Solvability and adjoint equation?

[jollage]

Hi all,

I post this thread in the Calculus, but no one seems to be interested. I realize that's a wrong place, So I repost here, could the moderators remove that thread, https://www.physicsforums.com/showthread.php?t=704643. Thank you.

Basically I don't understand the following procedure discussing the solvability of the differential equation. Please have a look and comment.

The operator $\mathcal{L}$ is defined as $\mathcal{L}=\mathcal{L}(\phi_0;k,\omega,X)$, where $\phi_0$ is the variable and $k,\omega,X$ are the parameters.

Now take the derivative w.r.t. k of the homogeneous equation $\mathcal{L}=0$, we have $\mathcal{L}(∂_k\phi_0;k,\omega,X)=-\mathcal{L}_k(\phi_0;k,\omega,X)-\mathcal{L}_{\omega}(\phi_0;k,\omega,X)∂_k\omega$, where it assumes $\omega$ is dependent on $k$ while $X$ independent on $k$.

Then the author writes that the solvability of the above differential equation requires that the right hand side be orthogonal to the homogeneous solution to the adjoint equation of $\mathcal{L}=0$.

I've came across such solvability criterion now and then (I guess it is common knowledge), but never truly understand what it really means and where it comes from. Also, what's the role of the adjoint equation? Could you help me? Thanks a lot in advance.

 

[gtoguy97]

The solvability criterion you mentioned is basically derived from Fredholm's alternative theorem, which states that a system of linear equations is solvable if and only if the right-hand-side of its equation is orthogonal to the kernel (the vector space spanned by the solutions of the homogeneous equation). In this case, the adjoint equation of \mathcal{L}=0 is used to find the kernel. In general, the adjoint equation is obtained by taking the transpose of the coefficients of \mathcal{L}, and the solutions of the adjoint equation are used to construct the so-called "dual basis", which by definition is orthogonal to the kernel of the original equation. The orthogonality of the right-hand-side of the equation with respect to the kernel is then equivalent to the orthogonality of the dual basis with respect to the right-hand-side of the equation. Thus, the solvability criterion is established.