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On commuting differential operators. (English) Zbl 0953.34073
Let $$F$$ be an ordinary differential field of characteristic $$0$$, $$D$$ be a differentiation of $$F$$, $$C$$ be its field of constants and $$R=F[D]$$ be a ring of linear differential operators over $$F$$(in other terms a skew polynomial ring). The author discusses in the paper operators from $$R$$ which have certain symmetry properties (so-called algebro-geometric expressions). Roughly speaking (Burchnall-Chaundy theorem), $$L\in R\setminus F$$ is algebro-geometric if the centralizer $$Z_R(L)$$ is not standard (i.e. $$Z_R(L)\neq C[M]$$ for some $$M\in R$$). G. M. Bergman [Trans. Am. Math. Soc. 137, 327-344 (1969; Zbl 0175.31501)] proved that for a free algebra over $$k$$ (related closely with a skew polynomial ring) any such centralizer is a polynomial ring of one indeterminate over $$k$$. Thus the existence of algebro-geometric elements isn’t an ordinary phenomenon. But when $$F$$ is a strongly normal extension of $$C$$ such elements exist. For example, $$L=D^2 - 2/x^2$$ is a simplest algebro-geometric operator over $$F=C(x)$$. I. Schur showed [Berl. Math. Ges. Sitzungsber. 4, 2-8 (1904; JFM 36.0387.01)] that a centralizer of any element $$L\in RF$$ is commutative. So a study of algebro-geometric operators of $$R$$ is equivalent to a study of commutative subrings of $$R$$.
The author obtains for the cases $$F=C(x)$$ and $$F=C(\exp(2\pi x/p))$$ the following main result (theorem 1): Suppose that the coefficients of the differential expression $L=D^n+q_{n-2}D^{n-2} +\dots+ q_0$ are either rational functions, which are bounded at infinity, or else meromorphic, simply periodic functions with period $$p$$, which remain bounded as $$\mid \operatorname{Im}(x/p)\mid$$ tends to infinity. If, regardless of $$z \in C$$, all solutions to the differential equation $$Ly=zy$$ are meromorphic then $$L$$ is algebro-geometric.
We note that according to K. Yosida’s result [Jap. J. Math. 9, 231-232 (1933; Zbl 0007.20801)] the linear differential operators over $$C(x)$$ satisfying conditions of theorem 1 have a fundamental system of solutions ( not just one solution) of the form $r(x)\exp(\lambda x)(\lambda \in C, r(x)\in C(x)).$ So these operators are just equivalent (over $$C(x)$$) to the operators with constant coefficients.
Then the author proves that if the family of equations $$Ly=zy$$ permits a general solution of special kind then $$L$$ is algebro-geometric. Using this solution one can easily receive a linear system of differential equations the solvability of which in $$F$$ means that the operator $$L$$ is algebro-geometric.

##### MSC:
 34M05 Entire and meromorphic solutions to ordinary differential equations in the complex domain 37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 34M15 Algebraic aspects (differential-algebraic, hypertranscendence, group-theoretical) of ordinary differential equations in the complex domain
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