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Mat. Zametki, 2016, Volume 99, Issue 2, Pages 283–287 (Mi mz10670)  

This article is cited in 3 scientific papers (total in 3 papers)

Common Eigenfunctions of Commuting Differential Operators of Rank $2$

V. S. Oganesyan

Lomonosov Moscow State University

Abstract: Commuting differential operators of rank $2$ are considered. With each pair of commuting operators a complex curve called the spectral curve is associated. The genus of this curve is called the genus of the commuting pair. The dimension of the space of common eigenfunctions is called the rank of the commuting operators. The case of rank $1$ was studied by I. M. Krichever; there exist explicit expressions for the coefficients of commuting operators in terms of Riemann theta-functions. The case of rank $2$ and genus $1$ was considered and studied by S. P. Novikov and I. M. Krichever. All commuting operators of rank $3$ and genus $1$ were found by O. I. Mokhov. A. E. Mironov invented an effective method for constructing operators of rank $2$ and genus greater than $1$; by using this method, many diverse examples were constructed. Of special interest are commuting operators with polynomial coefficients, which are closely related to the Jacobian problem and many other problems. Common eigenfunctions of commuting operators with polynomial coefficients and smooth spectral curve are found explicitly in the present paper. This has not been done so far.

Keywords: commuting differential operators of rank $2$, common eigenfunctions, spectral curve, confluent Heun equation.

Funding Agency Grant Number
Ministry of Education and Science of the Russian Federation НШ-4833.2014.1


DOI: https://doi.org/10.4213/mzm10670

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English version:
Mathematical Notes, 2016, 99:2, 308–311

Bibliographic databases:

UDC: 517.926.4
Received: 27.02.2015

Citation: V. S. Oganesyan, “Common Eigenfunctions of Commuting Differential Operators of Rank $2$”, Mat. Zametki, 99:2 (2016), 283–287; Math. Notes, 99:2 (2016), 308–311

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. E. Mironov, “Self-adjoint commuting differential operators of rank two”, Russian Math. Surveys, 71:4 (2016), 751–779  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    2. V. S. Oganesyan, “Commuting Differential Operators of Rank 2 with Rational Coefficients”, Funct. Anal. Appl., 52:3 (2018), 203–213  mathnet  crossref  crossref  isi  elib
    3. V. S. Oganesyan, “The AKNS hierarchy and finite-gap Schrödinger potentials”, Theoret. and Math. Phys., 196:1 (2018), 983–995  mathnet  crossref  crossref  adsnasa  isi  elib
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