### The Norm Estimate of the Difference Between the Kac Operator and Schrödinger Semigroup II: The General Case Including the Relativistic Case

**Takashi Ichinose**

*(Kanazawa University)*

**Satoshi Takanobu**

*(Kanazawa University)*

#### Abstract

More thorough results than in our previous paper in Nagoya Math. J. are given on the $L_p$-operator norm estimates for the Kac operator $e^{-tV/2} e^{-tH_0} e^{-tV/2}$ compared with the Schrödinger semigroup $e^{-t(H_0+V)}$. The Schrödinger operators $H_0+V$ to be treated in this paper are more general ones associated with the Lévy process, including the relativistic Schrödinger operator. The method of proof is probabilistic based on the Feynman-Kac formula. It differs from our previous work in the point of using

*the Feynman-Kac formula*not directly for these operators, but instead through*subordination*from the Brownian motion, which enables us to deal with all these operators in a unified way. As an application of such estimates the Trotter product formula in the $L_p$-operator norm, with error bounds, for these Schrödinger semigroups is also derived.Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-47

Publication Date: January 26, 2000

DOI: 10.1214/EJP.v5-61

#### References

- P. R. Chernoff,
Note on product formulas for operator semigroups,
*J. Funct. Anal.***2**, (1968), 238--242. Math. Review 37 #6793 - A. Doumeki, T. Ichinose and H. Tamura,
Error bounds on exponential product formulas for Schrödinger
operators,
*J. Math. Soc. Japan***50**, (1998), 359--377. Math. Review 99i:47086 - W. Feller,
*An Introduction to Probability Theory and its Applications*, Vol. 2, 2nd ed., John Wiley, (1971). Math. Review 42 #5292 - B. Helffer,
Correlation decay and gap of the transfer operator (English),
*Algebra i Analiz*(*St. Petersburg Math. J.*)**8**, (1996), 192--210. Math. Review 97j:81063 - B. Helffer,
Around the transfer operator and the Trotter-Kato formula,
*Operator Theory: Advances and Appl.***78**, 161--174, Birkhäuser, (1995). Math. Review 96k:82013 - T. Ichinose, Essential selfadjointness of the Weyl quantized relativistic
hamiltonian,
*Ann. Inst. H. Poincaré, Phys. Théor.***51**, (1989), 265--297. Math. Review 91a:81043 - T. Ichinose, Remarks on the Weyl quantized relativistic Hamiltonian,
*Note di Math.***12**, (1992), 49--67. Math. Review 95a:81056 - T. Ichinose and S. Takanobu, Estimate of the difference between the Kac
operator and the Schrödinger semigroup,
*Commun. Math. Phys.***186**, (1997), 167--197. Math. Review 99e:47052 - T. Ichinose and S. Takanobu, The norm estimate of the difference between
the Kac operator and the Schrödinger semigroup: A unified approach
to the nonrelativistic and relativistic cases,
*Nagoya Math. J.***149**, (1998), 53--81. Math. Review 99i:47075 - T. Ichinose and H. Tamura, Error bound in trace norm for Trotter-Kato
product formula of Gibbs semigroups,
*Asymp. Anal.***17**, (1998), 239--266. Math. Review 2000a:47098 - T. Ichinose and T. Tsuchida, On Kato's inequality for the Weyl quantized
relativistic Hamiltonian,
*Manuscripta Math.***76**, (1992), 269--280. Math. Review 93k:81040 - N. Ikeda and S. Watanabe,
*Stochastic Differential Equations and Diffusion Processes*, 2nd ed., North-Holland/Kodansha, (1989). Math. Review 90m:60069 - K. It^o,
*Introduction to Probability Theory*, Cambridge Univ. Press, (1984). Math. Review 86k:60001 - K. It^o and H. P. McKean, Jr.,
*Diffusion Processes and their Sample Paths*, Springer-Verlag, (1965). Math. Review 33 #8031 - M. Kac, Mathematical mechanisms of phase transitions,
*Statistical Physics, Phase Transitions and Superfluidity,*Vol.1,*Brandeis University Summer Institute in Theoretical Physics,*1966 (ed. by M. Chrétien, E.P. Gross and S. Deser), 241--305, Gordon and Breach, (1968). Math. Review number not available - M. Reed and B. Simon,
*Methods of Modern Mathematical Physics, I: Functional Analysis*, revised and enlarged ed., Academic Press, (1980). Math. Review 85e:46002 - K. Sato,
*Lévy Processes and Infinitely Divisible Distribution*, Cambridge Univ. Press, (1999). Math. Review number not available - S. Takanobu, On the error estimate of the integral kernel for the Trotter
product formula for Schrödinger operators,
*Ann. Probab.***25**, (1997), 1895--1952. Math. Review 99i:60134 - H. F. Trotter, On the product of semi-groups of operators,
*Proc. Amer. Math. Soc.***10**, (1959), 545--551. Math. Review 21 #7446

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