Can Factorization Always Solve Boundary Hermitian Eigenvalue Problems?

[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 $$\lambda _{n} = s.s^{*}$$

 

[gtoguy97]

, where s= \langle \phi _{2} \mid D_{1} \mid \phi _{1} \rangle ?No, this is not always possible. In general, it is not possible to find two operators D_1 and D_2 such that L=D_1D_2 since the boundary condition of the problem may be incompatible with the desired operators. For example, if the boundary condition involves a Dirichlet or Neumann condition, then the operators D_1 and D_2 would be incompatible with the boundary condition. Furthermore, in general, the eigenfunctions of L are not of the form \Psi (x)= \phi_{2} (x) \phi _{1}(x) and the eigenvalues of L are not of the form \lambda _{n} = s.s^{*} .