Gelfand–Levitan–Marchenko equation

Remarks

Example: the reflectionless data $r = 0$, $N = 1$, give $F(x) = \beta e^{-\kappa x}$. Try $K(x, y) = L(x) e^{-\kappa y}$; the equation becomes $L(x) + \beta e^{-\kappa x} + L(x) \frac{\beta}{2\kappa} e^{-2\kappa x} = 0$. With $w(x) = \frac{\beta}{2\kappa} e^{-2\kappa x}$ this gives $K(x, x) = -\frac{2\kappa w}{1 + w}$, and since $w' = -2\kappa w$, $$u(x) = -2 \frac{d}{dx} K(x, x) = -\frac{8 \kappa^2 w}{(1 + w)^2} = -2\kappa^2 \operatorname{sech}^2\!\big(\kappa (x - x_0)\big),$$ where $e^{2\kappa x_0} = \frac{\beta}{2\kappa}$. This potential has the single bound state energy $-\kappa^2$ and reflects nothing.

Non-example: the equation $K(x, y) + F(x - y) + \int_x^\infty K(x, z) F(z - y) \, dz = 0$ has kernels $F(x - y)$ and $F(z - y)$ that depend on differences; in the Gelfand–Levitan–Marchenko equation they must depend on the sums $x + y$ and $z + y$.

Several conventions are in use, with other signs and with data on the left instead of the right. The Gelfand–Levitan equation of Gelfand–Levitan theory is a related equation, with an integral over $(0, x)$, that recovers a potential on a half-line from its spectral function.