Category Theorems for Schrödinger Semigroups

Abstract

Stimulated by the category theorems of Eisner and Serény in the setting of unitary and isometric C_0 -semigroups on separable Hilbert spaces, we prove category theorems for Schrödinger semigroups. Specifically, we show that, to a given class of Schrödinger semigroups, Baire generically the semigroups are strongly stable but not exponentially stable. We also present a typical spectral property of the corresponding Schrödinger operators.

Slow dynamics for self-adjoint semigroups and unitary evolution groups

“…It has been shown in [1] (Proposition 3.1) that for every negative self-adjoint operator T , each η ∈ rng T and each ψ ∈ T −1 {η}, one has for t > 0,…”