Finite propagation speed and kernel estimates for Schrödinger operators

Remling, Christian

doi:10.1090/S0002-9939-07-08857-0

Finite propagation speed and kernel estimates for Schrödinger operators
HTML articles powered by AMS MathViewer

by Christian Remling PDF

Proc. Amer. Math. Soc. 135 (2007), 3329-3340 Request permission

Abstract:

We point out finite propagation speed phenomena for discrete and continuous Schrödinger operators and discuss various types of kernel estimates from this point of view.

References

Jean-Marc Bouclet, François Germinet, and Abel Klein, Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 132 (2004), no. 9, 2703–2712. MR 2054797, DOI 10.1090/S0002-9939-04-07431-3
Jeff Cheeger, Mikhail Gromov, and Michael Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geometry 17 (1982), no. 1, 15–53. MR 658471
E. W. Cheney, Introduction to approximation theory, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222517
Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
François Germinet, Alexander Kiselev, and Serguei Tcheremchantsev, Transfer matrices and transport for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 3, 787–830 (English, with English and French summaries). MR 2097423
François Germinet and Abel Klein, Operator kernel estimates for functions of generalized Schrödinger operators, Proc. Amer. Math. Soc. 131 (2003), no. 3, 911–920. MR 1937430, DOI 10.1090/S0002-9939-02-06578-4
Alexander Kiselev, Imbedded singular continuous spectrum for Schrödinger operators, J. Amer. Math. Soc. 18 (2005), no. 3, 571–603. MR 2138138, DOI 10.1090/S0894-0347-05-00489-3
Peter D. Lax and Ralph S. Phillips, Scattering theory, 2nd ed., Pure and Applied Mathematics, vol. 26, Academic Press, Inc., Boston, MA, 1989. With appendices by Cathleen S. Morawetz and Georg Schmidt. MR 1037774
Michael Reed and Barry Simon, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York-London, 1972. MR 0493419
Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
Christian Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2002), no. 2, 323–394. MR 1943095, DOI 10.1016/S0022-1236(02)00007-1
Christian Remling, Universal bounds on spectral measures of one-dimensional Schrödinger operators, J. Reine Angew. Math. 564 (2003), 105–117. MR 2021036, DOI 10.1515/crll.2003.086
Theodore J. Rivlin, An introduction to the approximation of functions, Blaisdell Publishing Co. [Ginn and Co.], Waltham, Mass.-Toronto, Ont.-London, 1969. MR 0249885
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Adam Sikora, On-diagonal estimates on Schrödinger semigroup kernels and reduced heat kernels, Comm. Math. Phys. 188 (1997), no. 1, 233–249. MR 1471338, DOI 10.1007/s002200050163
Barry Simon, Schrödinger operators in the twentieth century, J. Math. Phys. 41 (2000), no. 6, 3523–3555. MR 1768631, DOI 10.1063/1.533321
A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238