## The Annals of Probability

### On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators

Satoshi Takanobu

#### Abstract

The error estimate of the integral kernel for the Trotter product formula for Schrödinger operators is shown. A basic tool for doing so is the Feynman-Kac formula based on the pinned Brownian motion. This formula enables us to express the integral kernel in handleable form and hence estimate it. As a consequence the Trotter product formula in the $L_p$ operator norm is obtained.

#### Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1895-1952.

Dates
First available in Project Euclid: 7 June 2002

https://projecteuclid.org/euclid.aop/1023481116

Digital Object Identifier
doi:10.1214/aop/1023481116

Mathematical Reviews number (MathSciNet)
MR1487441

Zentralblatt MATH identifier
0898.60064

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60H10 60J35 60J65

#### Citation

Takanobu, Satoshi. On the error estimate of the integral kernel for the Trotter product formula for Schrödinger operators. Ann. Probab. 25 (1997), no. 4, 1895--1952. doi:10.1214/aop/1023481116. https://projecteuclid.org/euclid.aop/1023481116

#### References

• 1 DIA, B. O. and SCHATZMAN, M. 1997. An estimate on the Kac transfer operator. J. Funct. Anal. 145 108 135.
• 2 DOUMEKI, A. ICHINOISE, T. and TAMURA, H. 1997. Error bound on exponential product formulas for Schrodinger operators. J. Math. Soc. Japan. To appear. ¨
• 3 HELFFER, B. 1996. Correlation decay and gap of the transfer operator. Algebra i AnalizSt. Petersburg Math. J. 8 192 210.
• 4 HELFFER, B. 1995. Around the transfer operator and the Trotter Kato formula. Oper. Theory Adv. Appl. 78 161 174.
• 5 ICHINOSE, T. and TAKANOBU, S. 1997. Estimate of the difference between the Kac operator and the Schrodinger semigroup. Comm. Math. Phys. To appear. ¨
• 6 ICHINOSE, T. and TAMURA, H. 1997. Error bound in trace norm for Trotter Kato product formula of Gibbs semigroups. Asymptotic Analysis. To appear.
• 7 IKEDA, N. and WATANABE, S. 1988. Stochastic Differential Equations and Diffusion Processes, 2nd ed. North-Holland, Amsterdam.
• 8 REED, M. and SIMON, B. 1980. Methods of Modern Mathematical Physics I. Functional Analysis, rev. ed. Academic Press, New York.
• 9 SIMON, B. 1979. Functional Integration and Quantum Physics. Academic Press, New York.
• 10 SIMON, B. 1982. Schrodinger semigroups. Bull. Amer. Math. Soc. 7 447 526. ¨