Singular finite-gap operators and indefinite metrics

In many problems the ‘real’ spectral data for periodic finite-gap operators (consisting of a Riemann surface with a distingulished ‘point at infinity’, a local parameter near this point, and a divisor of poles) generate operators with singular real coefficients. These operators are not self-adjoint in an ordinary Hilbert space of functions of a variable $x$ (with a positive metric). In particular, this happens for the Lamé operators with elliptic potential $n(n+1)\wp(x)$, whose wavefunctions were found by Hermite in the nineteenth century. However, ideas in [1]–[4] suggest that precisely such Baker–Akhiezer functions form a correct analogue of the discrete and continuous Fourier bases on Riemann surfaces. For genus $g>0$ these operators turn out to be symmetric with respect to an indefinite (not positive definite) inner product described in this paper. The analogue of the continuous Fourier transformation is an isometry in this inner product. A description is also given of the image of this Fourier transformation in the space of functions of $x\in\mathbb R$.
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