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Spectrally meromorphic operators and non-linear systems. (English. Russian original) Zbl 1320.47046

Russ. Math. Surv. 69, No. 5, 924-926 (2014); translation from Usp. Mat. Nauk, Ser. Mat. 69, No. 5, 163-164 (2014).
Spectrally meromorphic operators on \(\mathbb R\) of the form \[ L = \partial_x + \sum_{n\geq i \geq 2} a_{n-i}\partial_x^{n-i} \] with coefficients meromorphic near the real axis are considered. Here, spectrally meromorphic means that all solutions of the eigenvalue equation \(L\psi=\lambda \psi\) are meromorphic near the singularities of \(L\). This operator is considered on a specific function space consisting of meromorphic functions with possible poles at a fixed discrete set of real points, and the restriction that all corresponding Laurent expansions contain no term of the form \(x^{-1}\).
For such operators, the spectrum is determined and, moreover, the number of negative squares of the inner product \([f,g]:= \int f(x)g(x)\,dx\) is described, where the integral is taken over \(\mathbb R\) outside a neighbourhood of the singularities and along a contour around them.

MSC:

47E05 General theory of ordinary differential operators
34L05 General spectral theory of ordinary differential operators
47B50 Linear operators on spaces with an indefinite metric
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