Periodic limit of inverse scattering

I t is well known that a p‐periodic potential Q(x) can be reconstructed from spectral data of the corresponding Hill operator −(d2/dx2) + Q(x) in terms of a Riemann θ‐function. We regard the periodic potential Q(x) as the pointwise limit of a scattering potential QN,c(x) (defined to equal Q(x) when −Np ≦ x ≦ Np, to equal zero when x < Np) and to equal c2 when x > (Np) as N → ∞ and c2 → ∞. The scattering potential QN,c(x) can be recovered from the scattering data of the corresponding Schrödinger operator in terms of a Dyson determinant according to a well known‐theory. We derive the Riemann θ‐function corresponding to the periodic potential Q(x) by taking the above limit of the Dyson determinant for the scattering potential. We first calculate the scattering data of the potential QN,c(x) through recursive formulas in terms of the left transmission and reflection coefficients T and R of the potential which is equal to Q(x) when 0 ≦ x ≦ p and equal to zero otherwise. We use these data to express the Dyson determinant of QN,c(x). We then expand the Dyson determinant into a Fredholm series and compute the main contributions to the expansion in the asymptotic limit N → ∞ and c2 → ∞ using a method developed by Lax, Levermore, and Venakides in their study of the small dispersion limit of the initial value problem of Korteweg‐de Vries equation. The computation of the leading order contributions reduces to a quadratic functional maximization problem constrained by a positivity condition and by a mass quantization condition. The solutions to this maximization problem constitute the differentials on a Riemann surface, the main ingredients for the Riemann θ‐function corresponding to the periodic potential. The limit of the Dyson determinant for QN,c(x) as N → ∞ and c2 → ∞ is shown to equal the exact Riemann θ‐function corresponding to the periodic potential Q(x) times an exponential function with exponent being a quadratic polynomial in x. Our calculation includes the correct phase shifts of the θ‐function. © 1993 John Wiley & Sons, Inc. Copyright © 1993 Wiley Periodicals, Inc., A Wiley Company

Duke Scholars

Author Stephanos Venakides Mathematics