Abstract
We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this. Topics in this second part include absence of embedded eigenvalues, trace class scattering, Kato smoothness, the quantum adiabatic theorem and Kato’s ultimate Trotter Product Formula.
Highlights
We review the work of Tosio Kato on the mathematics of non-relativistic quantum mechanics and some of the research that was motivated by this
Just before those warnings is a summary of the organization of the full paper which includes the following about Part 2
(9) In [322], Kato states this equivalence in stages since, as the title of the paper indicates, his focus is on controlling certain non-self-adjoint operators
Summary
In 1950, Kato published a paper in a physics journal (denoted as based on a presentation in 1948) on the quantum adiabatic theorem. We will see that the Kato dynamics defines a notion of parallel transport on the natural vector bundle over the manifold of all k-dimensional subspaces of a Hilbert space, H, and so a connection This connection is called the Berry connection and its holonomy is the Berry phase (when k = 1). Once new quantum mechanics was discovered, Born and Fock [65] in 1928 discussed what they called the quantum adiabatic theorem, essentially Theorem 17.2 for simple eigenvalues with a complete set of (normalizable) eigenfunctions. It was 20 years before Kato found his wonderful extension (and more than 30 years before Berry made the breakthrough). See [29, 362, 31]
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