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On the \(\Psi\)-function for finite-gap potentials. (English. Russian original) Zbl 1098.34580
Russ. Math. Surv. 57, No. 1, 165-167 (2002); translation from Usp. Mat. Nauk 57, No. 1, 167-168 (2002).
From the text: \(\theta\)-function formulas for the solutions of spectral problems and finite-gap potentials leave open the question of the explicit representation of the \(\Psi\)-function in terms of potentials. S. P. Novikov [Funct. Anal. Appl. 8, 236–246 (1974); translation from Funkts. Anal. Prilozh. 8, No. 3, 54–66 (1974; Zbl 0299.35017)] established a relationship between the spectral problem (1) \(\Psi"-u(x)\Psi=\lambda \Psi\) and stationary solutions of higher equations in the KdV hierarchy if the spectrum of operator (1) has a finite number of forbidden zones. As a result of the subsequent generalization of the concept of finite-gap potentials, this term is currently used for the potentials for which the \(\Psi\)-function is a Baker-Akhiezer function on an algebraic curve of finite genus. In this paper we note that the original stationary treatment naturally leads to expressions for the \(\Psi\)-function in terms of an arbitrary finite-gap potential and for the algebraic curve in terms of leading coefficients of its polar decomposition.

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
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