On the s-meromorphic OD operators

TL;DR

This paper studies s-meromorphic ordinary differential operators, showing their properties, relation to algebro-geometric operators, and how they can be approximated by rank one operators, with implications for spectral theory.

Contribution

It introduces and analyzes the class of s-meromorphic OD operators, establishing their properties, adjoint relations, and approximation by algebro-geometric rank one operators.

Findings

All s-meromorphic operators are associated with meromorphic solutions for all eigenvalues.

The adjoint of an s-meromorphic operator is also s-meromorphic.

Such operators can be approximated by algebro-geometric rank one operators on finite intervals.

Abstract

We consider linear spectral-meromorphic (s-meromorphic) OD operators at the real axis such that all local solutions to the eigenvalue problems are meromorphic for all $\lambda$. By definition, rank one algebro-geometrical operator $L$ admit an OD operator $A$ such that $[L,A]=0$ and rank of this commuting pair is equal to one. All of them are s-meromorphic. In particular, second order ``singular soliton'' operators satisfy to this condition. Operator $L^+$ formally adjoint to s-meromorphic operator $L$ is also s-meromorphic. For singular eigenfunctions of operators $L,L^+$ following scalar product $<f,g>=\int_R f\bar{g}dx$ is well-defined such that $<Lf,g>=<f,L^+g>$ avoiding isolated singular points. For the case $L=L^+$ this formula defines indefinite inner product on the spaces of singular functions $f,g\in F_L$ associated with operator $L$. They are $C^{\infty}$ outside of singularities and have isolated singularities of the same type as eigenfunctions $Lf=\lambda f$. Every s-meromorphic operator can be approximated by algebro-geometric rank one operators in any finite interval