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



On properties of Wronskians of ODE
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.


