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Singular soliton operators and indefinite metrics. (English) Zbl 1319.34153

The authors consider Sturm-Liouville equations \[ L\Psi \equiv -\Psi''(x)+u(x)\Psi (x)=\lambda \Psi (x) \] whose local solutions are meromorphic in \(x\) for all \(\lambda\). Such are operators \(L\), for which there exists a linear differential operator \(A\) of odd order such that \([L,A]=0\). In particular, this class contains operators with singular finite-gap and singular soliton potentials. The operators are considered on certain classes of functions with singularities, where they are symmetric with respect to an indefinite inner product with a finite number of negative squares. A spectral theory of these operators is developed both for periodic and rapidly decreasing potentials.
Connections with the time dynamics provided by the KdV hierarchy are studied. As the authors say, “the right analog of Fourier transform on Riemann Surfaces with good multiplicative properties (the R-Fourier transform) is a partial case of this theory. The potential has a pole in this case at \(x=0\) with asymptotics \(u\sim g(g+1)/x^2\). Here \(g\) is the genus of spectral curve”.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47E05 General theory of ordinary differential operators
47B50 Linear operators on spaces with an indefinite metric
47B25 Linear symmetric and selfadjoint operators (unbounded)
33E10 Lamé, Mathieu, and spheroidal wave functions
34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain
34B24 Sturm-Liouville theory
34L05 General spectral theory of ordinary differential operators
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References:

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