Symmetry of a regular SturmLiouville eigenvalue problem
 Notation


For a matrix $M$ by $M^\top$ we denote the matrix transpose of $M$.

In definite integrals instead of $\int_a^b f(x) dx$ we briefly write $\int_a^b f\, dx$ assuming that the reader understands which function of $x$ is being integrated.
 I. Linear algebra preliminaries


A linear algebra statement. Let $B$ be a $k\!\times\!(2k)$ matrix of rank $k$ and let $Q$ be $(2k)\!\times\!(2k)$ such that $Q^{\top}\!Q = I_{2k}$. Assume that $BQB^\top = 0_{k}$. Then ${\mathbf y}^\top Q {\mathbf x} = 0$ whenever $B{\mathbf x} = 0$ and $B{\mathbf y} = 0$.
Proof. Assume that $BQB^\top = 0_{k}$. This equality implies that the column space ${\rm Col}\bigl( QB^\top\!\bigr)$ is a subspace of the null space ${\rm Nul}\,B$. Since the rank of $B$ is $k$, by the RankNullity theorem, the dimension of the null space of $B$ is $k$. Since the rank of $B$ is $k$, the rank of $B^\top$ is $k$ and, since the matrix $Q$ is invertible, the rank of $QB^\top$ is also $k$. Thus, ${\rm Col}\bigl( QB^\top\!\bigr)$ and ${\rm Nul}\,B$ have the same dimension $k$. Together with ${\rm Col}\bigl( QB^\top\!\bigr) \subseteq {\rm Nul}\,B$ the equal dimension implies that ${\rm Col}\bigl( QB^\top\!\bigr) = {\rm Nul}\,B$. Now assume that ${\mathbf x}, {\mathbf y} \in {\rm Nul}\,B$. By the last equality we conclude that there exist vectors ${\mathbf u}, {\mathbf v} \in {\mathbb R}^k$ such that ${\mathbf x} = QB^\top {\mathbf u}$ and ${\mathbf y} = QB^\top {\mathbf v}$. Now calculate
\[
{\mathbf y}^\top\! Q {\mathbf x} = \bigl(QB^\top {\mathbf v} \bigr)^\top Q QB^\top {\mathbf u} =
{\mathbf v}^\top\! B Q^\top Q QB^\top {\mathbf u} = {\mathbf v}^\top\! BQB^\top {\mathbf u} = 0.
\]
In the last step we used the assumption $BQB^\top = 0_{k}$.

The above linear algebra statement formulated as an equivalence. Let $A$ and $B$ be $k\!\times\!(2k)$ matrices of rank $k$ such that $BA^\top = 0_{k}$. Let $Q$ be $(2k)\!\times\!(2k)$ matrix such that $Q^{\top}\!Q = I_{2k}$. Then $BQB^\top = 0_{k}$ if and only if $A Q A^\top = 0_{k}$.
Proof. Assume that $Q^{\top}\!Q = I_{2k}$, $BA^\top = 0_{k}$ and $BQB^\top = 0_{k}$. These two assumptions imply
${\rm Col}\bigl( A^\top\!\bigr) \subseteq {\rm Nul}\,B$, and ${\rm Col}\bigl( QB^\top\!\bigr) \subseteq {\rm Nul}\,B$, respectively. As in the proof of the above statement, we conclude that the dimensions of the spaces ${\rm Col}\bigl(A^\top\!\bigr)$, ${\rm Col}\bigl( QB^\top\!\bigr)$ and ${\rm Nul}\,B$ all equal to $k$. Thus,
\[
{\rm Col}\bigl( A^\top\!\bigr) = {\rm Nul}\,B \quad \text{and} \quad {\rm Col}\bigl( QB^\top\!\bigr) = {\rm Nul}\,B,
\]
and consequently
\[
{\rm Col}\bigl( A^\top\!\bigr) = {\rm Col}\bigl( QB^\top\!\bigr).
\]
This equality, the fact that the columns of $A^\top$ ($k$ of them) are linearly independent and the fact that the columns of $QB^\top$ ($k$ of them) are linearly independent, imply that there exists an invertible $k\!\times\! k$ matrix $S$ such that
\[
A^\top = QB^\top S.
\]
Then $A = S^\top\!BQ^\top$. So, we calculate, using the assumptions $Q^\top Q = I_{2k}$ and $B QB^\top = 0_k$,
\[
A Q A^\top = S^\top\!BQ^\top Q QB^\top S = S^\top\!B QB^\top S = 0_{k}.
\]
Thus we proved $A Q A^\top = 0_k$.
The implication: $Q^{\top}\!Q = I_{2k}$, $BA^\top = 0_{k}$ and $AQA^\top = 0_{k}$ imply $B QB^\top = 0_k$ follows by observing that $AB^\top = 0_{k}$ and applying the already proved implication with with the roles of $A$ and $B$ reversed.
 II. A SturmLiouville differential operator


