2015/10/22 by P. G. Grinevich, Grinevich, P. G., S. P. Novikov +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Physical sciences #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Meromorphic and Entire Functions #math-ph #math.MP
paper · pdf · doi:10.48550/arxiv.1510.06770
LaTeX, 8 pages
arxiv created 2015/10/22 · openalex publication_date 2015/10/22 · arxiv updated 2018/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 λ. 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>=∫R fgdx 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∈ FL associated with operator L. They are C∞ outside of singularities and have isolated singularities of the same type as eigenfunctions Lf=λf. Every s-meromorphic operator can be approximated by algebro-geometric rank one operators in any finite interval