Spectrally meromorphic operators and non-linear systems
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.
- Publication:
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Russian Mathematical Surveys
- Pub Date:
- October 2014
- DOI:
- 10.1070/RM2014v069n05ABEH004922
- arXiv:
- arXiv:1409.6349
- Bibcode:
- 2014RuMaS..69..924G
- Keywords:
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- Mathematics - Functional Analysis;
- Mathematical Physics;
- Nonlinear Sciences - Exactly Solvable and Integrable Systems
- E-Print:
- 5 pages, two references and new results are added