Advertisement

On Topics in Spectral and Stochastic Analysis for Schrödinger Operators

  • Michael Demuth
Part of the Mathematical Physics Studies book series (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.

Keywords

Stochastic Analysis Wave Operator Continuous Semigroup Singularity Region Large Time Behaviour 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aizenman, H.; Simon, B: Brownian motion and Harnack inequality for Schrödinger operators. Comm.Pure Appl.Math., Vol. XXXV, 209–273 (1982).MathSciNetCrossRefGoogle Scholar
  2. [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).MathSciNetADSCrossRefGoogle Scholar
  3. [3]
    Baumgärtel, H; Demuth, M.: Decoupling by a projection. Rep.Math.Phys 15, 173–186 (1979).MathSciNetADSMATHCrossRefGoogle Scholar
  4. [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).MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Carmona, R.: Regularity properties of Schrödinger and Dirichlet semigroups. J.Funct.Anal. 33, 259–296 (1979).MathSciNetMATHCrossRefGoogle Scholar
  6. [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. [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).MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Constantin, P.: Scattering for Schrödinger operators in a class of domains with non-compact boundaries. J.Funct.Anal. 44, 87–119 (1981)MathSciNetCrossRefGoogle Scholar
  9. [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. [10]
    Davies, E.B.: Trace properties of the Dirichlet Laplacian. Math.Z. 188, 245–251 (1985).MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Davies, E.B.; van den Berg, M.: Heat flow out of regions in ℝn, Preprint 1988.Google Scholar
  12. [12]
    Davies, E.B.: Heat kernels and spectral theory.Cambridge Univ. Press, 1988 (to appear).Google Scholar
  13. [13]
    Demuth, M.: On transformations in the Feynman-Kac-formula and quantum mechanical N-body systems. Math.Nachr. 122, 109–118 (1985).MathSciNetMATHCrossRefGoogle Scholar
  14. [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. [15]
    Demuth, M.; van Casteren, J.: On spectral theory for Feller generators. Prep.Univ.Instelling Antwerpen, 88–18 (1988).Google Scholar
  16. [16]
    Demuth, M.; van Casteren, J.: On differences of heat semigroups. Prep.Univ.Instelling Antwerpen, 88–13 (1988).Google Scholar
  17. [17]
    Fridman, A.: Stochastic differential equations and applications, Vol.1, Academic Press, 1975.Google Scholar
  18. [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. [19]
    Hempel, R.; Voigt, J.: The spectrum of a Schrödinger operator in LP(ℝv) is p-independent. Comm.Math.Phys. 104, 243–250 (1986).MathSciNetADSMATHCrossRefGoogle Scholar
  20. [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. [21]
    Kirsch, W.; Simon, B.: Universal lower bounds on eigenvalues splitting for one-dimensional Schrödinger operators. Comm.Math.Phys. 97, 453–460 (1985).MathSciNetADSMATHCrossRefGoogle Scholar
  22. [22]
    Kirsch, W.; Simon, B.: Comparison theorems for a gap of Schrödinger operators. J.Funct.Anal. 75, 396–410 (1987).MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Le-Gall, J.-F.: Sur une conjecture de M.Kac. Prob.Th.Rel.Fields 78, 389–402 (1988).MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Leinfelder, H.: Gauge invariance of Schrödinger operators and related spectral properties. J.Operator Theory 9, 163–179 (1983).MathSciNetMATHGoogle Scholar
  25. [25]
    Park, Y.M.: Bounds on exponentials of local number operators in quantum statistical mechanics. Comm.Math.Phys. 94, 1–33 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
  26. [26]
    Portenko,N.I.: Diffusion processes with unbounded drift coefficient (Russian). Teor.Verojatnost i Primenen. 20, 29–39 (1975).MathSciNetGoogle Scholar
  27. [27]
    Ray, D.B.: On spectra of second-order differential operators. Trans.Amer.Math.Soc. 77, 299–321 (1954).MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Simon, B.: Functional integration and quantum physics. Academic Press 1979.Google Scholar
  29. [29]
    Simon, B.: Schrödinger semigroups. Bull.Amer.Math.Soc. 7, 447–526 (1982).MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Simon, B.: Brownian motion, LP-properties of Schrödinger operators and the localization of binding. J.Funct.Anal. 35, 215–229 (1980).MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Simon, B.: Large time behaviour of the LP-norm of Schrödinger semigroups. J.Funct.Anal. 40, 66–83 (1981).MathSciNetMATHCrossRefGoogle Scholar
  32. [32]
    Spitzer, F.: Electrostatic capacity, heat flow, and Brownian motion. Z.Wahrscheinlichkeitsth.u.verw. Gebiete 3, 110–121 (1964).MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    van Casteren, J.: Generators of strongly continuous semigroups. Pitman, 1985.Google Scholar
  34. [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. [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. [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).MathSciNetMATHCrossRefGoogle Scholar
  37. [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).MathSciNetADSCrossRefGoogle Scholar
  38. [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).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • Michael Demuth
    • 1
  1. 1.Institute of MathematicsAcademy of Sciences of GDRBerlinGDR

Personalised recommendations