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Grinevich (1), Russia, S.Novikov (2) ((1) Landau Institute for Theoretical Physics, Steklov Math Institute, USA","submitted_at":"2014-09-22T21:38:25Z","abstract_excerpt":"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. 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