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TROTTER–KATO PRODUCT FORMULA AND AN APPROXIMATION FORMULA FOR A PROPAGATOR IN SYMMETRIC OPERATOR IDEALS

Part of: Special classes of linear operators Groups and semigroups of linear operators, their generalizations and applications Linear spaces and algebras of operators

Published online by Cambridge University Press:  16 June 2023

MEIRAM AKHYMBEK Open the ORCID record for MEIRAM AKHYMBEK [Opens in a new window]
MEIRAM AKHYMBEK*
Affiliation:
Institute of Mathematics and Mathematical Modeling, 28 Shevchenko Street, 050010 Almaty, Kazakhstan
*
e-mail: m.akhymbek@gmail.com
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Abstract

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Keywords

Trotter–Kato product formulaevolution equationpropagatorsymmetric operator idealslogarithmic submajorisation

MSC classification

Primary: 47L20: Operator ideals
Secondary: 47B25: Symmetric and selfadjoint operators (unbounded) 47D03: Groups and semigroups of linear operators
Type
PhD Abstract
Information
Bulletin of the Australian Mathematical Society , Volume 108 , Issue 1 , August 2023 , pp. 173 - 174
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

The Trotter–Kato product formula is a mathematical clarification of path integration in quantum theory [Reference Simon4]. It gives a precise meaning to Feynman’s path integral representation of the solutions to Schrödinger equations with time-dependent potentials. In this thesis, we consider the Trotter–Kato product formula in an arbitrary symmetrically F-normed ideal closed with respect to the logarithmic submajorisation.

An abstract nonautonomous evolution equation is widely used in various fields of mathematics and quantum mechanics. For example, the Schrödinger equation and linear partial differential equations of parabolic or hyperbolic type [Reference Phillips3, Reference Vuillermot, Wreszinski and Zagrebnov5]. The second problem we consider is the existence of the propagator for such an equation and its approximation formula in an arbitrary symmetric Banach ideal. The approximation formula in the autonomous case corresponds to the Trotter product formula.

Some of this research has been published in [Reference Akhymbek and Levitina1, Reference Akhymbek and Zanin2].

Footnotes

Thesis submitted to the University of New South Wales in September 2022; degree approved on 4 November 2022; supervisors Fedor Sukochev, Dmitriy Zanin, Galina Levitina.

References

Akhymbek, M. and Levitina, G., ‘Trotter–Kato product formula in symmetric F-normed ideals’, Studia Math. 266 (2022), 167–191.CrossRefGoogle Scholar
Akhymbek, M. and Zanin, D., ‘Approximation formula for a propagator in symmetrically normed ideals’, J. Math. Anal. Appl. 522(2) (2023), Article no. 126996.CrossRefGoogle Scholar
Phillips, R. S., ‘Perturbation theory for semi-groups of linear operators’, Trans. Amer. Math. Soc. 74 (1953), 199–221.CrossRefGoogle Scholar
Simon, B., Functional Integration and Quantum Physics, Pure and Applied Mathematics, 86 (Academic Press, New York–London, 1979).Google Scholar
Vuillermot, P.-A., Wreszinski, W. F. and Zagrebnov, V. A., ‘A general Trotter–Kato formula for a class of evolution operators’, J. Funct. Anal. 257(7) (2009), 2246–2290.CrossRefGoogle Scholar