Koopman and Transfer Operator Techniques from the Perspective of Quantum Theory

Dimitrios Giannakis () and Michael Montgomery ()
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Dimitrios Giannakis: Dartmouth College, Department of Mathematics
Michael Montgomery: Dartmouth College, Department of Mathematics

Chapter 105 in Operator Theory, 2026, pp 3211-3245 from Springer

Abstract: Abstract The study of mathematical connections between operator-theoretic formulations of classical dynamics and quantum mechanics began at least as early as the 1930s in work of Koopman and von Neumann and was developed in later decades by many authors, often independently, into a framework now broadly known as Koopman–von Neumann representation of classical dynamics. This chapter surveys aspects of this framework for measure-preserving ergodic dynamical systems and connects it with recent approximation techniques for Koopman and transfer operators that are amenable to data-driven numerical implementation. In broad terms, these methods are based on representations of (i) classical observables as elements of an algebra of operators acting on a Hilbert space; and (ii) classical probability measures as elements of the state space of that algebra, with lifted versions of the Koopman and transfer operators inducing dynamical evolution of observables and states, respectively. A common theme underlying the techniques surveyed here is the use of reproducing kernel Hilbert spaces with coalgebra structure (the so-called reproducing kernel Hilbert algebras) that aids the quantum representation of classical objects, as well as the use of Fock spaces to build approximation schemes with high expressivity and structure preservation properties (notably, preservation of positivity and multiplicativity of composition operators). Applications to quantum algorithms for approximating the Koopman evolution of observables in systems with pure point spectra are also discussed.

Keywords: Koopman operators; Transfer operators; Quantum information science; Koopman–von Neumann representation; Reproducing kernel Hilbert spaces (search for similar items in EconPapers)
Date: 2026
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DOI: 10.1007/978-3-032-16356-1_125

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