Reflectionless Schrodinger operators and Marchenko parametrization

Abstract

Let $T_q=-d^2/dx^2 +q$ be a Schr\"odinger operator in the space $L_2(\mathbb{R})$. A potential $q$ is called reflectionless if the operator $T_q$ is reflectionless. Let $\mathcal{Q}$ be the set of all reflectionless potentials of the Schr\"odinger operator, and let $\mathcal{M}$ be the set of nonnegative Borel measures on $\mathbb{R}$ with compact support. As shown by Marchenko, each potential $q\in\mathcal{Q}$ can be associated with a unique measure $\mu\in\mathcal{M}$. As a result, we get the bijection $\Theta\colon \mathcal{Q}\to \mathcal{M}$. In this paper, we show that one can define topologies on $\mathcal{Q}$ and $\mathcal{M}$, under which the mapping $\Theta$ is a homeomorphism.