# Linear Differential Operators

## Linear Differential Operators

Linear differential operators on a function space \(U\).

\(\langle x|L|u\rangle = L_x u(x)\), \(L_x=\sum_n a_n(x)\frac{d^n}{dx^n}\).

Want so solve \(L|u\rangle = |s\rangle\).

Also consider \(L|u_0\rangle = 0\) which defines the nullspace of \(N(L)\).

### Invertible

Assume we have some \(L^{-1}L=\mathbb{I}\) on \(N(L)^\dagger\) then \(u\rangle = L^{-1}|s\rangle + |u_0\rangle\) 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 \(|u_0\rangle\) and which \(L^{-1}\).

#### Particular Solution

\(L^{-1}|s\rangle\)

#### Homogeneous Solution

\(|u_0\rangle\).

## Definitions

### Adjoint

\(\langle u|A^\dagger|v\rangle = \langle v|A|u\rangle^*=\langle Au|v\rangle\)

### Self Adjoint

\(\langle v|L|u\rangle = \langle u|L|v\rangle^*\)

I.e. \(\int_a^b dx w(x)v(x)^* L_xu(x) = \int_a^b dx w(x)u(x)(L_xv(x))^*\).

### Aside

\(\langle x|L|u\rangle \equiv L_x u(x)\).

### Theorem

For second order differential operators, \(L_x=a(x)\frac{d^2}{dx^2}+b(x)\frac{d}{dx}+c(x)\), \(a(x)>0\), \(a,b,c\in\mathbb{R}\). Is self adjoint with respect to the weight \(w(x)=\frac{1}{a(x)}\exp\int_{x_0}^xdx'\frac{b(x')}{a(x')}\) and Hermitian for any of these boundary conditions on \([a,b]\): Dirchlet, Neumann, Robin, Periodic.