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Ground state representation of the infinite one-dimensional Heisenberg ferromagnet

II. An explicit plancherel formula

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Abstract

In its ground state representation, the infinite, spin 1/2 Heisenberg chain provides a model for spin wave scattering, which entails many features of the quantum mechanicalN-body problem. Here, we give a complete eigenfunction expansion for the Hamiltonian of the chain in this representation, forall numbers of spin waves. Our results resolve the questions of completeness and orthogonality of the eigenfunctions given by Bethe for finite chains, in the infinite volume limit.

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References

  1. Babbitt, D. G., Thomas, L. E.: An explicit Plancherel theorem for the ground state representation of the Heisenberg chain. Proc. Nat. Acad. Sci., to appear (1977)

  2. Bethe, H.: Zur Theorie der Metalle. I. Eigenwerte und Eigenfunktionen der linearen Atomkette. Z. Physik71, 205–226 (1931)

    Google Scholar 

  3. Hepp, K.: Scattering theory in the Heisenberg ferromagnet. Phys. Rev. B5, 95–97 (1972)

    Google Scholar 

  4. Thomas, L. E.: Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. I. J. Math. Anal. Appl., to appear

  5. Babbitt, D. G., Thomas, L. E.: Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. II. Completeness and orthogonality (a preprint available from the authors)

  6. Streater, R. F.: Spin wave scattering. In: Scattering theory in mathematical physics (ed. J. A. Lavita, P. Marchand), pp. 273–298. Boston: D. Reidel 1974

    Google Scholar 

  7. Wortis, M.: Bound states of two spin waves in the Heisenberg ferromagnet. Phys. Rev.132, 85–97 (1963)

    Google Scholar 

  8. Yang, C. N., Yang, C. P.: One dimensional chain of anisotropic spin-spin interactions. I. Proof of Bethe's hypothesis for ground state in a finite system. Phys. Rev.150, 221–137 (1966)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of California, 90024, Los Angeles, CA, USA

    Donald Babbitt

  2. Department of Mathematics, University of Virginia, 22903, Charlottesville, VA, USA

    Lawrence Thomas

Authors
  1. Donald Babbitt
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  2. Lawrence Thomas
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Additional information

Communicated by E. Lieb

Research supported in part by NSF Grant No. MCS-76-05857

Research supported in part by NSF Grant No. MCS-74-07313-A02

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Babbitt, D., Thomas, L. Ground state representation of the infinite one-dimensional Heisenberg ferromagnet. Commun.Math. Phys. 54, 255–278 (1977). https://doi.org/10.1007/BF01614088

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  • DOI: https://doi.org/10.1007/BF01614088

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