The trace formula for Schrödinger operators on the line

Abstract

This paper discusses certain aspects of the spectral and inverse spectral problems for the Schrödinger operator\(L(q) = - \frac{{d^2 }}{{dx^2 }} + q(x)\), for q(x)∈C(ℝ), the space of bounded continuous functions. The trace formula of the title is the relation $$q(0) = \lambda _0 + \sum\limits_{j = 1}^\infty {(\lambda _{2j} + \lambda _{2j - 1} - 2\mu _j )} $$ with appropriate choices of {λj} ∞j=0 and {µj} ∞j=1 , which is a familiar relation in the theory of Hill's equation. We characterize the set ℐ\( \subseteq \) C(ℝ) of potentials for which this holds. Further extensions of the theory of Hill's equation are also obtained. From the spectrum σ(L(q)) a torusT(q) is constructed, which is in general infinite dimensjional; everyq(x)∈ℐ can be mapped to a continuous path onT(q), described by the auxiliary spectrum {µj} ∞j=1 . Under certain geometrical conditions on σ(L(q)) this path is the orbit of aC 1 vector field onT(q), and the mapping extends to one from the hull ℋ\((q) = \overline {\{ q(x + \xi );\xi \in \mathbb{R}\} } \) to the closure of this orbit. In particular ℋ(q) is compact. These results have applications in the theory of Schrödinger operators with ergodic potentials.