On properties of Wronskians of ODE

Question

Hello everyone,

I'm currently working over a certain class of ODE of the form

$D_N \; \phi(x) = \lambda^N \phi(x) \quad, \quad N>1$

where $D_N = \delta_N \delta_{N-1} \cdots \delta_1$ and $\delta_k = (\frac{d}{dx}-A_k(x)), \quad 1 \leq k \leq N$.

For various reasons my interest at the moment is focused on a class of objects, that I call $m$-order Wronskians defined as $ \quad w_m = \det[W^{(m)}] \quad $ with the entries of the $m \times m$ matrix $W^{(m)}$ being

$ W^{(m)}_{ij} = \frac{d^{i-1}}{dx^{i-1}}\phi_j(x) \qquad 1 \leq i \leq m\ , \quad 1\leq j \leq m $

where $\left\lbrace \phi_j(x) \right\rbrace_{j=1}^N$ is a set of $N$ independent solutions of the ODE.

WHAT I KNOW:

1 - The $N$-order Wronskian $w_N$ is a nonzero constant, given I set $\sum_{i=1}^NA_i=0$ (which means that the coefficient of the $(N-1)$-order derivative in the ODE vanishes). This obviously reflect the independence of the solutions $\phi_j$ .

2- The $(N-1)$-order Wronskian $z(x) = w_{N-1}$ is a solution of the adjoint equation

$ D_N^{\dagger} \; z(x) = \lambda^N z(x) \quad $ where $\quad D_N^{\dagger} = \delta^{\dagger}_1 \delta^{\dagger}_2 \cdots \delta^{\dagger}_N \quad$ and $\quad \delta^{\dagger}_k = (\frac{d}{dx}+A_k(x))$

WHAT I WOULD LIKE TO KNOW:

some argument led me to argue that $m$-order Wronskians ($\forall m\ ,\ 1\leq m \leq N-1$) should satisfy some kind of ODE or pseudo-ODE (with which I mean equations that might contain both differential and integral operators). Sadly I could neither demonstrate this nor find any reference concerning this subject

Does someone know some reference to this subject?

Thank you very much for your time.

Stefano

Answer 1

I don't have a reference, a while back I learned in a paper by Percy Deift something that my help; it is easy to fill in the details with a bit of linear algebra. Write the equation as a system $\frac{d}{dx}F = M F \quad $ where $F$ is a function of $x$ taking values in $R^{n}$, and $M$ is a square matrix valued function. If $F_1, F_2$ are solutions, then $G= F_1 \wedge F_2$ the wedge product of the two solutions satisfies $\frac{d}{dx}G = M_2 G$ where $M_2 \colon R^{n} \wedge R^{n} \to R^{n} \wedge R^{n}$ is the linear map such that $M_2(a \wedge b) = Ma \wedge b + a \wedge Mb$ for all $a, b \in R^{n}$. The components of $G$ in the standard basis of $R^{n} \wedge R^{n}$ are the 2-minors of $\{\frac{d^{i-1}}{dx^{i-1}}\phi_j(x)\}_{1 \leq i \leq n\ , 1\leq j \leq n}$ ; i.e. the 2 order Wronskian derived from $F_1, F_2$ is one of the coordiantes of $G$. The other Wronskians appear in higher wedge products of solutions.