Interlacing and oscillation for Sturm–Liouville problems with separated and coupled boundary conditions

An extension is given to the known inequalities which interlace the eigenvalues corresponding to separated and coupled boundary conditions for the problem - ( py ′ ) ′ + qy = λ ry on [ a , b ] , assuming 1 / p , q and r ∈ L 1 ( [ a , b ] ) . The key is a new interlacing principle for intervals of eigenvalues corresponding to one parameter sets of boundary conditions. Application is given to eigenfunction oscillation.