Spectral Meromorphic Operators and Nonlinear Systems

TL;DR

This paper investigates a class of spectral-meromorphic differential operators with meromorphic coefficients, exploring their properties, self-adjointness, and relation to algebraic operators in integrable systems like KdV.

Contribution

It introduces the concept of s-meromorphic operators, analyzes their spectral properties, and connects them to algebraic operators in integrable systems, extending previous work on singular finite gap operators.

Findings

All algebraic operators in the Burchnall-Chaundy-Krichever ring are s-meromorphic.

Symmetric s-meromorphic operators are self-adjoint with respect to an indefinite inner product.

The study relates s-meromorphic operators to singular finite gap and algebrogeometric Schrödinger operators.

Abstract

We study here class of 1D spectral-meromorphic (s-meromorphic) OD operators $L=\partial_x^n+\sum_{n-2\geq i\geq 0}a_{n-2-i}\partial_x^i$ with meromorphic coefficients $a_j$ near $x\in R$ such that all eigenfunctions $L\psi=\alpha\psi$ are $x$--meromorphic near $x\in R$ for all $\alpha$. Symmetric $s$-meromorphic operators are self-adjoint with respect to indefinite inner product well-defined for some special spaces of singular functions. In particular, all algebraic operators $L$--i.e. operators entering Burchnall-Chaundy-Krichever (BChK) rank one commutative rings -- are s-meromorphic. For KdV system corresponding algebraic operator $L=-\partial_x^2+u(x,t)$ is called singular finite gap, singular soliton or algebrogeometric Schrodinger operator. This special case was already studied by the present authors in the recent works.