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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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