Gelfand–Levitan theory

Definition

Let $q \colon [0, \infty) \to \R$ be continuous, let $h \in \R$, and consider the Schrödinger operator $-y'' + q y$ on $[0, \infty)$ with the boundary condition $y'(0) = h y(0)$. For $\lambda \in \C$, let $\varphi(x, \lambda)$ be the solution of $-y'' + q y = \lambda y$ with $\varphi(0, \lambda) = 1$ and $\varphi'(0, \lambda) = h$. Let $\rho$ be the spectral function of the operator. This is the nondecreasing function for which $f \mapsto \int_0^\infty f(x) \varphi(x, \lambda) \, dx$ extends to a unitary map from $L^2(0, \infty)$ onto $L^2(\R, d\rho)$. Let $\rho_0(\lambda) = \frac{2}{\pi}\sqrt{\lambda}$ for $\lambda \ge 0$ and $\rho_0(\lambda) = 0$ for $\lambda < 0$; this is the spectral function for $q = 0$ and $h = 0$. Set $$F(x, t) = \int_{-\infty}^{\infty} \cos(\sqrt{\lambda}\, x) \cos(\sqrt{\lambda}\, t) \, d(\rho - \rho_0)(\lambda),$$ where the integral converges in a suitable regularised sense. The Gelfand–Levitan equation is the linear integral equation for an unknown function $K(x, t)$: $$K(x, t) + F(x, t) + \int_0^x K(x, s) F(s, t) \, ds = 0, \qquad 0 \le t < x.$$ Gelfand–Levitan theory is the method that recovers $q$ and $h$ from $\rho$ by solving this equation and setting $$q(x) = 2 \frac{d}{dx} K(x, x), \qquad h = K(0, 0).$$