A SturmLiouville differential operator. Let $a, b$ be real numbers, $a \lt b$. The assumptions about the coefficients:

$p$ is a continuous piecewise smooth function on the interval $[a,b]$ and $p(x) \gt 0$ for all $x \in [a,b]$,

$q$ is a piecewise continuous function on $[a,b]$,
We define a SturmLiouville differential operator $L$ by
\[
(Ly)(x) =  \bigl(p(x)y'(x)\bigr)' +q(x)y(x), \qquad a \leq x \leq b.
\]
The operator $L$ is defined for functions $y$ which are continuous on $[a,b]$ and such that the function $py'$ is a continuous piecewise smooth function. The set of all such functions $y$ we denote by ${\mathcal D}(L)$.

Lagrange's identity For all $u, v \in {\mathcal D}(L)$ we have
\[
L(u) v  u L(v) = \bigl((pu')v + u (pv') \bigr)'.
\]
The proof of this identity is by verification. The Fundamental theorem of calculus applied to Lagrange's identity yields Green's formula.

Green's formula For all $u, v \in {\mathcal D}(L)$ we have
\begin{align*}
\int_a^b (Lu) v dx  \int_a^b u (Lv) dx & = \Bigl.\bigl((pu')v + u (pv') \bigr)\Bigr_a^b \\
& = (pu')(b)v(b) + u(b) (pv')(b) + (pu')(a)v(a)  u(a) (pv')(a) \\
& =
\left[\!\begin{array}{c} v(a) \\ (pv')(a) \\ v(b) \\ (pv')(b) \end{array} \right]^\top
\left[\!\begin{array}{rrrr} 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \end{array} \right]
\left[\!\begin{array}{c} u(a) \\ (pu')(a) \\ u(b) \\ (pu')(b) \end{array} \right].
\end{align*}
We introduce the following notation
\[
{\mathsf b}(y) = \left[\!\begin{array}{c} y(a) \\ (py')(a) \\ y(b) \\ (py')(b) \end{array} \right], \qquad {\mathsf Q} = \left[\!\begin{array}{rrrr} 0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \end{array} \right].
\]
Here $(py')(a)$ means $p(a)y'(a)$. With this notation, Green's formula reads
\[
\int_a^b (Lu) v dx  \int_a^b u (Lv) dx =
{\mathsf b}(v)^\top\! {\mathsf Q} \, {\mathsf b}(u).
\]

Notice that the matrix ${\mathsf Q}$ is orthogonal: ${\mathsf Q}^\top \! {\mathsf Q} = {\mathsf Q} {\mathsf Q}^\top = I_4$.
 III. A regular SturmLiouville eigenvalue problem


A regular SturmLiouville eigenvalue problem. Let $a, b$ be real numbers, $a \lt b$. The assumptions about the coefficients:

$p$ is a continuous piecewise smooth function on the interval $[a,b]$ and $p(x) \gt 0$ for all $x \in [a,b]$,

