Duke Mathematical Journal

Applications of a commutation formula

P. A. Deift

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Duke Math. J. Volume 45, Number 2 (1978), 267-310.

First available in Project Euclid: 20 February 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A10: Spectrum, resolvent
Secondary: 34G10: Linear equations [See also 47D06, 47D09] 76Q05: Hydro- and aero-acoustics 78A45: Diffraction, scattering [See also 34E20 for WKB methods] 81E05 82A15


Deift, P. A. Applications of a commutation formula. Duke Math. J. 45 (1978), no. 2, 267--310. doi:10.1215/S0012-7094-78-04516-7. http://projecteuclid.org/euclid.dmj/1077312819.

Export citation


  • [1] M. Adler and J. Moser, On a class of polynomials connected with the Korteweg-de Vries equation, Mathematics Research Centre preprint, University of Wisconsin, May 1977.
  • [2] S. Albeverio, R. Høegh-Krohn, and L. Streit, Energy forms, Hamiltonians and distorted Brownian paths, Univ. Bielefeld, Zentrum für Interdisziplinäre Forschung preprint, 1976.
  • [3] V. Bargmann, Remarks on the determination of a central field of force from the elastic scattering phase shifts, Physical Rev. (2) 75 (1949), 301–303.
  • [4] V. Bargmann, On the connection between phase shifts and scattering potential, Rev. Modern Physics 21 (1949), 488–493.
  • [5] M. Š. Birman, Perturbation of the spectrum of a singular elliptic operator under variation of the boundary and boundary conditions, Soviet Math. Dokl. 2 (1961), 326–328.
  • [6] M. Š. Birman, Scattering problems for differential operators with constant coefficients, Funkcional. Anal. i Priložen. 3 (1969), no. 3, 1–16.
  • [7] Alwyn C. Scott, F. Y. F. Chu, and David W. McLaughlin, The soliton: a new concept in applied science, Proc. IEEE 61 (1973), 1443–1483.
  • [8] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955.
  • [9] M. M. Crum, Associated Sturm-Liouville systems, Quart. J. Math. Oxford Ser. (2) 6 (1955), 121–127.
  • [10] P. A. Deift, Classical scattering theory with a trace condition, Princeton university (1976). In preparation as a book for Princeton University Press.
  • [11] P. Deift and B. Simon, On the decoupling of finite singularities from the question of asymptotic completeness in two body quantum systems, J. Functional Analysis 23 (1976), no. 3, 218–238.
  • [12] P. A. Deift and E. Trubowitz, Inverse scattering on the line, to be submitted to Comm. Pure App. Math.
  • [13] L. D. Faddeyev, The inverse problem in the quantum theory of scattering, J. Mathematical Phys. 4 (1963), 72–104.
  • [14] L. D. Faddeev, Properties of the $S$-matrix of the one-dimensional Schrödinger equation, Amer. Math. Soc. Trans. Ser. 2 65 (1964), 139–166.
  • [15] C. S. Gardner, J. M. Greene, M. D. Kruskal, and M. R. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett. 19 (1967), 1095–1097.
  • [16] I. C. Gohberg and M. G. Kreĭ n, Introduction to the theory of linear nonselfadjoint operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969.
  • [17] T. Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966.
  • [18] T. Kato, Scattering theory with two Hilbert spaces, J. Functional Analysis 1 (1967), 342–369.
  • [19] M. G. Krein, Doklady Akad. Nauk S.S.S.R. 113(1957), 970.
  • [20] A. Lenard, Generalization of the Golden-Thompson inequality $\rm Tr(e\spAe\spB)\geq \rm Tr\,e\spA+B$, Indiana Univ. Math. J. 21 (1971/1972), 457–467.
  • [21] H. P. McKean and E. Trubowitz, Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points, Comm. Pure Appl. Math. 29 (1976), no. 2, 143–226.
  • [22] M. Reed and B. Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, 1972.
  • [23] M. Reed and Simon. B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1975.
  • [24] M. Reed and B. Simon, The scattering of classical waves from inhomogeneous media, Math. Z. 155 (1977), no. 2, 163–180.
  • [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. IV, Academic Press, New York, to appear, 1978.
  • [26] F. Reisz and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.
  • [27] S. Sakai, $C\sp*$-algebras and $W\sp*$-algebras, Springer-Verlag, New York, 1971.
  • [28] W. A. Stinespring, A sufficient condition for an integral operator to have a trace, J. Reine Angew. Math. 200 (1958), 200–207.
  • [29] C. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971/72), 469–480.
  • [30] I. M. Gel'fand and B. M. Levitan, On the determination of a differential equation from its spectral function, Izvestiya Akad. Nauk SSSR. Ser. Mat. 15 (1951), 309–360.
  • [31] J. Von Neumann and E. P. Wigner, Über merkwüdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465–467.