The boundary control approach to inverse spectral theory

We establish connections between four approaches to inverse spectral problems: the classical Gelfand–Levitan theory, the Simon theory, the approach proposed by Remling, and the Boundary Control method. We show that the Boundary Control approach provides simple and physically motivated proofs of the central results of other theories. We demonstrate also the connections between the dynamical and spectral data and derive the local version of the classical Gelfand–Levitan equations. In this paper we consider the Schrödinger operator (0.1) H = −∂2 x + q (x) on L (R+) ,R+ := [0,∞), with a real-valued locally integrable potential q and Dirichlet boundary condition at x = 0. Let dρ(λ) be the spectral measure corresponding to H, and m(z) be the (principal or Dirichlet) Titchmarsh-Weyl mfunction. 1. Three approaches to inverse spectral theory In this section we give a brief review of three different approaches to inverse problems for the operator (0.1): the Gelfand–Levitan theory, the Simon theory and the Remling approach. In the next section we describe the Boundary Control method and its connections with the other approaches. 1.1. Gelfand–Levitan theory. Determining the potential q from the spectral measure is the main result of the seminal paper by Gelfand and Levitan [16]. To formulate the result let us define the following functions: σ(λ) = { ρ(λ)− 2 3π λ 3 2 , λ > 0, ρ(λ), λ < 0 (1.1) F (x, t) = ∫ ∞ −∞ sin √ λx sin √ λt λ dσ(λ). (1.2) Let φ(x, λ) be a solution to the equation −φ′′ + q(x)φ = λφ, x > 0, (1.3) 1991 Mathematics Subject Classification. 34B20, 34E05, 34L25, 34E40, 47B20, 81Q10.