The curiosity in inverse difficulties of spectral research has elevated significantly lately because of the functions to special non-linear equations in mathematical physics. This monograph is dedicated to the particular idea of inverse difficulties and techniques in their resolution for the Sturm-Liouville case. Chapters 1--6 comprise proofs that are, in lots of situations, very diverse from these recognized past. Chapters 4--6 are dedicated to inverse difficulties of quantum scattering idea with cognizance being inquisitive about actual purposes. Chapters 7--11 are in accordance with the author's fresh study at the concept of finite- and infinite-zone potentials. A bankruptcy discussing the purposes to the Korteweg--de Vries challenge is usually incorporated. This monograph is critical analyzing for all researchers within the box of arithmetic and physics.

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