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, Volume 76, Issue 1, pp 367–395 | Cite as

Schrödinger operators with highly singular oscillating potentials

  • Karl-Theodor Sturm


We investigate the Feynman-Kac semigroupP t V and its densityp V(t,.,.),t>0, associated with the Schrödinger operator −1/2Δ+V on ℝd\{0}.V will be a highly singular, oscillating potential like
$$V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$$
with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ℝ such that for allt, x, y p V(t, x, y)≤γ·p(βt, x, y)·eat.

On the other hand, also conditions are derived which imply thatp V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.


Potential Versus Compatibility Condition Boundedness Condition Schr6dinger Operator Kato Class 
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  1. [1]
    M. Aizenman, B. Simon:Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math.35, 209–273 (1982)zbMATHMathSciNetGoogle Scholar
  2. [2]
    S. Albeverio, Ph. Blanchard, Zh. Mat:Feynman-Kac semigroups in terms of signed smooth measures. BiBoS-Preprint Nr. 424. Bielefeld 1990Google Scholar
  3. [3]
    M. B. Baetman, K. Chadan:Scattering theory with highly singular oscillating potentials. Ann. Inst. H. PoincaréA 24, 1–16 (1976)Google Scholar
  4. [4]
    P. Baras, J.A. Goldstein:Remarks on the inverse square potential, in quantum mechanics. In “Differential Equations” (ed. by I. Knowles, R.T. Lewis). Elsevier Science Publishers B.V. (North Holland) 1984Google Scholar
  5. [5]
    Ph. Blanchard, Zh. Ma:Semigroups of Schrödinger operators with potentials givenby Radon measures. To appear in “Stochastic Processes—Physics and Geometry” (ed. by S. Albeverio et al.). World Scient. SingaporeGoogle Scholar
  6. [6]
    Ph. Blanchard, Zh. Ma:Smooth measures and Schrödinger semigroups. BiBoS-Preprint Nr. 295. Bielefeld 1987Google Scholar
  7. [7]
    Ph. Blanchard, Zh. Ma:New Results on the Schrödinger semigroups with potentials given by signed smooth measures. In Lecture Notes Math.1444, 213–243. Springer 1990Google Scholar
  8. [8]
    A. Boukricha:The Schrödinger equation with an isolated singularity. In “Infinite dimensional analysis and stochastic processes” (ed. by Ph. Blanchard, S. Albeverio). Bielefeld 1983Google Scholar
  9. [9]
    M. Brelot:Etude de l'équation de la chaleur Δu=c(M) u(M), c(M)≥0, au voisinage d'un point singulier du coefficient. Ann. Sci. Ecole Norm. Sup.48, 153–246 (1931)zbMATHMathSciNetGoogle Scholar
  10. [10]
    M. Combescure, J. Ginibre:Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials. Ann. Inst. H. PoincaréA 24, 17–29 (1976)MathSciNetzbMATHGoogle Scholar
  11. [11]
    Ch. L. Fefferman:The uncertainty principle. Bull. Amer. Math. Soc.9, 129–206 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    H. Kalf, U.-W. Schmincke, J. Walter, R. Wüst:On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials. In Lecture Notes Math.448, 182–226. Springer 1975Google Scholar
  13. [13]
    B. Simon:Schrödinger Semigroups. Bull. Amer. Math. Soc.7, 447–526 (1982)zbMATHMathSciNetGoogle Scholar
  14. [14]
    K.-Th. Sturm:Störung von Hunt-Prozessen durch signierte additive Funktionale. Thesis, Erlangen 1989Google Scholar
  15. [15]
    K.-Th. Sturm:Schrödinger Semigroups on Manifolds. Preprint, Erlangen 1991Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Karl-Theodor Sturm
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

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