I know for the self-adjoint operators there is a robust theory for counting the eigenvalues based in Sturm-Liouville theory, even for differential operators of order greater than 2 (Greenberg has a lot of works in this line).
I'm trying to count the negative eigenvalues of the operator $=\partial_x^4 −\partial_x^2+\partial_x^2(())$ with $()$ smooth and rapidly decaying as $\to±\infty$. This operator is non-self-adjoint, the continuum spectrum is [0,∞), and the eigenvalues are real. But I'm not sure if the classical Sturm-Liouville Theory can be applied in this case.
In seeking out references for the theory of Sturm-Liouville theory in non-self-adjoint operator with order greater than 2, I have come across papers over particular results but nothing that clear my doubt. Thus my question:
Is there a book or paper that goes over the basics of non-self adjoint operators?
Is the hypothesis of self-adjoint essential in the Sturm-Liouville Theory?