Fuchsian singular point

Fuchsian singularity of a system

A singular point $t=t_*$ of a system linear of first order ordinary differential equations with meromorphic coefficients $$\dot z=A(t)z,\qquad z\in\CC^n,\quad t\in (\CC,t_*)\tag {LS}$$ is called Fuchsian (or Fuchsian singularity), if the matrix of coefficients $A(t)$ is defined in a punctured neighborhood $(\CC,t_*)$ of the point $t_*$ has a first order pole at this point [IY, Sect. 16].

A Fuchsian singularity is always regular, but not vice versa (L. Sauvage, 1886, see [IY, Theorem 16.10]).

By a suitable holomorphic gauge transformation $z=H(t)w$ with a holomorphic invertible matrix function $H(t)\in\operatorname{GL}(n,\CC)$, $t\in(\CC,t_*)$, a system (LS) near a Fuchsian singular point $t=t_*$ can be always brought into a polynomial normal form. To describe this form, assume for simplicity that $t_*=0$ and denote by $\lambda_1,\dots,\lambda_n$ the (complex) eigenvalues of the residue matrix $A_0=\lim_{t\to 0}tA(t)\in\operatorname{Mat}(n,\CC)$. The collection of eigenvalues is called non-resonant if $\lambda_i-\lambda_j\notin\NN=\{1,2,\dots\}$, [IY, Sect. 16C].

In the non-resonant case the Fuchsian system is locally gauge equivalent to the Euler system $$\dot w=t^{-1}A_0\cdot w,\tag E$$ with the residue matrix $A_0$ which can be assumed in the Jordan normal form (upper triangular). In the resonant case the normal form is polynomial, $$ \dot w=t^{-1}(A_0+tA_1+\dots+t^d A_d)w, \quad AS_0,A_1,\dots,A_d\in\operatorname{Mat}(n,\CC),\tag{NF} $$ where the (constant) matrix coefficient are all upper triangular, and $A_k$ may containg a nonzero term in the $(i,j)$th position only if $\lambda_i-\lambda_j=k\in\NN$ [IY, Theorems 16.15, 16.16].

The normal form (NF) is explicitly integrable. Denote by $\varLambda=\operatorname{diag}\{\lambda_1,\dots,\lambda_n\}$ the diagonal matrix with the eigenvalues of $A_0$ on the diagonal and by $N=(A(1)-A_0)+A_1+\cdots+A_d$ the nilpotent constant matrix. Then the matrix exponent $t^N=\exp (N\ln t)=E+N\ln t+\tfrac1{2!}N^2\ln^2t+\cdots$ is a finite sum (matrix polynomial in $\ln t$ of degree $\leqslant d$). The (multivalued) matrix function $X(t)=t^{\varLambda}t^N$ (in the specified order) is a fundamental matrix solution for the system (1), see [IY, Lemma 16.18].

Fuchsian singularity of a linear $n$th order differential equation

Let $$ L=a_0(t)\partial^n+a_{n-1}(t)\partial^{n-1}+\cdots+a_{n-1}(t)\partial +a_n(t),\qquad \partial =\frac d{dt},\tag L $$ be a linear ordinary differential operator of order $n$ with coefficients $a_0(\cdot),\dots,a_n(\cdot)$ meromorhic in some domain $U\subseteq\CC$. The linear homogeneous equation $$ Ly=0, \qquad y=y(t),\ t\in U,\tag{LE} $$ does not change if the operator $L$ is replaced by any other operator $L'=\phi(t)L$ with a meromorphic coefficient $\phi$.

A point $t_*\in U$ is non-singular for the equation (LE), if all ratios $\frac{a_j(t)}{a_0(t)}$ have a removable singularity at $t_*$ (i.e., admit a holomorphic extension to this point). Otherwise $t_*$ is a singular point. This point is called Fuchsian singular point, or Fuchsian singularity, see [IY, Sect. 19], if $$ \forall j=1,\dots,n\quad (t-t_*)^j\frac{a_j(t)}{a_0(t)}\text{ is holomorphic at }t_*.\tag{FC} $$

Example (Euler equation).

