Skip to main content
SpringerLink
Log in

Schrödinger Operators with \(A_{\infty }\) Potentials

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We study the heat kernel p(x,y,t) associated to the real Schrödinger operator H=−Δ + V on \(L^{2}(\mathbb {R}^{n})\), n≥1. Our main result is a pointwise upper bound on p when the potential \(V \in A_{\infty }\). In the case that \(V\in RH_{\infty }\), we also prove a lower bound. Additionally, we compute p explicitly when V is a quadratic polynomial.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Auscher, P., Ben Ali, B.: Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials. Ann. Inst. Fourier 57 (6), 1975–2013 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, J.M.: Shorter notes: Strongly continuous semigroups, weak solutions, and the variation of constants formula. Proc. Amer. Math. Soc. 63(2), 370–373 (1977)

    MathSciNet  MATH  Google Scholar 

  4. Beals, R.: A note on fundamental solutions. Comm. Part. Diff. Eq. 24, 369–376 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berndtsson, B.: \(\bar {\partial }\) and Schrödinger operators. Math. Z. 221, 401–413 (1996)

    MathSciNet  MATH  Google Scholar 

  6. Boggess, A., Raich, A.: Heat kernels, smoothness estimates and exponential decay. J. Fourier Anal. Appl. 19, 180–224 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Christ, M.: On the \(\bar {\partial }\) equation in weighted L 2 norms in \({\mathbb {C}}^{1}\). J. Geom. Anal. 1(3), 193–230 (1991)

    Article  MathSciNet  Google Scholar 

  8. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)

  9. Fefferman, C.: The uncertainty principle. Bull. Amer. Math. Soc. 9, 129–206 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. Haslinger, F.: Szegö kernels for certain unbounded domains in \({{\mathbb C}}\sp 2\). Travaux de la Conférence Internationale d’Analyse Complexe et du 7e Séminaire Roumano-Finlandais (1993). Rev. Roumaine Math. Pures Appl. 39, 939–950 (1994)

    MathSciNet  Google Scholar 

  11. Halfpap, J., Nagel, A., Wainger, S.: The Bergman and Szegö kernels near points of infinite type. Pacific J. Math. 246(1), 75–128 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Klaus-Jochen, E., Nagel, R.: A Short Course on Operator Semigroups. Springer (2006)

  13. Kurata, K.: An estimate on the heat kernel of magnetic Schrödinger operators and uniformly elliptic operators with non-negative potentials. J. London Math. Soc. 62(3), 885–903 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  15. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nagel, A.: Vector fields and nonisotropic metrics. In: Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., pp 241–306. Princeton University Press (1986)

  17. Raich, A.: Heat equations in \({\mathbb {R}}\times {\mathbb {C}}\). J. Funct. Anal. 240(1), 1–35 (2006)

    Article  MathSciNet  Google Scholar 

  18. Raich, A.: One-parameter families of operators in \({\mathbb {C}}\). J. Geom. Anal. 16(2), 353–374 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Raich, A.: Pointwise estimates of relative fundamental solutions for heat equations in \({\mathbb {R}}\times {\mathbb {C}}\). Math. Z. 256, 193–220 (2007)

    Article  MathSciNet  Google Scholar 

  20. Raich, A.: Heat equations and the weighted \(\bar {\partial }\)-problem. Commun. Pure Appl. Anal. 11(3), 885–909 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Raich, A., Tinker, M.: The Szegö kernel on a class of noncompact CR manifolds of high codimension. Complex Var. Elliptic Equ. 60(10), 1366–1373 (2015). doi:10.1080/17476933.2015.1015531

    Article  MathSciNet  MATH  Google Scholar 

  22. Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier 45, 513–546 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)

  24. van den Berg, M.: Gaussian bounds for the Dirichlet heat kernel. J. Funct. Anal. 88, 267–278 (1990)

  25. Visser, M.: Van Vleck determinants: Geodesic focusing in Lorentzian spacetimes. Phys. Rev. D 47, 2395–2402 (1993)

    Article  MathSciNet  Google Scholar 

  26. Zhang, Q.S., Zhao, Z.: Estimates of global bounds for some Schrödinger heat kernels on manifolds. Illinois J. Math. 44(3), 566–572 (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, SCEN 309, 1 University of Arkansas, Fayetteville, AR, 72701, USA

    Andrew Raich

  2. Plano, TX, USA

    Michael Tinker

Authors
  1. Andrew Raich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael Tinker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrew Raich.

Additional information

The first author was partially supported by NSF grant DMS-1405100.

The main results were part of Tinker’s Ph.D. thesis which he completed under Raich’s supervision.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raich, A., Tinker, M. Schrödinger Operators with \(A_{\infty }\) Potentials. Potential Anal 45, 387–402 (2016). https://doi.org/10.1007/s11118-016-9556-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-016-9556-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation