# Sergei Yakovenko's blog: on Math and Teaching

## Wednesday, November 12, 2014

### Lecture 4 (Nov. 14, 2014)

Filed under: Analytic ODE course,lecture — Sergei Yakovenko @ 5:47
Tags: , ,

## Algebraic theory of linear ordinary differential operators

• Differential field $\Bbbk=\mathscr M(\mathbb C^1,0)$ of meromorphic germs of functions of one variable $t\in(\mathbb C^1,0)$ + derivation $\partial =\frac{\mathrm d}{\mathrm dt}$ produce noncommutative polynomials $\Bbbk[\partial]$: a polynomial $L=\sum_{j=0}^n a_j\partial ^{n-j}$ acts on $\Bbbk$ in a natural way.
• The equation $Lu=0$ only exceptionally rarely has a solution in $\Bbbk$, but one can always construct a differential extension of $\Bbbk$ which will contain solutions of this equation.
• Analytically solutions of the equation form a tuple of functions $(u_1,\dots,u_n)$ 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 $M\in\mathrm{GL}(n,\mathbb C)$ such that $\Delta (u_1,\dots,u_n)=(u_1,\dots, u_n)\cdot M$.
• Instead of $\partial$, any other derivation can  be used, in particular, the Euler derivation $\epsilon=t\partial$.
• Example. Equations with constant coefficients have the form $L=\sum c_j \partial^{n-j}$ with constant coefficients $c_j\in\mathbb C$. Such an operator can always be factorized into commuting factors, $L=c_0\,\prod_{\lambda_i\in\mathbb C} (\partial-\lambda_i)^{\nu_i}$ with $\sum\nu_i=n=\deg L$. A fundamental system of solutions consists of quasipolynomials $q_{ik}(t)=\mathrm e^{\lambda_i}t^k$, $0\leqslant k < \nu_i$. In a similar way the Euler operator has the form $L=\sum c_j\epsilon^{n-j}$ and its solutions are functions $u_{ik}=t^{\lambda_i}\ln^k t$, $k=0,1,\dots,\nu_i-1$ (look at the model equation $\epsilon^\nu u=0$).
• Weyl equivalence of of two operators. Two operators $L,M\in\Bbbk[\partial]$ of the same order are called Weyl equivalent, if there exist an operator $H\in\Bbbk[\partial]$ which maps any solution $u$ of the equation $Lu=0$ to a solution  $v=Hu$ of the equation $Mv=0$ isomorphically (i.e., no solution is mapped to zero).
The above definition means that the composition $MH$ vanishes on all solutions of $Lu=0$, hence must be divisible by $L$: $MH=KL$ for some $K\in\Bbbk[\partial]$.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 $H,L$: it is divisible by both $L$ and $H$.
• 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 $H, L$ are mutually prime, there exist two operators $U,V\in\Bbbk[\partial]$ such that $UL+VH=1$,  hence $LUL+LVH=L$. This identity means that $LVH$ is simultaneously divisible by $L$ and by $H$ (immediately). Hence $LVH$ is divisible by their least common multiple $KL=MH$: there exists an operator $W\in\Bbbk[\partial]$ such that $LVH=W\cdot MH=WMH$. But since the algebra $\Bbbk[\partial]$ is without zero divisors, the right factor $H$ can be cancelled, implying $LV=WM$, which means that $V$ maps solutions of $Mv=0$ into those of $Lu=0$.
• Different flavors of Weyl equivalence: regular (nonsingular) requiring $H, K$ be nonsingular or arbitrary.
• Theorem. Any nonsingular operator $L=\partial^n+\sum_1^n a_j \partial^j$ with holomorphic coefficients $a_j\in\mathscr O(\mathbb C,0)$, is regular Weyl equivalent to the operator $M=\partial^n$.
This result is analogous to the rectification theorem reducing any nonsingular system $\mathrm dX=\Omega X$ to $\mathrm dX=0$.
• Theorem. Any Fuchsian operator is Weyl equivalent to an Euler operator.
This is similar to the meromorphic classification of tame systems. The conjugacy $H$ 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 $\partial$ or in $\epsilon$, one can represent them as non-commutative (formal) Taylor series of the form $L=\sum_{k\geqslant 0}t^k p_k(\epsilon)$ with the coefficients $p_k\in\mathbb C[\epsilon]$ from the commutative algebra of univariate polynomials, but not commuting with the “main variable” $t$.

Such an operator is Fuchsian of order $n$, if and only if $\deg p_k\leqslant n$ for all $k=1,2,\dots$, and $\deg p_0=n$. The polynomial $p_0$ is the “eulerization” of $L$, and the series can be considered as a noncommutative perturbation of the Euler operator $L_0=p_0\in\mathbb C[\epsilon]$.

Definition. The operator $L=p_0+tp_1+\cdots$ is non-resonant, if no two roots of $p_0$ differ by a nonzero integer, $\lambda_i-\lambda_j\notin\mathbb Z^*$.

Theorem. A non-resonant Fuchsian operator is Weyl equivalent to its Euler part with the conjugacy $H$ being a Fuchsian operator, $H=h_0+th_1+\cdots$, $\deg h_0\leqslant n-1,\ \gcd(p_0,h_0)=1$.

## In search of the general theory (to be continued)

References.

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).