adjoint

Usage

adjoint L

Parameter	Type	Description
L	$\mathbbm{Q}[x,\frac d{dx}]$	A differential operator

Returns

Returns the adjoint operator of $L$, i.e.

$$L^\ast = \sum_{i=0}^n (-1)^i \frac{d^i}{dx^i}\cdot a_i$$

where

$$L = \sum_{i=0}^n a_i \frac{d^i}{dx^i}$$

and $\cdot$ is the product in $\mathbbm{Q}[x,\frac d{dx}]$.

Example

It can happen that $L^\ast$ has a nontrivial rational kernel, while $L$ has a trivial one:

1 --> L := x*D^3 + D^2 - x^2*D - 2*x;
2 --> LL := adjoint(L);
3 --> tex(LL);

$$-x\,D^{3}-2\,D^{2}+x^{2}\,D$$

So $L^\ast 1 = 0$, while $L y = 0$ has no nonzero rational solution:

4 --> K := kernel(L);
5 --> tex(K);

$$[~]$$

We also verify that $L^{\ast\ast} = L$:

6 --> tex(L - adjoint(LL));

$$0$$