arXiv subjects
Publications and source records attributed to Eva Lübcke.
We present the concept of Baker-Akhiezer functions on singular complex curves. For this purpose, we translate the algebraic presentation of such curves in [Se, Chapter~IV] into the analytic setting. Generalised divisors and their interplay with partial desingularisations are the fundament of the construction of Baker-Akhiezer functions.
The Burchnall-Chaundy theory concerns the classification of all pairs of commuting ordinary differential operators. We phrase this theory in the language of spectral data for integrable systems. In particular, we define spectral data for rank 1 commutative algebras $A$ of ordinary differential operators. We solve the inverse problem for such data, i.e. we prove that the algebra $A$ is (essentially) uniquely determined by its spectral data. The isomorphy type of $A$ is uniquely determined by the underlying spectral curve.
We consider a compact Riemann surface with a holomorphic involution, two marked fixed points of the involution and a divisor obeying an equation up to linear equivalence of divisors involving all this data. Examples of such data are Fermi curves of 2-dimensional periodic Schrödinger operators. We show that the equation has a solution if and only if the two marked points are the only fixed points of the involution.