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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
Aronsson, G.: Extension of functions satisfying Lipschitz conditions. Ark. Mat. 6, 551–561 (1967)
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)
Beals, R.: A note on fundamental solutions. Comm. Part. Diff. Eq. 24, 369–376 (1999)
Berndtsson, B.: \(\bar {\partial }\) and Schrödinger operators. Math. Z. 221, 401–413 (1996)
Boggess, A., Raich, A.: Heat kernels, smoothness estimates and exponential decay. J. Fourier Anal. Appl. 19, 180–224 (2013)
Christ, M.: On the \(\bar {\partial }\) equation in weighted L 2 norms in \({\mathbb {C}}^{1}\). J. Geom. Anal. 1(3), 193–230 (1991)
Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1989)
Fefferman, C.: The uncertainty principle. Bull. Amer. Math. Soc. 9, 129–206 (1983)
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)
Halfpap, J., Nagel, A., Wainger, S.: The Bergman and Szegö kernels near points of infinite type. Pacific J. Math. 246(1), 75–128 (2010)
Klaus-Jochen, E., Nagel, R.: A Short Course on Operator Semigroups. Springer (2006)
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)
Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)
Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165, 207–226 (1972)
Nagel, A.: Vector fields and nonisotropic metrics. In: Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., pp 241–306. Princeton University Press (1986)
Raich, A.: Heat equations in \({\mathbb {R}}\times {\mathbb {C}}\). J. Funct. Anal. 240(1), 1–35 (2006)
Raich, A.: One-parameter families of operators in \({\mathbb {C}}\). J. Geom. Anal. 16(2), 353–374 (2006)
Raich, A.: Pointwise estimates of relative fundamental solutions for heat equations in \({\mathbb {R}}\times {\mathbb {C}}\). Math. Z. 256, 193–220 (2007)
Raich, A.: Heat equations and the weighted \(\bar {\partial }\)-problem. Commun. Pure Appl. Anal. 11(3), 885–909 (2012)
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
Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier 45, 513–546 (1995)
Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993)
van den Berg, M.: Gaussian bounds for the Dirichlet heat kernel. J. Funct. Anal. 88, 267–278 (1990)
Visser, M.: Van Vleck determinants: Geodesic focusing in Lorentzian spacetimes. Phys. Rev. D 47, 2395–2402 (1993)
Zhang, Q.S., Zhao, Z.: Estimates of global bounds for some Schrödinger heat kernels on manifolds. Illinois J. Math. 44(3), 566–572 (2000)
Author information
Authors and Affiliations
Corresponding author
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
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-016-9556-z