# Category theorems for Schr\"odinger semigroups

## TL;DR

This paper extends category theorems to Schrödinger semigroups, showing that generically they are strongly stable but not exponentially stable, and explores typical spectral properties of related operators.

## Contribution

It introduces category theorems for Schrödinger semigroups, revealing generic stability properties and spectral characteristics, inspired by prior results in unitary and isometric semigroup settings.

## Key findings

- Schrödinger semigroups are generically strongly stable
- They are not exponentially stable in a Baire sense
- Spectral properties of Schrödinger operators are characterized

## Abstract

Stimulated by the category theorems of Eisner and Ser\'eny in the setting of unitary and isometric $C_0$-semigroups on separable Hilbert spaces, we prove category theorems for Schr\"odinger semigroups. Specifically, we show that, to a given class of Schr\"{o}dinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the corresponding Schr\"{o}dinger operators.