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


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.


Inverse Problem Riemann Surface Negative Energy Scattering Problem Real Potential 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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

Copyright information

© Plenum Publishing Corporation 1994

Authors and Affiliations

  • P. G. Grinevich

There are no affiliations available

Personalised recommendations