$q$ is a piecewise continuous function on $[a,b]$,

$w$ is a piecewise continuous function on $[a,b]$ and $w(x) \gt 0$ for all $x \in (a,b)$.
We consider the following eigenvalue problem: Find all $\lambda$ for which there exists a nonzero $y \in {\mathcal D}(L)$ such that
\begin{equation*}
 \bigl(p(x)y'(x)\bigr)' + q(x)y(x) = \lambda w(x) y(x), \qquad a \leq x \leq b,
\end{equation*}
and $y$ satisfies the boundary conditions
\begin{align*}
\beta_{11} y(a) + \beta_{12} (py')(a) + \beta_{13} y(b) + \beta_{14} (py')(b) & = 0, \\
\beta_{21} y(a) + \beta_{22} (py')(a) + \beta_{23} y(b) + \beta_{24} (py')(b) & = 0.
\end{align*}
We assume that the boundary conditions are linearly independent; that is neither boundary conditions is a scalar multiple of the other.
Setting
\[
{\mathsf B} = \left[\!\begin{array}{rrrr}
\beta_{11} & \beta_{12} & \beta_{13} & \beta_{14} \\
\beta_{21} & \beta_{22} & \beta_{23} & \beta_{24} \end{array} \right]
\]
we can write the boundary conditions simply as
\[
{\mathsf B} \, {\mathsf b}(y) = 0,
\]
where $0$ is a $2\!\times\!1$ matrix. Since we assume that the boundary conditions are linearly independent, the $2\!\times\!4$ matrix ${\mathsf B}$ has rank $2$.

To make the above eigenvalue problem resemble an eigenvalue problem for matrices we will introduce a differential operator whose domain includes the boundary conditions. We call this operator $S$ and define it on the domain
\[
{\mathcal D}(S) = \bigl\{ y \in {\mathcal D}(L) : {\mathsf B} \, {\mathsf b}(y) = 0 \bigr\}.
\]
That is, the domain of $S$ consists of all the functions $y$ which are continuous on $[a,b]$ and such that the function $py'$ is a continuous piecewise smooth function and which satisfy the boundary conditions.
We define the operator $S$ as follows
\[
S y = \frac{1}{w} L y = \frac{1}{w} \bigl( (py')' + qy \bigr), \qquad y \in {\mathcal D}(S).
\]

The above stated eigenvalue problem can now be restated in terms of the operator $S$ as follows: Find all $\lambda$ for which there exists a nonzero $u \in {\mathcal D}(S)$ such that
\[
S u = \lambda \, u.
\]
 IV. Symmetry of a regular SturmLiouville eigenvalue problem


Finally, we can discuss the symmetry of the operator $S$ with respect to the inner product defined by
\[
\langle u,v \rangle = \int_a^b u \, v\, w\, dx
\]
for piecewise continuous functions $u$ and $v$. The operator $S$ is said to be symmetric if
\[
\bigl\langle Su, v \bigr\rangle = \int_a^b (Su)\, v \, w \, dx = \int_a^b u\, (Sv)\, w \, dx \bigl\langle u, Sv \bigr\rangle
\]
for all $u,v \in {\mathcal D}(S)$. Using the definition of $S$ and Green's formula from II.3, we calculate
\begin{align*}
\bigl\langle Su, v \bigr\rangle  \bigl\langle u, Sv \bigr\rangle & = \int_a^b (Su)\, v\, w\, dx  \int_a^b u\, (Sv)\, w\, dx \\
& = \int_a^b (Lu)\, v\, dx  \int_a^b u\, (Lv)\, dx \\
& = {\mathsf b}(v)^\top\! {\mathsf Q} \, {\mathsf b}(u).
\end{align*}
Hence, the operator $S$ is symmetric if and only if
\[
{\mathsf b}(v)^\top\! {\mathsf Q} \, {\mathsf b}(u) = 0 \quad
\text{whenever} \quad {\mathsf B} \, {\mathsf b}(u) = 0 \ \ \text{and} \ \ {\mathsf B} \, {\mathsf b}(v) = 0.
\]
Recall that we discussed this relationship in the Linear algebra section. There we proved: If $B$ is $k\!\times\!(2k)$ matrix of rank $k$ and $Q$ is an orthogonal $(2k)\!\times\!(2k)$ matrix such that ${B}{Q}{B}^\top = 0$, then
\[
{\mathbf y}^\top {Q} {\mathbf x} = 0 \quad \text{whenever} \quad {B}{\mathbf x} = 0 \ \ \text{and} \ \ {B}{\mathbf y} = 0.
\]

