Skip to main content

A class of analytic perturbations for one-body Schrödinger Hamiltonians

Abstract

We study a class of symmetric relatively compact perturbations satisfying analyticity conditions with respect to the dilatation group inR n. Absence of continuous singular part for the Hamiltonians is proved together with the existence of an absolutely continuous part having spectrum [0, ∞). The point spectrum consists in R−{0} of finite multiplicity isolated energy bound-states standing in a bounded domain. Bound-state wave functions are analytic with respect to the dilatation group. Some properties of resonance poles are investigated.

This is a preview of subscription content, access via your institution.

References

  1. Amrein, W., Misra, Ph., Martin. B.: On the asymptotic condition of scattering theory. Helv. Phys. Acta43, 313 (1970).

    Google Scholar 

  2. Combes, J. M.: An algebraic approach to quantum scattering theory. Preprint — Marseille.

  3. Lavine, R. B.: Scattering theory for long range potentials. J. Functional Analysis5, 368 (1970).

    Google Scholar 

  4. -- Commutators and scattering theory, Part I: Repulsive interactions. Preprint Cornell University.

  5. -- Commutators and scattering theory, Part II: Class of one-body problem. Preprint Cornell University.

  6. Weidmann, J.: The virial theorem and its applications to the spectral theory of Schrödinger operators. Bull. Amer. Math. Soc.73, 452 (1967).

    Google Scholar 

  7. Lovelace, C.: Three particle systems and unstable particles. Scottish Universities Summer School, Moorhouse (1963).

    Google Scholar 

  8. Bottino, Longoni, A., Regge, T.: Potential scattering for complex energy and angular momentum. Nuovo Cimento23, 354 (1962).

    Google Scholar 

  9. Brown, L., Fivel, D., Lee, B., Sawyer, R.: Fredholm method in potential scattering and its application to complex angular momentum. Ann. Phys.23, 187 (1963).

    Google Scholar 

  10. Steinberg, S.: Meromorphic families of compact operators. Arch. for Rat. Mechanics and analysis.31, 5, 372 (1968).

    Google Scholar 

  11. Dunford-Schwartz: Linear operators I. New York: Interscience Publ. 1959.

    Google Scholar 

  12. Nelson, E.: Analytic vectors. Ann. Math.70, 3 (1959).

    Google Scholar 

  13. Combes, J. M.: Relatively compact interactions in many particle system. Appendix I. Commun. math. Phys.12, 283 (1969).

    Google Scholar 

  14. Kato: Perturbation theory for linear operators. Berlin-Heidelberg-New York: Springer 1966.

    Google Scholar 

  15. Wong, J.: On a condition for completness. J. Math. Phys.10, 1438 (1969).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aguilar, J., Combes, J.M. A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun.Math. Phys. 22, 269–279 (1971). https://doi.org/10.1007/BF01877510

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01877510

Keywords

  • Neural Network
  • Statistical Physic
  • Wave Function
  • Complex System
  • Nonlinear Dynamics