Estimating eigenvalues of second order linear ODE

Question

Suppose an ODE is given by a linear operator $L$ where $$L = P(x)\frac{d^2}{dx^2} + Q(x) \frac{d}{dx}$$ and we would like to find the eigenvalues of this operator so that $$Ly = \lambda y.$$ I am interested in theorems that either estimate $\lambda$ or can give a bound for $\lambda$. The only result I am aware of is when the ODE is a Sturm-Liouville problem, for which we can use the Rayleigh quotient. I'm not sure much can be said in general. What if we put more conditions on $P$ and $Q$ beyond them just being continuous, for example that they're trigonometric, polynomials, or periodic? Is the only way to determine $\lambda$ through direct computation?

Answer 1

Suppose that $P(x)>0$ and divide by it to see that the equation is equivalent to

$$y'' + \frac{Q}{P}y' = \lambda \frac{1}{P}y$$ Next, define $\mu = \exp\left({\int\dfrac{Q}{P}dx }\right)>0$. If we multiply through by $\mu$ and use the fact that $$(\mu y')' = \mu y'' + \frac{Q}{P}\mu y'$$ then the equation reduces to

$$(\mu y')' = \lambda \frac{\mu}{P}y$$ which is a Sturm Liouville equation.