I've posted this now in on /r/math on math.se to no avail. Maybe this problem is harder than I thought and you folks can help me out.

I'm working on a variational problem in elasticity which involves a hefty number of Lagrange multipliers. I have calculated the second variation to be $$ \int_{-h}^h ds ~\psi \left[ \frac{d^4 }{ds^4 } + \frac{d}{ds}(\Lambda(s) \frac{d}{ds} )\right] ~\psi $$

where $ \psi $ is the variation field, and $ \Lambda(s) $ is a Lagrange multiplier function. I understand that in order to evaluate stability of solutions to the Euler-Lagrange equations, I want to look at the spectrum of the differential operator in brackets. How do I do that when this unknown function shows up in the operator? I can numerically find extrema of the functional, which lead to a particular $\Lambda(s) $, but the only property $\Lambda(s) $ seems to have in general is that it is even. A recommendation for a text that could advise me about how to find spectra for such operators would also be welcome.