On the s-meromorphic OD operators

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

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