Self-Adjoint Differential Equations I

Let $\mathcal{L}$ be the second-order linear differential operator
which acts on a function $u(x)$ as

Define an adjoint operator $\bar{\mathcal{L}}$ by
If $\mathcal{L}=\bar{\mathcal{L}}$, $\mathcal{L}$ is said to be self-adjoint. One can immediately see that $\mathcal{L}=\bar{\mathcal{L}}$ if and only if \begin{equation}\label{eq:self-adjoint}p_0^\prime=p_1.\end{equation} Let $p(x)=p_0(x)$ and $q(x)=p_2(x)$. Then
Note that one can transform a non-self-adjoint 2nd-order linear differential operator to a self-adjoint one. The idea is similar to that of finding a integrating factor to transform a non-separable first-order linear differential equation to a separable one.

Suppose that \eqref{eq:ldo} is not self-adjoint, i.e. $p_1\ne p_0′$. Multiply $\mathcal{L}$ by $\frac{f(x)}{p_0(x)}$. Then
Suppose $\mathcal{L}’$ is self-adjoint. Then by \eqref{eq:self-adjoint}
That is,
If $p_1=p_0′$, then
&=\frac{1}{p_0}\exp(\ln p_0(x))\\
&=\frac{1}{p_0(x)}\cdot p_0\\
i.e. $f(x)=p_0(x)$ as expected.

Eigenfunctions, Eigenvalues

From separation of variables or directly from a physical problem, we have second-order linear differential equation of the form
\begin{equation}\label{eq:sl}\mathcal{L}u(x)+\lambda w(x)u(x)=0,\end{equation}
where $\lambda$ is a constant and $w(x)>0$ is a function called a density or weighting function. The constant $\lambda$ is called an eigenvalue and $u(x)$ is called an eigenfunction.

Example. [Schrödinger Equation]

The Schrödinger equation
is of the form \eqref{eq:sl}. Recall that $H$ is the Hamiltonian operator
where $V(x)$ is a potential. So $H$ is a second-order linear differential operator. The weight function $w(x)=-1$ and $E$ is energy as an eigenvalue. Clearly Schrödinger equation is self-adjoint.

Example. [Legendre's Equations]

Legendre’s equation
$$(1-x^2)y^{\prime\prime}-2xy’+n(n+1)y=0$$ is of the form \eqref{eq:sl}, where $\mathcal{L}y=(1-x^2)y^{\prime\prime}-2xy’$, $w(x)=1$, and $\lambda=n(n+1)$. Since $p_0^\prime=-2x=p_1$, Legendre’s equations are self-adjoint.

4 thoughts on “Self-Adjoint Differential Equations I

    1. Ian Leslie

      I believe there is a typo in the adjoint operator definition. In the second line of that definition the p1 multiplying u(x) should be primed.

  1. Pingback: Self-Adjoint Differential Equations II: Hermitian Operators | MathPhys Archive

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