Skip to main content
SpringerLink
Log in
Book cover

Recent Developments in Quantum Mechanics pp 223–242Cite as

On Topics in Spectral and Stochastic Analysis for Schrödinger Operators

  • Chapter

Part of the book series: Mathematical Physics Studies ((MPST,volume 12))

Abstract

It is given a certain overview on results in spectral theory for Schrödinger and generalized Schrödinger operators obtained in the last years by means of stochastic analysis, in particular by the use of the Feynman-Kac formulae.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aizenman, H.; Simon, B: Brownian motion and Harnack inequality for Schrödinger operators. Comm.Pure Appl.Math., Vol. XXXV, 209–273 (1982).

    Article  MathSciNet  Google Scholar 

  2. Angelescu, N.; Nenciu, G.: On the independence of the thermodynamic limit on the boundary conditions in quantum statistical mechanics Comm.Math.Phys. 29, 15–30 (1973).

    Article  MathSciNet  ADS  Google Scholar 

  3. Baumgärtel, H; Demuth, M.: Decoupling by a projection. Rep.Math.Phys 15, 173–186 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Berthier, A.M.; Gaveau, B.: Critère de convergence des fonctionelle de Kac et application en mécanique quantique et en géométrie. J.Funct.Anal. 29, 416–424 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  5. Carmona, R.: Regularity properties of Schrödinger and Dirichlet semigroups. J.Funct.Anal. 33, 259–296 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  6. Carmona, R.; Masters, W.Ch.; Simon, B.: Relativistic Schrödinger operators: Asymptotic behaviour of the eigenfunctions. Preprint Cal.Inst. of Techn. (1989), to be published in J.Funct.Anal.

    Google Scholar 

  7. Combes, J.M.; Weder, R.: New criterion for the existence and completeness of wave operators and applications to scattering of unbounded obstacles. Comm.Part.Equat. 6, 1179–1223 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  8. Constantin, P.: Scattering for Schrödinger operators in a class of domains with non-compact boundaries. J.Funct.Anal. 44, 87–119 (1981)

    Article  MathSciNet  Google Scholar 

  9. Cycon, H.L.; Froese, R.G.; Kirsch, W.; Simon, B.: Schrödinger operators with applications to quantum mechanics and global geometry. Textbooks in Math.Phys., Springer-Verlag, 1986.

    Google Scholar 

  10. Davies, E.B.: Trace properties of the Dirichlet Laplacian. Math.Z. 188, 245–251 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  11. Davies, E.B.; van den Berg, M.: Heat flow out of regions in ℝn, Preprint 1988.

    Google Scholar 

  12. Davies, E.B.: Heat kernels and spectral theory.Cambridge Univ. Press, 1988 (to appear).

    Google Scholar 

  13. Demuth, M.: On transformations in the Feynman-Kac-formula and quantum mechanical N-body systems. Math.Nachr. 122, 109–118 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  14. Demuth, M.: On spectral properties of semigroups with Dirichlet generators. In: Proceedings of Symp. “Part.Diff.Equat.”, Holzhau 1988,.Teubner-Texte zur Mathematik, Vol. 112, 52–62 (1988).

    Google Scholar 

  15. Demuth, M.; van Casteren, J.: On spectral theory for Feller generators. Prep.Univ.Instelling Antwerpen, 88–18 (1988).

    Google Scholar 

  16. Demuth, M.; van Casteren, J.: On differences of heat semigroups. Prep.Univ.Instelling Antwerpen, 88–13 (1988).

    Google Scholar 

  17. Fridman, A.: Stochastic differential equations and applications, Vol.1, Academic Press, 1975.

    Google Scholar 

  18. Ginibre, J.: Some applications of functional integration in statistical mechanics and quantum field theory. In: Statistical mechanics and quantum field theory, Les Houches 1970. Ed.C.DeWitt, R.Stora. Gordon and Breach, 327–427 (1971).

