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Extended class of Dubrovin’s equations related to the one-dimensional quantum three-body problem. (English) Zbl 0897.47055

Summary: The relation of the quantum one-dimensional three-body problems with zero-range interaction to the matrix Riemann-Hilbert problem with meromorphic coefficient is shown. The solution of this problem is discussed using the exact analytic diagonalization of the coefficient. The problem is reduced to the boundary value problem on the Riemann surface. The solution of this problem is expressed in terms of the Riemann theta-functions. An extended class of integrable Dubrovin’s type ordinary differential equations related to the one-dimensional quantum three-body problem is derived.

MSC:

47N50 Applications of operator theory in the physical sciences
81U10 \(n\)-body potential quantum scattering theory
30E25 Boundary value problems in the complex plane
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[1] McGuire, J.B., Study of exactly soluble one-dimensional N-body problems, J. math. phys., 5, 5, 622-636, (1964) · Zbl 0131.43804
[2] Gaudin, M., La fonction D’onde de Bethe, (1983), Masson Paris · Zbl 0509.60093
[3] McGuire, J.B.; Hurst, C.A., The scattering of three impenetrable particles in one dimension, J. math. phys., 13, 10, 1595-1607, (1972)
[4] McGuire, J.B.; Hurst, C.A., Three interacting particles in one dimension: an algebraic approach, J. math. phys., 29, 1, 155-168, (1988)
[5] Buslaev, V.S.; Merkuriev, S.P.; Salikov, S.P., On diffractional character of scattering in one-dimensional system of three-body quantum particles, () · Zbl 0616.47008
[6] Lipszyc, K., One-dimensional model of the rearrangement processes and Faddeev equations, J. math. phys., 15, 1, 133-138, (1974)
[7] Lipszyc, K., On the application of the Sommerfeld-maluzhinetz transformation to some one-dimensional three-particle problems, J. math. phys., 21, 1, 1092-1102, (1980)
[8] Gaudin, M.; Derrida, B., Solution exacte d’un probleme modele a trios corps, etat Lie, J. de phys., 36, 12, 1183-1197, (1975)
[9] Kurasov, P.B., On direct and inverse scattering problems in dimension one, Ph.D. thesis, (1993), Stockholm · Zbl 0801.47006
[10] Maluzhinetz, G.D., Dokl. akad. nauk. SSSR, 60, 3, 307-311, (1948)
[11] Maluzhinetz, G.D., Dokl. akad. nauk. SSSR, 1, 3, 226-234, (1955)
[12] Maluzhinetz, G.D., Inverse formula for zommerfeld integral, 118, 6, (1958)
[13] G.D. Maluzhinetz, 78 (3), 439-442 (1951).
[14] Maluzhinetz, G.D., Radiation of sound by the oscillating edges of an arbitrary sector. I 1099-1102, Akust. zhurnal, 1, 2, 144-164, (1955)
[15] Maluzhinetz, G.D., Radiation of sound by the oscillating edges of an arbitrary sector. II, Akust. zhurnal, 121, 3, 436-439, (1958)
[16] Maluzhinetz, G.D., Ann. der phys., 7, 6, 107-112, (1960), heft 1-2
[17] Maluzhinetz, G.D.; Tuzhilin, A.A., Diffraction of a flat sound wave on a thin semi-infinite, Zhurnal vych. mat. i mat. fiz., 10, 5, 1210-1227, (1970) · Zbl 0259.76041
[18] Tuzhilin, A.A., Differenz. uravnenij, 4, 4, 692-704, (1970)
[19] Tuzhilin, A.A., Differenz. uravnenij, 4, 6, 1048-1063, (1970)
[20] Leinaas, J.M.; Myrheim, J., Nuovo cimento, 37B, 1, 1-23, (1977)
[21] Rudin, G., The Sommerfeld-maluzhinetz method for quantum scattering in the Aharonov-Bohm gauge fields, ()
[22] Aharonov, Y.; Bohm, D., Phys. rev., 115, 485, (1959)
[23] Rudin, G.E., The Sommerfeld integral approach to the Aharonov-Bohm effect, Preprint IPRT #04/93, (1993), St. Petersburg
[24] Kuperin, Yu.A.; Romanov, R.V.; Rudin, G.E., Scattering on the hyperbolic plane in the Aharonov-Bohm gauge field, Lett. math. phys., 31, 271-278, (1994) · Zbl 0805.35084
[25] Wilczek, F., Phys. rev. lett., 48, 17, 1144-1146, (1982)
[26] Antoniou, I.; Prigogine, I., Intrinsic irreversibility and integrability of dynamics, Physica A, 192, 443, (1993)
[27] Merkuriev, S.P.; Faddeev, L.D., Quantum scattering theory for several particle systems, Math. phys. and appl. math., 11, (1993) · Zbl 0797.47005
[28] Kuperin, Yu.A.; Merkuriev, S.P., Selfadjoint extensions and scattering theory for several-body systems, Amer. math. soc. transl., 150, 2, 141-176, (1992) · Zbl 0756.35064
[29] Pavlov, B.S., Extensions theory and exactly solvable models, Russian math. surveys, 42, 127, (1987) · Zbl 0665.47004
[30] Gahov, F.D., Uspehi mat. nauk., 7, 4, 3, (1952)
[31] Dubrovin, B.A., Theta-functions and non-linear equations, Uspehi mat. nauk, 36, 2, 11, (1981) · Zbl 0478.58038
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