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
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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).

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