Suppose $p, q$ are functions that satisfy the properties for $$\frac{d}{dx} \Big( p(x) \frac{d\phi}{dx}\Big) + q(x)\phi + \lambda w(x) \phi = 0 \tag{1}$$ to be a Sturm-Liouville problem on some interval $[a,b]$ with Dirichlet boundary conditions.
It is well known that the eigenvalues of (1) can obtained by the Rayleigh quotient $$\lambda = -\frac{\int_a^b \Big[\phi \frac{d}{dx} \Big(p(x)\frac{d\phi}{dx}\Big) + q(x)\phi^2\Big] dx}{\int_a^b \phi^2 w(x)dx}$$ however this assumes that the solution to (1) are known so I do not see how this formula is very useful. What can one do if the solutions $\phi$ to (1) are not known? Can any estimates be derived?