 Sturm–Liouville theory

Sturm-Liouville theory studies weighted eigenvalue problems of the formwith boundary conditions making the operator self-adjoint. For a regular finite-interval problem, $p,w$ are positive and appropriate coefficient regularity is assumed. Integration by parts cancels the boundary terms and proves that eigenfunctions at distinct eigenvalues are orthogonal in the weighted inner product $\int w{y}_1\overline{y_2}$.

For $p=w=1$, $q=0$, $y(0)=y(L)=0$, the eigenfunctions are $\sin (n\pi x/L)$ with eigenvalues $(n\pi /L{)}^2$, $n\ge 1$. Their expansions solve wave equations and heat equations by reducing spatial behavior to independent modes. Singular endpoints require additional domain and boundary analysis; regular-problem conclusions cannot simply be transferred without those hypotheses.

Sturm–Liouville theory is a fundamental concept in the field of differential equations and mathematical physics. It deals with a specific type of second-order linear differential equation known as the Sturm–Liouville problem. This theory has applications in various areas, including quantum mechanics, vibration analysis, and heat conduction.