Tosio Kato’s work on non-relativistic quantum mechanics: part 2

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.

On the Absolutely Continuous Spectrum of Generalized Indefinite Strings

“…acting in L 2 (R) coincides with [0, ∞) as long as V is a real-valued potential in L 2 (R) (see also [31,33,37,40] for further results and references). A similar statement for multi-dimensional operators is known as Simon's conjecture [42,Conjecture 20.2] and still open, although this area has seen significant progress in the 1990s and 2000s. We are not attempting to give an overview of this topic (which would be a rather difficult task) but only point to the review articles [15] and [41,42], where further references can be found.…”

“…A similar statement for multi-dimensional operators is known as Simon's conjecture [42,Conjecture 20.2] and still open, although this area has seen significant progress in the 1990s and 2000s. We are not attempting to give an overview of this topic (which would be a rather difficult task) but only point to the review articles [15] and [41,42], where further references can be found. Our aim here is to investigate spectral types, or more precisely, the absolutely continuous part of the spectrum of generalized indefinite strings, introduced recently in [20].…”

On the absolutely continuous spectrum of generalized indefinite strings

“…acting in L 2 (R) coincides with [0, ∞) as long as V is a real-valued potential in L 2 (R) (see also [31,32,36,39] for further results and references). A similar statement for multi-dimensional operators is known as Simon's conjecture [42,Conjecture 20.2] and still open, although this area has seen significant progress in the 1990s and 2000s. We are not attempting to give an overview of this topic (which would be a rather difficult task) but only point to the review articles [15] and [40,41,42], where further references can be found.…”