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Adjoint Operator

Given a Second-Order Ordinary Differential Equation

{\tilde {\mathcal L}} u(x) \equiv p_0 {du^2\over dx^2} +p_1 {du \over dx} +p_2 u,
\end{displaymath} (1)

where $p_i \equiv p_i(x)$ and $u\equiv u(x)$, the adjoint operator ${\tilde {\mathcal L}}^\dagger$ is defined by
$\displaystyle {\tilde {\mathcal L}}^\dagger u$ $\textstyle \equiv$ $\displaystyle {d\over dx^2} (p_0 u)-{d \over dx} (p_1 u)+p_2 u$  
  $\textstyle =$ $\displaystyle p_0 {d^2 u\over dx^2} +(2{p_0}'-p_1) {du\over dx} + ({p_0}''-{p_1}'+p_2)u.$  

Write the two Linearly Independent solutions as $y_1(x)$ and $y_2(x)$. Then the adjoint operator can also be written
{\tilde {\mathcal L}}^\dagger u = \int (y_2 {\tilde {\mathca...
= {\left[{{p_1\over p_0} ({y_1}'y_2-y_1{y_2}')}\right]}.
\end{displaymath} (3)

See also Self-Adjoint Operator, Sturm-Liouville Theory

© 1996-9 Eric W. Weisstein