Abstract
A rigorous proof is given of the orthogonality and the completeness of the Bethe Ansatz eigenstates of theN-body Hamiltonian of the nonlinear Schroedinger model on a finite interval. The completeness proof is based on ideas of C.N. Yang and C.P. Yang, but their continuity argument at infinite coupling is replaced by operator monotonicity at zero coupling. The orthogonality proof uses the algebraic Bethe Ansatz method or inverse scattering method applied to a lattice approximation introduced by Izergin and Korepin. The latter model is defined in terms of monodromy matrices without writing down an explicit Hamiltonian. It is shown that the eigenfunctions of the transfer matrices for this model converge to the Bethe Ansatz eigenstates of the nonlinear Schroedinger model.
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Babbitt, D., Thomas, L.E.: Commun. Math. Phys.54, 255–278 (1977)
Babbitt, D., Thomas, L.E.: J. Math. Phys.19, 1699–1704 (1978)
Babbit, D., Thomas, L.E.: J. Math. An. & Appl.72 305–328 (1979)
Babbit, D., Thomas, L.E., Gutkin: Lett Math. Phys.20, 91–99 (1990)
Baxter, R.J.: Ann. Phys.70, 323–337 (1972)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. New York: Academic Press 1982
Berg, M. van den, Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.118, 61–85 (1988)
Berg, M. van den, Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.127, 41–69 (1990)
Berg, M. van den, Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.128, 231–245 (1990)
Bergknoff, H., Thacker, H.B.: Phys. Rev.D19, 3666–3681 (1979)
Bethe, H.: Zeits. f. Phys.71, 205–226 (1931)
Craemer, D.B., Thacker, H.B., Wilkinson, D.: Phys. Rev.D21, 1523–1528 (1980)
Destri, C., Lowenstein, J.H.: Nucl. Phys.B205, 369–385 (1982)
De Vega, H.J.: Int. J. Mod. Phys.A4, 2371–2463 (1989)
Dorlas, T.C., Lewis, J.T., Pulé, J.V.: Commun. Math. Phys.124, 365–402 (1989)
Faddeev, L.D.: Sov. Sc. Rev.C1, 107–155 (1980)
Faddeev, L.D., Sklyanin, E.K., Takhtadzhyan, L.A.: Theor. Math. Phys.40 (1979)
Faddeev, L.D., Takhtadzhyan, L.A.: Russ. Math. Surv.1
Gaudin, M.: La fonction d'onde de Bethe, Paris: Masson, 1983
Girardeau, M.: J. Math. Phys.1, 516–523 (1960)
Izergin, A.G., Korepin, V.E.: Sov. Phys. Dokl.26, 653–654 (1981)
Izergin, A.G., Korepin, V.E.: Nucl. Phys.B205 [FS5], 401–413 (1982)
Izergin, A.G., Korepin, V.E., Smirnov, F.A.: Theor. Math. Phys.48, 773–776 (English) (1981)
Korepin, V.E.: Commun. Math. Phys.86, 391–418 (1982)
Landau, L.D., Lifschitz, E.M.: Statistical Physics, Vol. 5 of: ‘Course of Theoretical Physics’, London: Pergamon Press 1958
Lieb, E.H., Liniger, W.: Phys. Rev.130, 1605–1624 (1963)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. (Revised and enlarged edition), New York: Academic Press 1980
Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis and Self-Adjointness. New York: Academic Press 1975
Reed, M., Simon, B.: Methods of Modern Mathematical Physics IV: Analysis of Operators. New York: Academic Press 1978
Simon, B.: J. Funct. Anal.28, 377–385 (1978)
Sklyanin, E.K.: Sov. Phys. Dokl.24, 107–109 (1979)
Takahashi, M.: Progr. Theor. Phys.46, 401–415 (1971)
Thacker, H.B., Wilkinson, D.: Phys. Rev.D19, 3660–3665 (1979)
Thacker, H.B.: Rev. Mod. Phys.53, 253–285 (1981)
Thomas, L.E.: J. Math. Anal. & Appl.59, 392–414 (1977)
Yang, C.N., Yang, C.P.: J. Math. Phys.10, 1115–1122 (1969)
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Dorlas, T.C. Orthogonality and completeness of the Bethe Ansatz eigenstates of the nonlinear Schroedinger model. Commun.Math. Phys. 154, 347–376 (1993). https://doi.org/10.1007/BF02097001
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DOI: https://doi.org/10.1007/BF02097001