Operator theory
Linear operators can be further generalised to a wider class of
differential operators, \({\cal L}\) which act on eigenfunctions
\(y_i(x)\) which have eigenvalues \(\lambda_i\), and weighted by
a weight function \(\rho(x)\).
[Sturm-Liouville Operator] The Sturm-Liouville operator is defined as
\[{\cal L} = - \qty( p(x) \dv[2]{x} + r(x) \dv{x} + q(x) )\]
[Sturm-Liouville Equation] The Sturm-Liouville equation is a
differential equation of the form
\[{\cal L} y = \lambda \rho(x) y\]
These can be simplified if \(r(x) = p^{\prime}(x)\), when
\[ \begin{align}\begin{aligned}{\cal L} = - \qty( \dv{x} \qty( p(x) \dv{x} ) + q(x) )\\and so\end{aligned}\end{align} \]
\[{\cal L} y = - (py^{\prime})^{\prime} - qy\]
Provided we set approporiate conditions on the functions
\(p(x), q(x)\), and, \(\rho(x)\), and use appropriate boundary
conditions on a range \([a,b]\), we can make a number of assertions
about the SL-equation.
[Properties of the Sturm-Liouville Operator]
[the:hermitiansloper] The Sturm-Liouville Operator is Hermitian over the
range \([a,b]\)
An operator is Hermitian over the range \([a,b]\) if
\[\int_a^b f^{*}(x) \qty[ {\cal L} g(x)] \dd{x} = \int_a^b
\qty[{\cal L} f(x)]^{*} g(x) \dd{x}\]
In the case of the SL operator,
\[ \begin{align}\begin{aligned}{\cal L} y = - \qty( p y^{\prime})^{\prime} - qy\\Applying :math:`{\cal L}` to :math:`y_i`, and premultiplying by\end{aligned}\end{align} \]
\(y_i^{*}\), then integrating over \([a,b]\),
\[ \begin{align}\begin{aligned}\begin{split} \begin{aligned}
\int_b^a y_i^{*} {\cal L} y_i \dd{x} &= - \int_a^b \qty( y_i^{*} (p y_j^{\prime})^{\prime} + y_i^{*} q y_j ) \dd{x} \\
&= - \int_a^b y_i^{*} \qty(p y^{\prime}_j)^{\prime} \dd{x} - \int_a^b y_i^{*} q y_j \dd{x}
\end{aligned}\end{split}\\The first integral can be integrated by parts to yield\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \int_a^b y_i^{*} \qty(p y^{\prime}_j)^{\prime} \dd{x} = -
\qty[y_i^{*} p y^{\prime}_j]_a^b + \int_a^b \qty(y_i^{*})^{\prime} p
y_j^{\prime} \dd{x}\\We set the boundary conditions to set\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned}\qty[y_i^{*} p y^{\prime}_j]_a^b = 0\\This leaves the integral, which can be solved by integrating again,\end{aligned}\end{align} \]
\[ \begin{align}\begin{aligned} \int_a^b \qty(y_i^{*})^{\prime} p y_j^{\prime} \dd{x} =
\qty[y_i^{*} p y^{\prime}_j]_a^b = 0 - \int_a^b \qty(
\qty(y_i^{*})^{\prime} p )^{\prime} y_j \dd{x}\\Rearranging, and returning to the original integral,\end{aligned}\end{align} \]
\[\begin{split}\begin{aligned}
\int_b^a y_i^{*} {\cal L} y_i \dd{x} &= - \int_a^b \qty[ y_j \qty( p (y_i^{*})^{\prime} )^{\prime} + y_j q y_i^{*}] \dd{x} \\
\int_a^b y_i^{*} \qty( {\cal L} y_j ) \dd{x} &= \int_a^b \qty( {\cal L} y_i)^{*} y_j \dd{x}
\end{aligned}\end{split}\]
The eigenvalues of the Sturm-Liouville Operator are real.
q From theorem [the:hermitiansloper] the operator is known to be
Hermitian. From theorem [the:eigenvaluehermitian] we know that the
eigenvalues of a Hermitian matrix are real.
The eigenvalues of the Sturm-Liouville Operator form an ordered set
\(\lambda_1 < \lambda_2 < \cdots < \lambda_n\).
The eigenfunctions of the Sturm-Liouville Operator, \(y_i(x)\) have
\(i-1\) zeros over the range \([a,b]\).
The normalised eigenfunctions, \(y_i(x)\) of the Sturm-Liouville
operator form an orthogonal basis,
\[\braket{y_i}{y_j} = \delta_{ij}\]