The Weyl symbol of Schr\"odinger semigroups
Laurent Amour, Lisette Jager, Jean Nourrigat

TL;DR
This paper analyzes the Weyl symbol of Schr"odinger semigroups, providing estimates on its behavior and derivatives, especially in high dimensions, and establishing its membership in a specific symbol class.
Contribution
It introduces dimension-independent estimates for the Weyl symbol of Schr"odinger semigroups with nonnegative potentials, extending understanding in high-dimensional analysis.
Findings
Derived $L^{ abla}$ norm estimates for the Weyl symbol.
Proved dimension-independent derivative estimates for the symbol.
Established the symbol's membership in a specific high-dimensional symbol class.
Abstract
In this paper, we study the Weyl symbol of the Schr\"odinger semigroup , , , on , with nonnegative potentials in . Some general estimates like the norm concerning the symbol are derived. In the case of large dimension, typically for nearest neighbor or mean field interaction potentials, we prove estimates with parameters independent of the dimension for the derivatives . In particular, this implies that the symbol of the Schr\"odinger semigroups belongs to the class of symbols introduced in [1] in a high-dimensional setting. In addition, a commutator estimate concerning the semigroup is proved.
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Spectral Theory in Mathematical Physics · Advanced Mathematical Physics Problems
