an:02076322
Zbl 1098.34579
Brezhnev, Yu. V.
On Dubrovin equations for finite-gap operators
EN
Russ. Math. Surv. 57, No. 2, 415-417 (2002); translation from Usp. Mat. Nauk 57, No. 2, 191-192 (2002).
00095928
2002
j
34L40 34L99 37K20
From the text: In [\textit{N. V. Ustinov} and \textit{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.
Zbl 1098.34580