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

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.

DOI: https://doi.org/10.4213/rm9307

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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

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
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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.
2. Grinevich P.G., Novikov S.P., “Singular solitons and indefinite metrics”, Dokl. Math., 83:1 (2011), 56–58
3. O. Chalykh, P. Etingof, “Orthogonality relations and Cherednik identities for multivariable Baker–Akhiezer functions”, Adv. Math., 238 (2013), 246–289
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.
5. P.G. Grinevich, S.P. Novikov, “Singular soliton operators and indefinite metrics”, Bull. Braz. Math. Soc. (N.S.), 44:4 (2013), 809–840
6. P. G. Grinevich, S. P. Novikov, “On $\mathbf{s}$-meromorphic ordinary differential operators”, Russian Math. Surveys, 71:6 (2016), 1143–1145
7. P. G. Grinevich, S. P. Novikov, “Singular solitons and spectral meromorphy”, Russian Math. Surveys, 72:6 (2017), 1083–1107
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