Singular Finite-Gap Operators and Indefinite Metric

TL;DR

This paper studies singular finite-gap operators with real spectral data, revealing their symmetry in indefinite inner products and establishing an analog of Fourier transform within this framework.

Contribution

It introduces a framework where singular finite-gap operators are symmetric in indefinite inner products, extending Fourier analysis concepts to Riemann surface-based operators.

Findings

Operators are symmetric in an indefinite inner product space.

An isometry acts as an analog of the Fourier transform in this setting.

Real spectral data can lead to operators with singular coefficients.

Abstract

Many "real" inverse spectral data for periodic finite-gap operators (consisting of Riemann Surface with marked "infinite point", local parameter and divisors of poles) lead to operators with real but singular coefficients. These operators cannot be considered as self-adjoint in the ordinary (positive) Hilbert spaces of functions of x. In particular, it is true for the special case of Lame operators with elliptic potential $n(n+1)\wp(x)$ where eigenfunctions were found in XIX Century by Hermit. However, such Baker-Akhiezer (BA) functions present according to the ideas of works by Krichever-Novikov (1989), Grinevich-Novikov (2001) right analog of the Discrete and Continuous Fourier Bases on Riemann Surfaces. It turns out that these operators for the nonzero genus are symmetric in some indefinite inner product, described in this work. The analog of Continuous Fourier Transform is an isometry in this inner product. In the next work with number II we will present exposition of the similar theory for Discrete Fourier Series