The Direct and Inverse Scattering Transforms for a Sturm-Liouville Equation on Finite Interval as Smooth Maps. The Geometry of a Set of Finitely-Zoned Potentials

Abstract

Here we consider from the slightly new point of view the very old problem of finding the potential u from the spectral data of Sturm-Liouville operator $Q = - {\partial ^2}/\partial {x^2} + u\left( x \right)$ on [0, 1] with some boundary conditions. Two potentials u₁ and u₂, which are near in some sense should have the spectral data, which are near in some other sense. Here we will discuss these notions of neighborhood of spectral data for different notions of neighborhood for potentials and three classical sets of spectral data:

• i) the tied spectrum (i.e., with $\psi \left( 0 \right) = \psi \left( 1 \right) = 0$) and the norming numbers (i.e., the L₂-norms of the tied eigenfunctions with $\psi \prime \left( 0 \right) = 1$);

• ii) the tied and semitied (i.e., with $\psi \left( 0 \right) = \psi \prime \left( 1 \right) = 0$) spectrum;

• iii) the periodic and antiperiodic spectrum, the tied spectrum and some sequence of discrete parameters (signs).

Professor I.S. Zakharevich was an invited lecturer but unfortunately he could not attend the meeting.