Linear Differential Operators

Linear differential operators on a function space $U$ .

$⟨ x | L | u ⟩ = L_{x} u (x)$ , $L_{x} = \sum_{n} a_{n} (x) \frac{d^{n}}{d x^{n}}$ .

Want so solve $L | u ⟩ = | s ⟩$ .

Also consider $L | u_{0} ⟩ = 0$ which defines the nullspace of $N (L)$ .

Invertible

Assume we have some $L^{- 1} L = I$ on $N (L)^{†}$ then $u ⟩ = L^{- 1} | s ⟩ + | u_{0} ⟩$ is the general solution. Note that since the null space has vectors that disappeared, then we must include some linear combination of those in the inverse.

To completely fix a single solution, we must supply boundary conditions on the solution to select which $| u_{0} ⟩$ and which $L^{- 1}$ .

Particular Solution

$L^{- 1} | s ⟩$

Homogeneous Solution

$| u_{0} ⟩$ .

Definitions

Adjoint

$⟨ u | A^{†} | v ⟩ = ⟨ v | A | u ⟩^{*} = ⟨ A u | v ⟩$

Self Adjoint

$⟨ v | L | u ⟩ = ⟨ u | L | v ⟩^{*}$

I.e. $\int_{a}^{b} d x w (x) v (x)^{*} L_{x} u (x) = \int_{a}^{b} d x w (x) u (x) (L_{x} v (x))^{*}$ .

Aside

$⟨ x | L | u ⟩ \equiv L_{x} u (x)$ .

Theorem

For second order differential operators, $L_{x} = a (x) \frac{d^{2}}{d x^{2}} + b (x) \frac{d}{d x} + c (x)$ , $a (x) > 0$ , $a, b, c \in R$ . Is self adjoint with respect to the weight $w (x) = \frac{1}{a (x)} \exp \int_{x_{0}}^{x} d x^{'} \frac{b (x^{'})}{a (x^{'})}$ and Hermitian for any of these boundary conditions on $[a, b]$ : Dirchlet, Neumann, Robin, Periodic.

Linear Differential Operators