Abstract
We give a tight upper bound for Schrödinger-type perturbations of integral kernels.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adamchik, V.: On Stirling numbers and Euler sums. J. Comput. Appl. Math. 79(1), 119–130 (1997)
Bogdan, K., Byczkowski, T.: Potential theory of Schrödinger operator based on fractional laplacian. Probab. Math. Stat. 20(2), 293–335 (2000)
Bogdan, K., Hansen, W., Jakubowski, T.: Time-dependent Schrödinger perturbations of transition densities. Stud. Math. 189(3), 235–254 (2008)
Bogdan, K., Jakubowski, T.: Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Commun. Math. Phys. 271(1), 179–198 (2007)
Bogdan, K., Sztonyk, P.: Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian. Stud. Math. 181(2), 101–123 (2007)
Böttcher, B.: Construction of time-inhomogeneous Markov processes via evolution equations using pseudo-differential operators. J. Lond. Math. Soc. 78, 605–621 (2008)
Branson, D.: Stirling numbers and Bell numbers: their role in combinatorics and probability. Math. Sci. 25(1), 1–31 (2000)
Chen, Z.-Q., Song, R.: General gauge and conditional gauge theorems. Ann. Probab. 30(3), 1313–1339 (2002)
Cheon, G.-S., El-Mikkawy, M.E.A., Seol, H.-G.: New identities for Stirling numbers via Riordan arrays. J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 13(4):311–318 (2006)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. Robert E. Krieger, Melbourne (1981) (Based on notes left by Harry Bateman, With a preface by Mina Rees, with a foreword by E. C. Watson, Reprint of the 1953 original)
Evans, K.P., Jacob, N.: Feller semigroups obtained by variable order subordination. Rev. Mat. Complut. 20(2), 293–307 (2007)
Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete mathematics, 2nd edn. In: A foundation for Computer Science. Addison-Wesley, Reading (1994)
Grzywny, T., Ryznar, M.: Two-sided optimal bounds for Green functions of half-spaces for relativistic α-stable process. Potential Anal. 28(3), 201–239 (2008)
Gulisashvili, A.: Classes of time-dependent measures, non-homogeneous Markov processes, and Feynman-Kac propagators. Trans. Am. Math. Soc. 360(8), 4063–4098 (2008)
Gulisashvili, A., van Casteren, J.A.: Non-autonomous Kato Classes and Feynman-Kac Propagators. World Scientific, Hackensack (2006)
Hansen, W.: Global comparison of perturbed Green functions. Math. Ann. 334(3), 643–678 (2006)
Jakubowski, T.: The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Stat. 22(2), 419–441 (2002)
Kim, P., Song, R.: Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts. J. Math. Anal. Appl. 332(1), 57–80 (2007)
Kulczycki, T.: Properties of Green function of symmetric stable processes. Probab. Math. Stat. 17(2), 339–364 (1997)
Liang, W.: An identity of Stirling numbers of the second kind. Sci. Mag. 2(2), 40–43 (2006)
Liskevich, V., Vogt, H., Voigt, J.: Gaussian bounds for propagators perturbed by potentials. J. Funct. Anal. 238(1), 245–277 (2006)
Stós, A.: Symmetric α-stable processes on d-sets. Bull. Pol. Acad. Sci. Math. 48(3), 237–245 (2000)
Sun, P.: Product of uniform distribution and Stirling numbers of the first kind. Acta Math. Sin. Engl. Ser. 21(6), 1435–1442 (2005)
Tanabe, H.: Equations of evolution. In: Monographs and Studies in Mathematics, vol. 6. Pitman (Advanced Publishing Program), Boston (1979) (Translated from the Japanese by N. Mugibayashi and H. Haneda)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by grants KBN, YSP and fellowship of CCRRDT Pays de la Loire.
Rights and permissions
About this article
Cite this article
Jakubowski, T. On Combinatorics of Schrödinger Perturbations. Potential Anal 31, 45–55 (2009). https://doi.org/10.1007/s11118-009-9123-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-009-9123-y