{'fraction_non_alphanumeric': 0.02760180995475113, 'fraction_numerical': 0.0013574660633484162, 'mean_word_length': 4.727979274611399, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Dubrovin equations for finite-gap operators

It is only recently that the following question has been raised: how can one carry over the constructions relating to the Dubrovin equations (DEs) in the theory of ﬁnite-gap potentials of the Schr¨odinger operator [1] to arbitrary spectral problems? One has in mind the equations at the zeros of the Ψ-function and the trace formulae. Although the physical interpretation of these objects as (generalized) analogues of scattering data is not all that obvious, or simply does not exist, the above problem is of interest in its own right, if only because the well-known Novikov equations which arise in the general theory of ﬁnite-gap integration are (in all likelihood in the most general case) completely integrable ﬁnite-dimensional dynamical systems, and the reduction of the DEs to a Jacobi inversion problem brings about their Liouville integrability. We add that if an algebraic curve covers elliptic curves, then using the trace formulae one can produce solutions in elliptic functions, really new ﬁnite-gap potentials, and various applications to non-linearpartial diﬀerential equations. In the note [2] 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 Ψ-formula, the algebraic curve, the Novikov equations, and their integrals. Once Ψ is known it is natural to expect that the equations at its zeros γ k ( x ) should be obtainable on the basis of elementary considerations. This happens to be the case, and we shall show how to solve the problem algorithmically in the presence of additional features: trace formulae and the Abel transformation. We do not discuss here the separate question of the exact correspondence between the algebraic curve and the divisor of the zeros { γ k } on the one hand, and boundary conditions of Dirichlet type Ψ( x o ) = Ψ( x o + Ω) = 0 on the other, in the context of the recovery of the potential from these data as scattering data on the interval for arbitrary spectral problems. The family { γ k } is taken as the starting point.