Let $\lambda_{1}<\lambda_{2}<\ldots \lambda_{n} \ldots$ and $y_{1}(x), y_{2}(x), \ldots y_{n}(x) \ldots$ be the eigenvalues and corresponding eigenfunctions for the Sturm-Liouville system

$$\mathcal{L} y_{n}=\lambda_{n} w(x) y_{n},$$

where

$$\mathcal{L} y \equiv \frac{d}{d x}\left(-p(x) \frac{d y}{d x}\right)+q(x) y,$$

with $p(x)>0$ and $w(x)>0$. The boundary conditions on $y$ are that $y(0)=y(1)=0$.

Show that two distinct eigenfunctions are orthogonal in the sense that

$$\int_{0}^{1} w y_{n} y_{m} d x=\delta_{n m} \int_{0}^{1} w y_{n}^{2} d x .$$

Show also that if $y$ has the form

$$y=\sum_{n=1}^{\infty} a_{n} y_{n},$$

with $a_{n}$ being independent of $x$, then

$$\frac{\int_{0}^{1} y \mathcal{L} y d x}{\int_{0}^{1} w y^{2} d x} \geq \lambda_{1}$$

Assuming that the eigenfunctions are complete, deduce that a solution of the diffusion equation,

$$\frac{\partial y}{\partial t}=-\frac{1}{w} \mathcal{L} y$$

that satisfies the boundary conditions given above is such that

$$\frac{1}{2} \frac{d}{d t}\left(\int_{0}^{1} w y^{2} d x\right) \leq-\lambda_{1} \int_{0}^{1} w y^{2} d x .$$