Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Spectral and scattering theory for perturbations of the Carleman operator


Author: D. R. Yafaev
Original publication: Algebra i Analiz, tom 25 (2013), nomer 2.
Journal: St. Petersburg Math. J. 25 (2014), 339-359
MSC (2010): Primary 47A40; Secondary 47B25
DOI: https://doi.org/10.1090/S1061-0022-2014-01294-0
Published electronically: March 12, 2014
MathSciNet review: 3114858
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The spectral properties of the Carleman operator (the Hankel operator with the kernel $ h_{0}(t)=t^{-1}$) are studied; in particular, an explicit formula for its resolvent is found. Then, perturbations are considered of the Carleman operator $ H_{0}$ by Hankel operators $ V$ with kernels $ v(t)$ decaying sufficiently rapidly as $ t\to \infty $ and not too singular at $ t=0$. The goal is to develop scattering theory for the pair $ H_{0}$, $ H=H_{0} +V $ and to construct an expansion in eigenfunctions of the continuous spectrum of the Hankel operator $ H$. Also, it is proved that, under general assumptions, the singular continuous spectrum of the operator $ H$ is empty and that its eigenvalues may accumulate only to the edge points 0 and $ \pi $ in the spectrum of $ H_{0}$. Simple conditions are found for the finiteness of the total number of eigenvalues of the operator $ H$ lying above the (continuous) spectrum of the Carleman operator $ H_{0}$, and an explicit estimate of this number is obtained. The theory constructed is somewhat analogous to the theory of one-dimensional differential operators.


References [Enhancements On Off] (What's this?)

  • 1. R. Beals, P. Deift, and C. Tomei, Direct and inverse scattering on the line, Math. Surveys Monogr., vol. 28, Amer. Math. Soc., Providence, RI, 1988. MR 954382 (90a:58064)
  • 2. V. S. Buslaev and L. D. Faddeev, Formulas for traces for a singular Sturm-Liouville differential operator, Dokl. Akad. Nauk SSSR 132 (1960), 451-454; English transl., Soviet Math. Dokl. 1 (1960), 451-454. MR 0120417 (22:11171)
  • 3. L. D. Faddeev, Properties of the $ S$-matrix of the one-dimensional Schrödinger equation, Trudy Mat. Inst. Steklov 13 (1964), 314-336. English. transl., Amer. Math. Soc. Transl. Ser. 2, vol. 65, Amer. Math. Soc., Providence, RI, 1967, pp. 139-166. MR 0178188 (31:2446)
  • 4. J. S. Howland, Spectral theory of self-adjoint Hankel matrices, Michigan Math. J. 33 (1986), no. 2, 145-153. MR 837573 (87i:47035)
  • 5. -, Spectral theory of operators of Hankel type. I, II, Indiana Univ. Math. J.  41 (1992), no. 2, 409-426, 427-434. MR 1183350 (94a:47041)
  • 6. S. T. Kuroda, Scattering theory for differential operators, J. Math. Soc. Japan  25 (1973), no. 1, 75-104; no. 2, 222-234. MR 0326435 (48:4779); MR 0326436 (48:4780)
  • 7. J. Östensson and D. R. Yafaev, Trace formula for differential operators of an arbitrary order, A panorama of modern operator theory and related topics, 541-570, Oper. Theory Adv. Appl., vol. 218, Birkhäuser/Springer, Basel, 2012. MR 2931945
  • 8. V. V. Peller, Hankel operators and their applications, Springer-Verlag, New York, 2002. MR 1949210 (2004e:47040)
  • 9. S. R. Power, Hankel operators on Hilbert space, Research Notes in Math., 64, Pitnam, Boston, 1982. MR 666699 (84e:47037)
  • 10. D. R. Yafaev, Mathematical scattering theory. General theory, Transl. Math. Monogr., 105, Amer. Math. Soc., Providence, RI, 1992. MR 1180965 (94f:47012)
  • 11. -, Spectral and scattering theory of fourth order differential operators, Spectral theory of differential operators, 265-299, Amer. Math. Soc. Transl. Ser. 2, 225, Amer. Math. Soc., Providence, RI, 2008. MR 2490556 (2011c:34214)
  • 12. -, Mathematical scattering theory. Analytic theory, Math. Surveys Monogr., 158, Amer. Math. Soc., Providence, RI, 2010. MR 2598115 (2012d:47033)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 47A40, 47B25

Retrieve articles in all journals with MSC (2010): 47A40, 47B25


Additional Information

D. R. Yafaev
Affiliation: IRMAR, Université de Rennes I, Campus de Beaulieu, 35042 Rennes Cedex, France
Email: yafaev@univ-rennes1.fr

DOI: https://doi.org/10.1090/S1061-0022-2014-01294-0
Keywords: Hankel operators, resolvent kernels, absolutely continuous spectrum, eigenfunctions, wave operators, scattering matrix, resonances, discrete spectrum, total number of eigenvalues
Received by editor(s): September 20, 2012
Published electronically: March 12, 2014
Dedicated: In memory of Vladimir Savel’evich Buslaev
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society