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On Dubrovin equations for finite-gap operators. (English. Russian original) Zbl 1098.34579
Russ. Math. Surv. 57, No. 2, 415-417 (2002); translation from Usp. Mat. Nauk 57, No. 2, 191-192 (2002).
From the text: In [N. V. Ustinov and Yu. V. Brezhnev, Russ. Math. Surv. 57, No. 1, 165–167 (2002); translation from Usp. Mat. Nauk 57, No. 1, 167–168 (2002; Zbl 1098.34580)] the following universal property of finite-gap potentials was discovered: they form a class for which the spectral problem is integrable in quadratures. There it is shown how to obtain all the ingredients of the direct spectral problem: the \(\Psi\)-formula, the algebraic curve, the Novikov equations, and their integrals. Once \(\Psi\) is known it is natural to expect that the equations at its zeros \(\gamma_k(x)\) should be obtainable on the basis of elementary considerations. This happens to be the case, and we show how to solve the problem algorithmically in the presence of additional features: trace formulae and the Abel transformation.

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L99 Ordinary differential operators
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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