The equation $$ t^n y^{(n)}+c_1t^{n-1}\,y^{(n-1)}+\cdots+c_{n-1}\,y'+c_n\,y=0,\qquad c_1,\dots,c_n\in\CC,\tag{EE} $$ has a Fuchsian singularity at the origin $t_*=0$.

Fuchsian condition in the Euler-like form.

The Fuchsian condition at the oriigin looks more naturally if instead of the "powers" of $\partial=\frac d{dt}$ the linear operator $L$ is expanded in the "powers" of the Euler operator $\mathscr E=t\frac d{dt}$ (in the general case one has to use the translated operator $\mathscr E_{t_*}=(t-t_*)\frac d{dt}$ to check the condition at $t_*\in\CC$). The expression for $L$ can be transformed using the Leibnitz rule $\mathscr E(fg)=f\cdot\mathscr Eg+g\cdot\mathscr Ef$ to the form with meromorphic coefficients collected to the left of the powers of $\mathscr E^j$, $$ L=a_0(t)(t^{-1}\mathscr E)^n+a_1(t)(t^{-1}\mathscr E)^{n-1}+\cdots +a_{n-1}(t)t^{-1}\mathscr E+a_n(t)=b_0(t)\mathscr E^n+b_1(t)\mathscr E^{n-1}+\cdots+b_{n-1}(t)\mathscr E+b_n(t) \tag{EL} $$ with uniquely defined meromorphic at $t_*=0$ coefficients $b_0,\dots,b_n$. In terms of these coefficients the Fuchsian condition (FC) looks completely analogous to the nonsingularity condition: $$ \text{(FC)}\iff\frac{b_j(t)}{b_0(t)}\text{ is holomorphic at }t=0\quad\forall j=1,\dots,n. $$ In some sense, the Fuchsian condition means that the equation is no more singular than the Euler equation. The transformation $$ z_1=y,\quad z_2=\mathscr Ey,\quad z_3=\mathscr E^2y,\quad \dots,\quad z_n=\mathscr E^{n-1}y\tag{T} $$ reduces the equation $Ly=0$ to a system of $n$ first order linear differential equations of the form (LS); if the origin was a Fuchsian point in the sense of (FC), then the corresponding system will have a first order pole at the origin, i.e., will be Fuchsian in the previous sense.

The polynomial $$ \lambda^n+ c_1\lambda^{n-1}+\cdots+c_{n-1}\lambda+c_n\in\CC[\lambda],\qquad c_j=\lim_{t\to0}\frac{b_j(t)}{b_0(t)}, $$ is called the characteristic polynomial and its roors are characteristic exponents of the Fuchsian singularity: they coincide with the eigenvalues of the residue matrix of the system obtained by the transformation (T).

Theorem (L. Fuchs, 1868, see [IY, Theorem 19.20])

A regular singular point of a linear $n$th order equation satisfies the Fuchs condition (FC).

Fuchsian conditions at infinity

The Fuchsian condition can also be formulated for the point $t_*=0$: the infinity is a Fuchsian singularity for the system (LS), resp., the equation (LE), if after the change of the independent variable $s=1/t$ the transformed system will have a Fuchsian singularity at the point $s=0$. For systems this means that the matrix function $A(t)$ admits a convergent expansion $A(t)=t^{-1}(A_0+t^{-1}A_1+\cdots+t^{-k}A_k+\cdots)$, and for the equations written in the Euler-like form (EL) the Fuchsian condition means that the ratios $\frac{b_j(t)}{b_0(t)}$ are holomorphic at infinity (have finite limits as $t\to\infty$).

References

[In]	E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944. MR0010757, especially Chapters XV and XIX.
[H]	P. Hartman, Ordinary differential equations, Birkhäuser, Boston, Mass., 1982, ISBN: 3-7643-3068-6, MR0658490, especially Chapter IV, Section 10.
[IY]	Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations, Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008. xiv+625 pp. ISBN: 978-0-8218-3667-5, MR2363178, especially Chapter III.