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Bounded solutions of KdV and non-periodic one-gap potentials in quantum mechanics. (English) Zbl 1382.37071

Summary: We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of \(N\)-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by \(2N\) real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.

MSC:

37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
81U15 Exactly and quasi-solvable systems arising in quantum theory
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