Sturm-Liouville equation ($-(p{y}^{\prime }{)}^{\prime }+qy=\lambda wy$)

A second-order differential equation in the displayed divergence form. Under the usual regular spectral assumptions, $p>0$, $w>0$, the coefficients have suitable regularity, and separated self-adjoint boundary conditions specify a Sturm-Liouville problem. Multiplying by $y$ and using zero endpoint values gives the Rayleigh quotient $\lambda =\int (p{y}^{\prime 2}+q{y}^2)dx/\int w{y}^{2}dx$ for a nonzero real eigenfunction. Sign conventions for the potential and eigenvalue vary; they must be read from the stated equation.