Domains of uniqueness for $C_0$-semigroups on the dual of a Banach space

TL;DR

This paper investigates the uniqueness domains of $C_0$-semigroups on dual Banach spaces under specific topologies, and applies results to the uniqueness of solutions for Schr"odinger and Fokker-Planck equations.

Contribution

It establishes that only a core can serve as the domain of uniqueness for $C_0$-semigroups on dual spaces with the topology of uniform convergence on compact sets, and applies this to differential operators.

Findings

Only a core can be the domain of uniqueness for such semigroups.

The generalized Schr"odinger operator is $L^ abla$-unique.

Weak solutions to the Fokker-Planck equation are $L^1$-unique.

Abstract

Let $({\cal X},\|\:.\:\|)$ be a Banach space. In general, for a $C_0$-semigroup \semi on $({\cal X},\|\:.\:\|)$, its adjoint semigroup \semia is no longer strongly continuous on the dual space $({\cal X}^{*},\|\:.\:\|^{*})$. Consider on ${\cal X}^{*}$ the topology of uniform convergence on compact subsets of $({\cal X},\|\:.\:\|)$ denoted by ${\cal C}({\cal X}^{*},{\cal X})$, for which the usual semigroups in literature becomes $C_0$-semigroups. The main purpose of this paper is to prove that only a core can be the domain of uniqueness for a $C_0$-semigroup on $({\cal X}^{*},{\cal C}({\cal X}^{*},{\cal X}))$. As application, we show that the generalized Schr\"odinger operator ${\cal A}^Vf={1/2}\Delta f+b\cdot\nabla f-Vf$, $f\in C_0^\infty(\R^d)$, is $L^\infty(\R^d,dx)$-unique. Moreover, we prove the $L^1(\R^d,dx)$-uniqueness of weak solution for the Fokker-Planck equation associated with ${\cal A}^V$.