# Boundary problem

1. Feb 18, 2010

### zetafunction

given a boundary Hermitian eigenvalue problem

$$L= -\frac{d}{dx}\left[p(x)\frac{dy}{ dx}\right]+q(x)y=\lambda w(x)y$$

with y=y(x) , in one dimension, can we always find two operators

$$D_{1} = \frac{d}{dx}+f(x)$$ and $$D_{2} = -\frac{d}{dx}+U(x)$$

so $$L= D_{1} D_{2}$$, with $$Adj( D_{1}) = D_{2}$$ $$Adj( D_{2}) = D_{1}$$ ???

Also the eigenfunctions of L are of the form $$\Psi (x)= \phi_{2} (x) \phi _{1}(x)$$

and the eigenvalues of L are of the form $$\lambda _{n} = s.s^{*}$$