06 - Adjoint Operators

Inner Product

Inner Product of two (suitably smooth) functions defined on an interval $[a,b]$ by

$$⟨u,v⟩:={\int }_a^{b}u(x)\overline{v(x)}dx$$

where $\overline{v(x)}$ denotes the complex conjugate of $v(x)$

Adjoint Operator

Let $L$ be a linear operator then

Corresponding adjoint operator $L^{\ast }$ is defined by inner product relation

$$⟨Ly,w⟩=⟨y,L^{\ast }w⟩$$

for all $y,w$ in suitable inner product space

Solving for the Linear Operator

1. Using integration by parts move the derivatives of operator from $y$ to $w$

2. Using the boundary conditions of $y$ ensure that the boundary terms vanish to get boundary conditions for $w$

Properties of Adjoint Operator

1. Calculate adjoint $L^{\ast }$ of operator $L$ without worrying about boundary conditions

2. If $L^{\ast }=L$ then operator $L$ is self-adjoint

3. Operator $L$ combined with homogenous boundary condition give problem in form

$$L+BC$$

So corresponding adjoint boundary conditions give adjoint problem

$$L^{\ast }+B{C}^{\ast }$$

4. If $L=L^{\ast }$ and $BC=B{C}^{\ast }$ then problem is fully self-adjoint

5. If $L=L^{\ast }$ but $BC\ne B{C}^{\ast }$ then problem is formally self-adjoint

General Form for Adjoint Operator

Let $L$ be a linear operator then

$$Ly=P_2{y}^{\prime \prime }+P_1{y}^{\prime }+P_{0}y⟺L^{\ast }w=(P_{2}w{)}^{\prime \prime }-(P_{1}w{)}^{\prime }+P_{0}w$$

Property for the Boundary

As

$$\begin{array}{rl}wLy-y{L}^{\ast }w& =w[P_2{y}^{\prime \prime }+P_1{y}^{\prime }+P_{0}y]-y[(P_{2}w{)}^{\prime \prime }-(P_{1}w{)}^{\prime }+P_{0}w]\\ & =[P_{2}w{y}^{\prime }-(P_{2}w{)}^{\prime }y+P_{1}wy{]}^{\prime }\end{array}$$

Hence

$$⟨Ly,w⟩-⟨y,L^{\ast }w⟩=[P_{2}w{y}^{\prime }-(P_{2}w{)}^{\prime }y+P_{1}wy{]}^{\prime }$$

Inner Product with Weighting Function

Let $r(x)$ be a weighting function then

$$⟨y_j,y_k{⟩}_r:={\int }_a^b{y}_j(x){\overline{y}}_k(x)r(x)dx$$

where $r>0$ (almost everywhere) on $[a,b]$