Theoretical and Mathematical Physics

, Volume 99, Issue 2, pp 599–605 | Cite as

Fast-decaying potentials on the finite-gap background and the\(\bar \partial - \)problem on the Riemann surfaces

  • P. G. Grinevich
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Abstract

The direct and the inverse ‘scattering problems’ for the heat-conductivity operator\(LP = \partial _y - \partial _x^2 + u(x,y)\) are studied for the following class of potentials:u(x,y)=u o (x,y)+u1(x,y), whereu o (x,y) is a nonsingular real finite-gap potential andu1(x,y) decays sufficiently fast asx2+y2→∞. We show that the ‘scattering data’ for such potentials is the\(\bar \partial - \) data on the Riemann surface corresponding to the potentialu o (x,y). The ‘scattering data’ corresponding to real potentials is characterized and it is proved that the inverse problem corresponding to such data has a unique nonsingular solution without the ‘small norm’ assumption. Analogs of these results for the fixed negative energy scattering problem for the two-dimensional time-independent Schrödinger operator\(LP = - \partial _x^2 - \partial _y^2 + u(x,y)\) are obtained.

Keywords

Inverse Problem Riemann Surface Negative Energy Scattering Problem Real Potential 

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References

  1. [1]
    Novikov S.P. and Veselov A.P., Sov. Math. Dokl.,30, 588–591; Sov. Math. Dokl.,30 (1984) 705–708.Google Scholar
  2. [2]
    Manakov S.V., Usp. Mat. Nauk,31 (1976), о 5, 245–246.Google Scholar
  3. [3]
    Ablowitz M.J., Jaacov D. Bar, and Fokas A.S., Stud. in Appl. Math.,69 (1983), о 2, 135–143.Google Scholar
  4. [4]
    Beals R. and Coifman R.R., Pseudodifferential Oper. and Appl. Proc. Symp. Notre Dame Ind., Apr. 2–5, 1984, Providence R.I., 1985, 45–70.Google Scholar
  5. [5]
    Grinevich, P.G. and Novikov R.G., Functional Anal. Appl.,19 (1985), о 4, 276–285.Google Scholar
  6. [6]
    Grinevich P.G. and Manakov S.V., Functional Anal. Appl.,20 (1986), о 2, 94–103.Google Scholar
  7. [7]
    Grinevich P.G. and Novikov S.P., Functional Anal. Appl.,22 (1988), о 1, 19–27.Google Scholar
  8. [8]
    Krichever, I.M., Sov. Math. Dokl.,17 (1976) 394–397.Google Scholar
  9. [9]
    Krichever I.M., Russian Math. Surveys,44:2 (1989), 145–225.Google Scholar
  10. [10]
    Dubrovin B.A., Krichever I.M., and Novikov S.P., Sov. Math. Dokl.17 (1976) 947–951.Google Scholar
  11. [11]
    Natanzon S.M., Functional. Anal. Appl.,22 (1988), о 1, 68–70; Functional Anal. Appl.,26 (1992) о 1, 13–20.Google Scholar
  12. [12]
    Kusnetsov E.A., and Mikhailov A.V., Sov. Phys. — JETP40 (1974), о 5, 855–859.Google Scholar
  13. [13]
    Krichever I.M., Functional. Anal. Appl.9 (1975), о 2, 161–163.Google Scholar
  14. [14]
    Bikbaev R.F. and Sharipov R.A., Theor. Math. Phys.,78 (1989), о 3, 244–252.Google Scholar
  15. [15]
    Rodin Yu.L., Physica D,24 (1987), о 1-3, 1–53.Google Scholar
  16. [16]
    Grinevich P.G., Functional Anal. Appl.,23 (1989), о 4, 79–80.Google Scholar
  17. [17]
    Faddeev L.D. Inverse problem of the quantum scattering theory II. — Sovremennye Problemy Matematiki, Vol. 3 (1974), VINITI, Moscow, transl. in Journal of Sov. Math.,5 (1976), о 3, 334–396.Google Scholar

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© Plenum Publishing Corporation 1994

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  • P. G. Grinevich

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