    Google Scholar 

  19. Hempel, R.; Voigt, J.: The spectrum of a Schrödinger operator in LP(ℝv) is p-independent. Comm.Math.Phys. 104, 243–250 (1986).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Khas’minskii, R.S.: On positive solutions of the equation Au + Vu=0 (Russian). Theor.Verojatnost.i Primenen. 4, 332–341 (1959).

    Google Scholar 

  21. Kirsch, W.; Simon, B.: Universal lower bounds on eigenvalues splitting for one-dimensional Schrödinger operators. Comm.Math.Phys. 97, 453–460 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Kirsch, W.; Simon, B.: Comparison theorems for a gap of Schrödinger operators. J.Funct.Anal. 75, 396–410 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  23. Le-Gall, J.-F.: Sur une conjecture de M.Kac. Prob.Th.Rel.Fields 78, 389–402 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  24. Leinfelder, H.: Gauge invariance of Schrödinger operators and related spectral properties. J.Operator Theory 9, 163–179 (1983).

    MathSciNet  MATH  Google Scholar 

  25. Park, Y.M.: Bounds on exponentials of local number operators in quantum statistical mechanics. Comm.Math.Phys. 94, 1–33 (1984).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  26. Portenko,N.I.: Diffusion processes with unbounded drift coefficient (Russian). Teor.Verojatnost i Primenen. 20, 29–39 (1975).

    MathSciNet  Google Scholar 

  27. Ray, D.B.: On spectra of second-order differential operators. Trans.Amer.Math.Soc. 77, 299–321 (1954).

    Article  MathSciNet  MATH  Google Scholar 

  28. Simon, B.: Functional integration and quantum physics. Academic Press 1979.

    Google Scholar 

  29. Simon, B.: Schrödinger semigroups. Bull.Amer.Math.Soc. 7, 447–526 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  30. Simon, B.: Brownian motion, LP-properties of Schrödinger operators and the localization of binding. J.Funct.Anal. 35, 215–229 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  31. Simon, B.: Large time behaviour of the LP-norm of Schrödinger semigroups. J.Funct.Anal. 40, 66–83 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  32. Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z.Wahrscheinlichkeitsth.u.verw. Gebiete 3, 110–121 (1964).

    Article  MathSciNet  MATH  Google Scholar 

  33. van Casteren, J.: Generators of strongly continuous semigroups. Pitman, 1985.

    Google Scholar 

  34. van Casteren, J.: On generalized Schrödinger semigroups. Proc.of ISAM 88, Markovsche Processe und Steuerungstheorie, Gaussig GDR, 11.-15. Jan.1988.

    Google Scholar 

  35. van Casteren, J.: Pointwise inequalities for Schrödinger semigroups. To appear in Lect.Notes Pure Appl.Math.; Preprint Univ.Instelling Antwerpen 87–27 (1987).

    Google Scholar 

  36. van den Berg, M.: On the spectrum of the Dirichlet Laplacian for hornshaped regions in 6t“ with infinite volume. J.Funct.Anal. 58, 150–156 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  37. van den Berg, M.: A uniform bound on trace etA for convex regions in tR’ with smooth boundaries. Comm.Math.Phys. 92, 525–530 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  38. van den Berg, M.: On the asymptotics of the heat equation and bounds on traces associate with Dirichlet Laplacian. J.Funct.Anal. 71, 279–293 (1987).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, Academy of Sciences of GDR, Mohrenstr. 39, 1086, Berlin, GDR

    Michael Demuth

Authors
  1. Michael Demuth
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University Paris VII, Paris, France

    Anne Boutet de Monvel

  2. Bucharest University, Bucharest, Romania

    Petre Dita

  3. Theoretical Physics Department, Institute of Atomic Physics, Bucharest, Romania

    Gheorghe Nenciu

  4. Bucharest University, Bucharest, Romania

    Radu Purice

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Demuth, M. (1991). On Topics in Spectral and Stochastic Analysis for Schrödinger Operators. In: Boutet de Monvel, A., Dita, P., Nenciu, G., Purice, R. (eds) Recent Developments in Quantum Mechanics. Mathematical Physics Studies, vol 12. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3282-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-94-011-3282-4_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-5449-2

  • Online ISBN: 978-94-011-3282-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation