Abstract
In preceding papers we have shown how the analysis of the transfer matrix method in statistical mechanics permits us to get a very natural result for the splitting for the transfer matrix. The purpose of this note is to analyze the link between this result and previous results obtained by J.Sjöstrand concerning the splitting between the two first eigenvalues of the Schrödinger operator. We present here improved results and analyze as a byproduct the convergence in the Trotter-Kato formula in a particular (non abstract but relatively general) case. As is known this is strongly related with the Feynman-Kac formula.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. Brunaud and B. Helffer: Un problème de double puits provenant de la théorie statistico-mécanique des changements de phase (ou relecture d’un cours de M. Kac), preprint LMENS, March 1991.
F. Daumer: Personal communication, June 1994.
J. Glimm, A. Jaffe: Quantum physics (a functional integral point of view), Springer Verlag, Second edition (1987).
R. Goodman: Nilpotent Lie groups: structure and applications to analysis. Lecture Notes in Mathematics, n° 562, Springer-Verlag (1966).
B. Helffer: Universal estimate of the gap for the Kac operator in the convex case, Comm. in Math. Physics, Vol-161, n° 3, (1994), p. 631–642.
B. Helffer: Spectral properties of the Kac operator in large dimension Proceeding of the Summer School in Vancouver, August 1993.
B. Helffer: Correlation decay and gap of the transfer operator, Proceedings of the conference in the Euler Institut in St Petersbourg (submitted in April 1994 ).
B. Helffer, J. Sjöstrand: On the correlation for Kac like models in the convex case, Report No. 9, 1992–93, Institut Mittag-Leffler, March 1993. To appear in Journal of statistical physics.
M. Kac: Mathematical mechanisms of phase transitions, Brandeis lectures (1966), Gordon and Breach.
G. Parisi: Statistical field theory, Frontiers in Physics, Addison-Wesley Publishing Company (1984)
M. Reed, B. Simon: Methods of modern mathematical physics, Tome I-IV, Functional Analysis, Academic Press 1972–79.
D. Robert: Autour de l’approximation semi-classique, Birkhäuser, Vo1.68, Progress in Mathematics, (1987)
Dzh. L. Rogava: Error bounds for Trotter-type formulas for Self-adjoint operators, Functional analysis and its applications (1993) p. 217–219.
B. Simon: The P(φ)2 Euclidean quantum field theory, Princeton Series in Physics (1974).
B. Simon: Functional integration and quantum physics, Pure and Applied mathematics n°86, Academic press, New York, (1979).
I.M. Singer, B. Wong, S.T. Yau and S.S.T. Yau: An estimate of the gap of the first two eigenvalues of the Schrodinger operator, Ann. Scuola Norm. Sup. Pisa(4), 12 (1985), p. 319–333.
J. Sjöstrand: Potential wells in high dimensions II, more about the one well case. preprint March 1991, Ann. Inst. Poincaré, Section Physique théorique, 58 (1)(1993).
J. Sjöstrand: Ferromagnetic integrals, correlations and maximum principles, Preprint Université Paris-Sud (1993).
C.J. Thompson: Mathematical Statistical Mechanics, Princeton University Press (1972).
H. Trotter: On the product of semigroups of operators, Proc.Amer. Math. Soc. 10 (1959) p. 549–551.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Helffer, B. (1995). Around the Transfer Operator and the Trotter-Kato Formula. In: Demuth, M., Schulze, BW. (eds) Partial Differential Operators and Mathematical Physics. Operator Theory Advances and Applications, vol 78. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9092-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-0348-9092-2_18
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9903-1
Online ISBN: 978-3-0348-9092-2
eBook Packages: Springer Book Archive