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Second Order Perturbation Theory for Embedded Eigenvalues

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Abstract

We study second order perturbation theory for embedded eigenvalues of an abstract class of self-adjoint operators. Using an extension of the Mourre theory, under assumptions on the regularity of bound states with respect to a conjugate operator, we prove upper semicontinuity of the point spectrum and establish the Fermi Golden Rule criterion. Our results apply to massless Pauli-Fierz Hamiltonians for arbitrary coupling.

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References

  1. Agmon S., Herbst I., Skibsted E.: Perturbation of embedded eigenvalues in the generalized N-body problem. Commun. Math. Phys. 122, 411–438 (1989)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Aguilar J., Combes J.M.: A class of analytic perturbation for one-body Schrödinger Hamiltonians. Commun. Math. Phys. 22, 269–279 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Amrein W., Boutetde Monvel A., Georgescu V.: C 0-groups, commutator methods and spectral theory of N-body Hamiltonians. Birkhäuser, Basel–Boston–Berlin (1996)

    Google Scholar 

  4. Bach V., Fröhlich J., Sigal I.M.: Quantum electrodynamics of confined non-relativistic particles. Adv. Math. 137, 299–395 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bach V., Fröhlich J., Sigal I.M., Soffer A.: Positive commutators and the spectrum of Pauli-Fierz Hamiltonian of atoms and molecules. Commun. Math. Phys. 207, 557–587 (1999)

    Article  ADS  MATH  Google Scholar 

  6. Balslev E., Combes J.M.: Spectral properties of many-body Schrödinger operators with dilation analytic interactions. Commun. Math. Phys. 22, 280–294 (1971)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Bruneau L., Dereziński J.: Pauli-Fierz Hamiltonians defined as quadratic forms. Rep. Math. Phys. 54, 169–199 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. Cattaneo L.: Mourre’s inequality and embedded bound states. Bull. Sci. Math. 129, 591–614 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cattaneo L., Graf G.M., Hunziker W.: A general resonance theory based on Mourre’s inequality. Ann. Henri Poincaré 7, 583–601 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. Davies E.B.: Linear operators and their spectra. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  11. Dereziński J., Gérard C.: Asymptotic completeness in quantum field theory. Massive Pauli-Fierz Hamiltonians. Rev. Math. Phys. 11, 383–450 (1999)

    MATH  Google Scholar 

  12. Dereziński J., Jakšić V.: Spectral theory of Pauli-Fierz operators. J. Funct. Anal. 180, 243–327 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Faupin, J., Møller, J.S., Skibsted, E.: Regularity of bound states. Rev. Math. Phys. (to appear)

  14. Fröhlich J., Griesemer M., Sigal I.M.: Spectral Theory for the Standard Model of Non-Relativistic QED. Commun. Math. Phys. 283, 613–646 (2008)

    Article  ADS  MATH  Google Scholar 

  15. Georgescu V., Gérard C.: On the virial theorem in quantum mechanics. Commun. Math. Phys. 208, 275–281 (1999)

    Article  ADS  MATH  Google Scholar 

  16. Georgescu V., Gérard C., Møller J.S.: Commutators, C 0–semigroups and resolvent estimates. J. Funct. Anal. 216, 303–361 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Georgescu V., Gérard C., Møller J.S.: Spectral theory of massless Pauli-Fierz models. Commun. Math. Phys. 249, 29–78 (2004)

    Article  ADS  MATH  Google Scholar 

  18. Golénia S.: Positive commutators, Fermi Golden Rule and the spectrum of 0 temperature Pauli-Fierz Hamiltonians. J. Funct. Anal. 256, 2587–2620 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Golénia S., Jecko T.: A New Look at Mourre’s Commutator Theory. Compl. Anal. Oper. Th. 1, 399–422 (2007)

    Article  MATH  Google Scholar 

  20. Hille, E., Phillips, R.S.: Functional Analysis and Semigroups. Providence, RI: Amer. Math. Soc., 1957

  21. Hunziker W., Sigal I.M.: The quantum N-body problem. J. Math. Phys. 41, 3448–3510 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Hübner M., Spohn H.: Spectral properties of the spin-boson Hamiltonian. Ann. Inst. Henri Poincaré 62, 289–323 (1995)

    MATH  Google Scholar 

  23. Jakšić V., Pillet C.A.: On a model for quantum friction, II. Fermi’s Golden Rule and dymamics at positive temperature. Commun. Math. Phys. 176, 619–644 (1996)

    Article  ADS  MATH  Google Scholar 

  24. Kato T.: Perturbation Theory for Linear Operators. Second edition. Springer-Verlag, Berlin (1976)

    Book  Google Scholar 

  25. Mourre, É.: Absence of singular continuous spectrum for certain selfadjoint operators. Commun. Math. Phys. 78, 391–408 (1980/81)

    Google Scholar 

  26. Møller, J.S., Rasmussen, M.G.: The Translation Invariant Massive Nelson Model: II. The continuous Spectrum Below the Two–boson Thershold. In preparation

  27. Møller J.S., Skibsted E.: Spectral theory of time-periodic many-body systems. Adv. in Math. 188, 137–221 (2004)

    Article  Google Scholar 

  28. Reed, M., Simon, B.: Methods of modern mathematical physics I-IV. New York, Academic Press, 1972-78

  29. Simon B.: Resonances in N-body quantum systems with dilation analytic potential and foundation of time-dependent perturbation theory. Ann. Math. 97, 247–274 (1973)

    Article  MATH  Google Scholar 

  30. Skibsted E.: Spectral analysis of N-body systems coupled to a bosonic field. Rev. Math. Phys. 10, 989–1026 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  31. Yosida K.: Functional analysis. Springer, Berlin (1965)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, UMR-CNRS 5251, Université de Bordeaux 1, 351 Cours de la Libération, 33405, Talence Cedex, France

    J. Faupin

  2. Institut for Matematiske Fag, Aarhus Universitet, Ny Munkegade, 8000, Aarhus C, Denmark

    J. S. Møller & E. Skibsted

Authors
  1. J. Faupin
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  2. J. S. Møller
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  3. E. Skibsted
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Correspondence to J. Faupin.

Additional information

Communicated by H. Spohn

Partially supported by the Center for Theory in Natural Sciences, Aarhus University.

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Faupin, J., Møller, J.S. & Skibsted, E. Second Order Perturbation Theory for Embedded Eigenvalues. Commun. Math. Phys. 306, 193–228 (2011). https://doi.org/10.1007/s00220-011-1278-x

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