Algebraic theory of linear ordinary differential operators
- Differential field of meromorphic germs of functions of one variable + derivation produce noncommutative polynomials : a polynomial acts on in a natural way.
- The equation only exceptionally rarely has a solution in , but one can always construct a differential extension of which will contain solutions of this equation.
- Analytically solutions of the equation form a tuple of functions analytic and multivalued in a punctured neighborhood of the origin. The multivaluedness is very special: the linear span remains the same after the analytic continuation, hence there exists a matrix such that .
- Instead of , any other derivation can be used, in particular, the Euler derivation .
- Example. Equations with constant coefficients have the form with constant coefficients . Such an operator can always be factorized into commuting factors, with . A fundamental system of solutions consists of quasipolynomials , . In a similar way the Euler operator has the form and its solutions are functions , (look at the model equation ).
- Weyl equivalence of of two operators. Two operators of the same order are called Weyl equivalent, if there exist an operator which maps any solution of the equation to a solution of the equation isomorphically (i.e., no solution is mapped to zero).
The above definition means that the composition vanishes on all solutions of , hence must be divisible by : for some .Note that the operator represented by each side of the above equality, is a non-commutative analog of the least common multiple of mutually prime polynomials : it is divisible by both and .
- Theorem. The Weyl equivalence is indeed an equivalence relationship: it is reflexive, symmetric and transitive.
The only thing that needs to be proved is the symmetry. Since are mutually prime, there exist two operators such that , hence . This identity means that is simultaneously divisible by and by (immediately). Hence is divisible by their least common multiple : there exists an operator such that . But since the algebra is without zero divisors, the right factor can be cancelled, implying , which means that maps solutions of into those of .
- Different flavors of Weyl equivalence: regular (nonsingular) requiring be nonsingular or arbitrary.
- Theorem. Any nonsingular operator with holomorphic coefficients , is regular Weyl equivalent to the operator .
This result is analogous to the rectification theorem reducing any nonsingular system to .
- Theorem. Any Fuchsian operator is Weyl equivalent to an Euler operator.
This is similar to the meromorphic classification of tame systems. The conjugacy may be non-Fuchsian.
- Missing part: a genuine analog of holomorphic classification of Fuchsian systems.
Poincare-Dulac-Fuchs classification of Fuchsian operators
Instead of representing operators as non-commutative polynomials in or in , one can represent them as non-commutative (formal) Taylor series of the form with the coefficients from the commutative algebra of univariate polynomials, but not commuting with the “main variable” .
Such an operator is Fuchsian of order , if and only if for all , and . The polynomial is the “eulerization” of , and the series can be considered as a noncommutative perturbation of the Euler operator .
Definition. The operator is non-resonant, if no two roots of differ by a nonzero integer, .
Theorem. A non-resonant Fuchsian operator is Weyl equivalent to its Euler part with the conjugacy being a Fuchsian operator, , .
In search of the general theory (to be continued)
The classical paper by Ø. Ore (1932) in which the theory of non-commutative polynomials was established, and the draft of the paper by Shira Tanny and S.Y., based on Shira’s M.Sc. thesis (Weizmann Institute of Science, 2014).