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Sturm-Liouville problem¶

aka S.L.P
definition

\begin{matrix} (r y^{'})^{'} + (q + λ p) y = 0 \\ p > 0 and r > 0 on (a, b) \end{matrix}

eigenvalue $λ$
- S.L.P has a nontrivial solution for $λ$ , then the $λ$ is an eigenvalue
eigenfunction
- the solution for the corresponding S.L.P is called eigenfunction

## Regular SLP

boundary condition

\begin{aligned} A_{1} y (a) + A_{2} y^{'} (a) = 0 \\ B_{1} y (b) + B_{2} y^{'} (b) = 0 \end{aligned}

in which,

\begin{aligned} A_{1}^{2} + A_{2}^{2} \neq 0 \\ B_{1}^{2} + B_{2}^{2} \neq 0 \end{aligned}

Periodic SLP¶

define on $[a, b]$
$r (a) = r (b)$
$y (a) = y (b)$ , $y^{'} (a) = y^{'} (b)$

Singular SLP¶

$r (a) = 0$ , or $r (b) = 0$ , (or both)

SL Theorem¶

if $ϕ$ is an eigenfunction, then $c ϕ$ is also an eigenfunction $\forall c \in ℝ$
$λ_{n} \neq λ_{m}$
let SLP define on $(a, b)$ , then

\begin{matrix} (ϕ_{n}, ϕ_{m}) = \int_{a}^{b} p (x) ϕ_{n} (x) ϕ_{m} (x) d x = 0 \end{matrix}

in which $p$ is weighted function

Eigenfunction expansion¶

With series solution, we can write any function in

\begin{array}{r} f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} \end{array}

likewise, we can also write

\begin{array}{r} f (x) = \sum_{m = 0}^{\infty} a_{m} y_{m} (x) \end{array}

in which $y$ is an eigenfunction, multiply this equation by $p (x) y_{n}$

\begin{matrix} p (x) f (x) y_{n} = \sum_{m = 0}^{\infty} a_{m} y_{n} y_{m} \end{matrix}

and integrate to get

\begin{aligned} (f, y_{m}) & = \sum_{n = 1}^{\infty} a_{n} (y_{n}, y_{m}) \\ = a_{m} (y_{m}, y_{m}) \\ a_{m} & = \frac{(f, y_{m})}{(y_{m}, y_{m})} = \frac{(f, y_{m})}{| | y_{m} | |^{2}} \end{aligned}