# Removing extraneous solutions from an eigenvalue equation

I have an eigenvalue problem of the form

$\left[ L_1 + \dfrac{L_2}{\Omega} + \dfrac{L_3}{\Omega^2} + \dfrac{\Omega-1}{\Omega+\eta}\right] \phi(x) = 0$

which I am trying to solve for the complex $\Omega$; here the various $L_i$ are operators in $x$. The technique I am using to solve the system requires me to multiply the equation with the LCM (lowest common multiple) $\Omega^2(\Omega+\eta)$ and re-write the system as

$\left[ \Omega^3\alpha + \Omega^2\beta + \Omega\gamma + \Delta \right]\phi(x) = 0$,

where $\alpha, \beta, \gamma, \Delta$ are some combination of $L_i$. Now I have two questions:

(1) The system in general has infinitely many solutions (e.g. for $L_2=0$, the solutions are a superposition of Hermite polynomials of all orders). Could the fictitious root, in particular $\Omega=-\eta$, somehow 'interact' with the physical solutions? Or would it not?

(2) In case they do interact, are there techniques to remove such extraneous roots?

Thank you

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