# zbMATH — the first resource for mathematics

An operator-theoretic proof of an estimate on the transfer operator. (English) Zbl 0946.47026
The authors address the question: How fast the Trotter-Kato type product formula $[\exp(tV/2n)\exp(-t\Delta/n)\exp(tV/2n)]^n$ converges to $$\exp\{(-\Delta +V)\}$$? Here $$\Delta$$ is a Laplacian on $$R^N$$ and $$V$$ is a potential. Using operator techniques they prove a fine estimate in terms of $$L^2$$-operator norm convergence and some smoothness conditions on $$V$$. Similar estimations were obtained by T. Ichinose and S. Takanobu [Nagoya Math. J. 149, 53-81 (1998; Zbl 0917.47041)] using probabilistic methods.

##### MSC:
 47D06 One-parameter semigroups and linear evolution equations 47D08 Schrödinger and Feynman-Kac semigroups
Full Text:
##### References:
 [1] Dia, B.O.; Schatzman, M., Commutateurs de certains semi-groupes holomorphes et application aux directions alternées, M2an, 30, 343-383, (1996) · Zbl 0853.47024 [2] Dia, B.O.; Schatzman, M., An estimate on the Kac transfer operator, J. funct. anal., 145, 108-135, (1997) · Zbl 0919.47031 [3] Doumeki, A.; Ichinose, T.; Tamura, H., Error bounds on exponential product formulas for Schrödinger operators, J. math. soc. Japan, 50, 359-377, (1998) · Zbl 0910.47031 [4] Helffer, B., Around the transfer operator and the trotter – kato formula, Oper. theory adv. appl., 78, 161-174, (1995) · Zbl 0835.47050 [5] Helffer, B., Correlation decay and gap of the transfer matrix, Algèbre anal., 8, 192-210, (1996) · Zbl 0866.35079 [6] Helffer, B., Recent results and open problems on Schrödinger operators, (), 11-162 · Zbl 1075.82524 [7] Hislop, P.; Segal, I.M., Introduction to spectral theory with applications to Schrödinger operators, (1996), Springer-Verlag New York [8] Ichinose, T.; Takanobu, S., Estimate of the difference between the Kac operator and the Schrödinger semigroup, Comm. math. phys., 186, 167-197, (1997) · Zbl 0912.47025 [9] Ichinose, T.; Takanobu, S., The norm estimate of the difference between the Kac operator and the Schrödinger semi-group: A unified approach to the nonrelativistic and relativistic cases, Nagoya math. J., 149, 53-81, (1998) · Zbl 0917.47041 [10] Ichinose, T.; Tamura, H., Error estimates in operator norm for trotter – kato product formula, Integral equations operator theory, 27, 195-207, (1997) · Zbl 0906.47028 [11] Ichinose, T.; Tamura, H., Error bound in trace norm for trotter – kato product formula of Gibbs semigroups, Asymptotic anal., 17, 239-266, (1998) · Zbl 0940.47036 [12] Kato, T., Perturbation theory for linear operators, (1966), Springer-Verlag Berlin · Zbl 0148.12601 [13] Reed, M.; Simon, B., Methods of modern mathematic physics. I. functional analysis, (1980), Academic Press New York [14] Shen, Z., L^p estimates for Schrödinger operators with certain potentials, Ann. inst. Fourier (Grenoble), 45, 513-546, (1995) · Zbl 0818.35021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.