About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 170397, 9 pages
http://dx.doi.org/10.1155/2014/170397
Research Article

Asymptotic Behaviors of the Eigenvalues of Schrödinger Operator with Critical Potential

1Mathematics and Statistics School, Henan University of Science and Technology, No. 263 Kaiyuan Road, Luolong District, Luoyang, Henan 471023, China
2College of Civil Engineering and Architecture, Zhejiang University, B505 Anzhong Building, 866 Yuhangtang Road, Hangzhou, Zhejiang 310058, China
3College of Food and Bioengineering, Henan University of Science and Technology, No. 263 Kaiyuan Road, Luo-Long District, Luolong, Henan 471023, China

Received 29 October 2013; Revised 8 January 2014; Accepted 9 January 2014; Published 13 March 2014

Academic Editor: Manuel Maestre

Copyright © 2014 Xiaoyao Jia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Jensen and G. Nenciu, “A unified approach to resolvent expansions at thresholds,” Reviews in Mathematical Physics, vol. 13, no. 6, pp. 717–754, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. M. Murata, “Asymptotic expansions in time for solutions of Schrödinger-type equations,” Journal of Functional Analysis, vol. 49, no. 1, pp. 10–56, 1982.
  3. J. Rauch, “Local decay of scattering solutions to Schrödinger's equation,” Communications in Mathematical Physics, vol. 61, no. 2, pp. 149–168, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  4. X. Wang, “Threshold energy resonance in geometric scattering,” in Proceedings of the Symposium Scattering and Spectral Theory, vol. 26, pp. 135–164, Matem atica Contempor anea, Recife, Brazil, 2004.
  5. X. Wang, “Asymptotic expansion in time of the Schrödinger group on conical manifolds,” AnnaLes de L'Institut Fourier, vol. 56, no. 6, pp. 1903–1946, 2006.
  6. X. Jia and Y. Zhao, “Coupling constant limits of Schrödinger operators with critical potentials,” Boundary Value Problems, vol. 2013, no. 1, 15 pages, 2013.
  7. X. Wang, “Number of eigenvalues for dissipative Schrödinger operators under perturbation,” Journal de Mathématiques Pures et Appliquées, vol. 96, no. 5, pp. 409–422, 2011.
  8. M. Klaus and B. Simon, “Coupling constant thresholds in nonrelativistic quantum mechanics, I. Short-range two-body case,” Annals of Physics, vol. 130, no. 2, pp. 251–281, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. S. Fassari and M. Klaus, “Coupling constant thresholds of perturbed periodic Hamiltonians,” Journal of Mathematical Physics, vol. 39, no. 9, pp. 4369–4416, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. C. Hainzl and R. Seiringer, “Asymptotic behavior of eigenvalues of Schrödinger type operators with degenerate kinetic energy,” Mathematische Nachrichten, vol. 283, no. 3, pp. 489–499, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. V. Vougalter, “On threshold eigenvalues and resonances for the linearized NLS equation,” Mathematical Modelling of Natural Phenomena, vol. 5, no. 4, pp. 448–469, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  12. A. Laptev and M. Solomyak, “On the negative spectrum of the two-dimensional Schrödinger operator with radial potential,” Communications in Mathematical Physics, vol. 314, no. 1, pp. 229–241, 2012.
  13. G. Rozenblum and M. Solomyak, “Spectral estimates for Schrödinger operators with sparse potentials on graphs,” Journal of Mathematical Sciences, vol. 176, no. 3, pp. 458–474, 2011. View at Publisher · View at Google Scholar · View at Scopus
  14. A. Pushnitski, “The Birman-Schwinger principle on the essential spectrum,” Journal of Functional Analysis, vol. 261, no. 7, pp. 2053–2081, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. M. Reed and B. Simon, Methods of Modern Mathematical Physics: Analysis of Operators, vol. 4, Academic Press, New York, NY, USA, 1978.