Non-periodic one-gap potentials of the Schrodinger operator

Non-periodic one-gap potentials of the Schrodinger operator

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Dmitry Zakharov, Courant Institute of Mathematical Sciences
Fine Hall 601

The spectral theory of the one-dimensional Schrodinger operator, and the corresponding Cauchy problem for the KdV equation, has been extensively studied for two cases of potentials: rapidly vanishing and periodic. The former leads to the method of the inverse spectral transform (IST), while the latter leads to the so-called finite gap solutions defined on an auxiliary algebraic curve. An important class of rapidly vanishing potentials is the class of reflectionless Bargmann potentials, which correspond to the n-soliton solutions of KdV. It was long believed that the closure of the set of n-soliton solutions would include the periodic potentials, but an effective description of this limit has been lacking. In our work, we consider a symmetric Riemann—Hilbert problem whose finite approximations are the rapidly vanishing n-soliton solutions of KdV. We show that the elliptic one-gap potentials of the Schrodinger operator can also be constructed from this Riemann—Hilbert problem. We also show that a generic solution of this Riemann—Hilbert problem is a non-periodic one-gap potential. Joint work with Vladimir Zakharov.