Sergei Yakovenko's blog: on Math and Teaching

Oscillatory behavior of Fuchsian equations

Semilocal theory

Consider a holomorphic linear equation in the unit disk $0<|t|\le 1$, having a unique Fuchsian singularity at the origin $t=0$. Such an equation can be always reduced to the form $Lu=0,\ L=\epsilon^n+a_1(t)\epsilon^{n-1}+\cdots+a_n(t)$, with holomorphic bounded coefficients $a_1,\dots,a_n\in\mathscr O(D)$, $D=\{|t|\leqslant 1\}$, $|a_k(t)|\leqslant A$.

The previous results imply that one can produce an explicit upper bound for the variation of argument of any nontrivial solution $u$ of the equation $Lu=0$ along the boundary of the unit disk $\partial D$: $\left.\mathrm{Var\,Arg\,}u(t)\right|_{t=1}^{t=\mathrm e^{2\pi \mathrm i}}\leqslant V_L=C\cdot n(A+1)$ for some universal constant $C$.

If the solution itself is holomorphic (e.g., in the case of apparent singularities), such bound would imply (by virtue of the argument principle) a bound for the number of zeros of $u$ in $D$. Unfortunately, solutions are usually ramified and the argument principle does not work. Denote by $\mathbf M$ the monodromy operator along the boundary.

Definition

The Fuchsian point is called quasiunipotent, if all eigenvalues $\mu_1,\dots,\mu_n$ of the matrix $\mathbf M$ have modulus one, $|\mu_k|=1$.

Theorem 1

The number of isolated roots of any solution of the equation $Lu=0$ in the Riemann domain $\Pi=\{0<|t|\leqslant 1,\ |\mathrm{Arg\,}t\le 2\pi\}$ having real coefficients $a_k(\mathbb R)\subseteq\mathbb R,\ k=1,\dots,n$ and a single quasiunipotent singularity at the origin does not exceed $(2n+1)(2V_L+1)$, where $V_L=Cn(A+1)$ is the parameter bounding the magnitude of coefficients of $L$.

The proof is based on a version of the flavor of the Rolle theorem for the “difference operators” $\mathbf P_\mu=\mu^{-1}\mathbf M-\mu\mathbf M^{-1}$ for any unit $\mu$ such that $\mu^{-1}=\bar\mu$:

$\#\{t\in\Pi:\ u(t)=0\}\leqslant \#\{t\in\Pi:\ \bigl(\mathbf P_\mu u\bigr)(t)=0\}+2V_L.$

A version of the Cayley-Hamilton theorem asserts that the (commutative) composition $\mathbf P=\prod_{\mu}\mathbf P_\mu$ over all eigenvalues of the monodromy operator (counted with their multiplicities) vanishes on all solutions of the real Fuchsian equation.

Global theory

A linear ordinary differential equatuib with rational coefficients from $\Bbbk=\mathbb C(t)$ can always be transformed to the form

$Lu=0,\qquad p_0(t)\partial^n+p_1(t)\partial^{n-1}+\cdots+p_n(t),\qquad p_0,\dots,p_n\in\mathbb C[t].\qquad (*)$

It may depend on additional parameters $\lambda=(\lambda_1,\dots,\lambda_r)\in\mathbb C^r$: if this dependence is rational, then we may assume that the coefficients of the operator are polynomials from $\mathbb C[t,\lambda]$. The new feature then will be appearance of singular perturbations: for some values of the parameters $\lambda=\lambda_*$ the leading coefficient $p_0(~\cdot~,\lambda_*)$ may vanish identically in $t$, meaning that the order of the corresponding equation drops down to a smaller value. Such phenomenon is known to cause numerous troubles of analytic nature.

Changing the independent variable $\tau=1/t$ allows to investigate the nature of singularity at the infinite point $t=\infty\in\mathbb C P^1$. The equation is called Fuchsian, if it is Fuchsian at each its singular point on the Riemann sphere $\mathbb C P^1=\mathbb C\cup\{\infty\}$.

Assume that infinity is non-singular (this can always be achieved by a Mobius transformation of the independent variable $t$). Then a Fuchsian equation with the singular locus $Z=\{z_1,\dots,z_m\}\subset\mathbb C$ can always be transformed to the form $Mu=0$, where $M$ is the operator

$M= E^n+q_1(t)E^{n-1}+\cdots+q_{n-1}(t)E+q_n(t),$

$E=E_Z=(t-z_1)\cdots(t-z_m)\partial$

(nonsingularity at infinity implies certain bounds on the degrees of the polynomials $p_k\in\mathbb C[t]$). However, the coefficients of this form depend in the rational way not only on the coefficients of the original equation (*), but also on the location of the points $\{z_1, \dots,z_m\}$.

Definition.

The slope of this operator (*) is defined as the maximum

$\displaystyle\angle L=\max_{k=1,\dots,n}\frac{\|p_k\|}{\|p_0\|}$

where the norm of a polynomial $p(t)=\sum_0^r c_j t^j\in\mathbb C[t]$ is the sum $\|p\|=\sum_j |c_j|$.

Simple inequalities:

1. Any polynomial $p(t)$ of known degree $d=\deg p$ and norm $M=\|p\|$ admits an explicit upper bound for $|p(t)|$ on any disk $\{|t|\leqslant R\}$: $|p(t)| \leqslant MR^d$ for $R>1$.
2. A polynomial of unit norm $\|p\|=1$ admits a lower bound for $|p(t)|$ for points distant from its zero locus $Z=\{t:\ p(t)=0\}$. More precisely,

$\displaystyle |p(t)|\geqslant 2^{-O(d)}\left(\frac rR\right)^d,\qquad r=\mathrm{dist }(t,Z),\quad R=|t|>1.$

We expect that for an equation having only quasiunipotent Fuchsian singular points, the number of isolated roots of solutions can be explicitly bounded in terms of $n=\mathrm{ord }L,\ d=\max\deg p_k$ and $B=\angle L$. Indeed, it looks like we can cut out circular neighborhoods of all singularities and apply Theorem 1.

The trouble occurs when singularities are allowed to collide or almost collide. Then any slit separating them will necessarily pass through the area where the leading coefficient $p_0$ is dangerously small.