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Uspekhi Mat. Nauk, 2009, Volume 64, Issue 4(388), Pages 45–72 (Mi umn9307)  

This article is cited in 7 scientific papers (total in 7 papers)

Singular finite-gap operators and indefinite metrics

P. G. Grinevicha, S. P. Novikovab

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b University of Maryland, College Park

Abstract: In many problems the ‘real’ spectral data for periodic finite-gap operators (consisting of a Riemann surface with a distingulished ‘point at infinity’, a local parameter near this point, and a divisor of poles) generate operators with singular real coefficients. These operators are not self-adjoint in an ordinary Hilbert space of functions of a variable $x$ (with a positive metric). In particular, this happens for the Lamé operators with elliptic potential $n(n+1)\wp(x)$, whose wavefunctions were found by Hermite in the nineteenth century. However, ideas in [1]–[4] suggest that precisely such Baker–Akhiezer functions form a correct analogue of the discrete and continuous Fourier bases on Riemann surfaces. For genus $g>0$ these operators turn out to be symmetric with respect to an indefinite (not positive definite) inner product described in this paper. The analogue of the continuous Fourier transformation is an isometry in this inner product. A description is also given of the image of this Fourier transformation in the space of functions of $x\in\mathbb R$.
Bibliography: 24 titles.

Keywords: spectral theory, singular finite-gap operators, Lamé potentials, indefinite Hilbert spaces, continuous Fourier–Laurent bases on Riemann surfaces, Calogero–Moser models.


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English version:
Russian Mathematical Surveys, 2009, 64:4, 625–650

Bibliographic databases:

Document Type: Article
UDC: 512.772+517.984
MSC: 35P05, 37K20
Received: 24.06.2009

Citation: P. G. Grinevich, S. P. Novikov, “Singular finite-gap operators and indefinite metrics”, Uspekhi Mat. Nauk, 64:4(388) (2009), 45–72; Russian Math. Surveys, 64:4 (2009), 625–650

Citation in format AMSBIB
\by P.~G.~Grinevich, S.~P.~Novikov
\paper Singular finite-gap operators and indefinite metrics
\jour Uspekhi Mat. Nauk
\yr 2009
\vol 64
\issue 4(388)
\pages 45--72
\jour Russian Math. Surveys
\yr 2009
\vol 64
\issue 4
\pages 625--650

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    This publication is cited in the following articles:
    1. Hemery A.D., Veselov A.P., “Whittaker–Hill equation and semifinite-gap Schrödinger operators”, J. Math. Phys. (N Y), 51:7 (2010), 072108, 17 pp.  crossref  mathscinet  zmath  adsnasa  isi  scopus
    2. Grinevich P.G., Novikov S.P., “Singular solitons and indefinite metrics”, Dokl. Math., 83:1 (2011), 56–58  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. O. Chalykh, P. Etingof, “Orthogonality relations and Cherednik identities for multivariable Baker–Akhiezer functions”, Adv. Math., 238 (2013), 246–289  crossref  mathscinet  zmath  isi  elib  scopus
    4. M. V. Feigin, M. A. Hallnäs, A. P. Veselov, “Baker-Akhiezer functions and generalised Macdonald-Mehta integrals”, J. Math. Phys., 54:5 (2013), 052106, 22 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. P.G. Grinevich, S.P. Novikov, “Singular soliton operators and indefinite metrics”, Bull. Braz. Math. Soc. (N.S.), 44:4 (2013), 809–840  crossref  mathscinet  zmath  isi  scopus
    6. P. G. Grinevich, S. P. Novikov, “On $\mathbf{s}$-meromorphic ordinary differential operators”, Russian Math. Surveys, 71:6 (2016), 1143–1145  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    7. P. G. Grinevich, S. P. Novikov, “Singulyarnye solitony i spektralnaya meromorfnost”, UMN, 72:6(438) (2017), 113–138  mathnet  crossref  mathscinet  elib
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