in the Heisenberg Ferromagnet , Ph. D. Thesis , Swiss Federal Institute of Technology Zurich , 1973 .
[12] M. Reed and B. Simon , Methods of modern mathematical physics IV . Analysis of operators , Academic Press , 1978 . MR 493421 | Zbl 0401.47001 · Zbl 0401.47001
[13] I.M. Sigal and A. Soffer , Local decay and propagation estimates for time-dependent and time-independent Hamiltonians , Princeton University preprint, 1988 .
[14] R.F. Streater , The Heisenberg Ferromagnet as a Quantum Field Theory , Comm. Math. Phys. , Vol. 6 , 1967 , p. 233 . Article | MR 228244 · Zbl 0155.32503
[15] R.F. Streater , Spin-wave scattering, in Scattering Theory in Mathematical Physics , J. A. LAVITA and J.-P. MARCHAND eds., Reidel , Dordrecht, The Netherland s, 1974 .
[16] G.J. Watts , Theory of Spin-Wave Scattering , Ph.D. Thesis Bedford College London , 1971 , unpublished. · JFM 26.0960.01
[17] M. Wortis , Bound States of two Spin Waves in the Heisenberg Ferromagnet , Phys. Rev. , Vol. 132 , 1963 , p. 85 . MR 155644
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.