# 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

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

$$\langle v|L|u\rangle = \langle u|L|v\rangle^*$$

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.

### Inverses of Differential Operators

Created: 2023-06-25 Sun 02:25

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