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**On a class of elliptic potentials of the Dirac operator.**
*(English.
Russian original)*
Zbl 0920.35122

Sb. Math. 188, No. 1, 115-135 (1997); translation from Mat. Sb. 188, No. 1, 109-128 (1997).

This paper addresses the question of constructing finite gap potentials of the Dirac operator:
\[
-i\left[\begin{matrix} 1&0\\ 0&-1\end{matrix}\right] +\left[\begin{matrix} 0&p(x,t)\\ q(x,t)&0\end{matrix}\right]
\]
which are moreover elliptic with respect to \(x\) (the time parameter \(t\) obeys the dynamics of the nonlinear Schrödinger equation and in ansatz I below, also the Calogero-Moser dynamics for the particles \(x_j(t)\), \(j=1,\dots,n\), where the potential has a pole as a function of \(x\)). By using an ansatz on the poles of the potential similar to the one used for the scalar operator case by I. M. Krichever [Funkts. Anal. Prilozh. 14, No. 4, 45-54 (1980; Zbl 0462.35080)] and other authors, the author constructs the desired type of solutions. However, since the spectral curve is not given explicitly in terms of these poles (Krichever had produced a Lax-pair matrix for the dynamics, whose determinant gives the spectral curve) but also through an ansatz, the condition could be empty.

In this paper, the author constructs new examples to show that the ansatz is not empty, and he does this in fact in each of the two following cases: Ansatz I, the elliptic curve has only one point covered by the two points ‘at infinity’ of the spectral curve (two classes of genus-2 spectral curves are explicitly exhibited); Ansatz II, the two points ‘at infinity’ of the spectral curve ar e mapped to two distinct points of the elliptic curve [spectral curves of this type had been exhibited by A. O. Smirnov, Sb. Math. 186, No. 8, 1213-1221 (1995); translation from Mat. Sb. 186, No. 8, 133-141 (1995; Zbl 0863.35029)]. The qualification ‘at infinity’ refers to the spectral parameter of the Dirac operator.

In this paper, the author constructs new examples to show that the ansatz is not empty, and he does this in fact in each of the two following cases: Ansatz I, the elliptic curve has only one point covered by the two points ‘at infinity’ of the spectral curve (two classes of genus-2 spectral curves are explicitly exhibited); Ansatz II, the two points ‘at infinity’ of the spectral curve ar e mapped to two distinct points of the elliptic curve [spectral curves of this type had been exhibited by A. O. Smirnov, Sb. Math. 186, No. 8, 1213-1221 (1995); translation from Mat. Sb. 186, No. 8, 133-141 (1995; Zbl 0863.35029)]. The qualification ‘at infinity’ refers to the spectral parameter of the Dirac operator.

Reviewer: E.Previato (Boston)