Frontmatter -- Contents -- Preface -- Introduction -- Chapter 1. transformation operators -- Chapter 2. inverse sturm-liouville problem on the half-line: determining the sturm-liouville operator from spectral function -- Chapter 3. determining the regular sturm-liouville operator from two spectra -- Chapter 4. the inverse problem of the quantum scattering theory -- Chapter 5. the inverse problem of the quantum scattering theory at fixed energy -- Chapter 6. the inverse problem of the scattering theory on the whole line -- Chapter 7. inverse sturm-liouville problem on the whole line from a spectral matrix function -- Chapter 8. finite-zone potentials -- Chapter 9. infinite-zone potentials -- Chapter 10. quasi-periodicity of finite-zone potentials -- Chapter 11. quasi-periodicity of infinite-zone potentials -- Chapter 12. relation between the sturm-liouville and the kortewegde vries equation -- References