Thus, if the "boundary" matrix ${\mathsf B}$ is $2\!\times\!4$ matrix of rank $2$, ${\mathsf Q}$ is an orthogonal matrix, that is ${\mathsf Q}^\top {\mathsf Q}$, and ${\mathsf B}{\mathsf Q}{\mathsf B}^\top = 0$, we deduce that
\[
{\mathbf y}^\top {\mathsf Q} {\mathbf x} = 0 \quad \text{whenever} \quad {\mathsf B}{\mathbf x} = 0 \ \ \text{and} \ \ {\mathsf B}{\mathbf y} = 0.
\]
This implies that the operator $S$ is symmetric whenever ${\mathsf B}{\mathsf Q}{\mathsf B}^\top = 0$; that is
\[
\bigl\langle Su, v \bigr\rangle = \int_a^b (Su)\, v\, w\, dx = \int_a^b u\, (Sv)\, w \, dx = \bigl\langle u, S v \bigr\rangle
\]
holds for all $u, v \in {\mathcal D}(S)$.
 V. Consequences of the symmetry


Here we assume that the matrix ${\mathsf B}$ in the eigenvalue problem III.1 satisfies ${\mathsf B}{\mathsf Q}{\mathsf B}^\top = 0$.

Orthogonality of eigenfunctions corresponding to the distinct eigenvalues. Let $\lambda$ and $\mu$ be distinct eigenvalues of the eigenvalue problem introduced in III.1 and let $u$ and $v$, respectively, be corresponding eigenfunctions. Then $u, v \in {\mathcal D}(S)$ and
\[
Su = \lambda u \quad \text{and} \quad Sv = \lambda v.
\]
We calculate
\[
\bigl\langle Su, v \bigr\rangle = \int_a^b Su \, v \, w \, dx = \lambda \int_a^b u \, v \, w\, dx \langle u, v \rangle
\]
and
\[
\bigl\langle u, S v \bigr\rangle = \int_a^b u \, Sv \, w \, dx = \mu \int_a^b u \, v \, w \, dx = \langle u, v \rangle.
\]
By IV.2 we have
\[
\int_a^b (Su)\, v\, w\, dx = \int_a^b u\, (Sv)\, w \, dx.
\]
Therefore the last three equalities imply
\[
\lambda \langle u, v \rangle = \mu \langle u, v \rangle.
\]
Consequently
\[
\bigl(\lambda  \mu \bigr) \langle u, v \rangle = 0.
\]
Since $\lambda \neq \mu$, the last equality yields
\[
\int_a^b\! u v w\, dx = \langle u, v \rangle = 0,
\]
that is the eigenfunctions are orthogonal with respect to the inner product introduced in IV.1.

All eigenvalues are real. Let $\lambda$ be an eigenvalue of the eigenvalue problem introduced in III.1 and let $u$ be a corresponding eigenfunction. Here we allow that $\lambda$ is possibly a complex number and that $u$ is a complex function. Then $u \in {\mathcal D}(S)$ and
\[
(pu')' + qu = \lambda w u.
\]
Since we work only with real functions, we did not emphasize that the functions $p$, $q$ and $w$ are real and that all entries of the matrix ${\mathsf B}$ are real. Taking the complex conjugate of the last displayed equality we get
\[
(p\overline{u}')' + q\overline{u} = \overline{\lambda} w \overline{u}.
\]
Here the bar above a symbol denotes its complex conjugate. Taking the complex conjugate in the boundary conditions we see that the function $\overline{u}$ also satisfies the boundary conditions;that is ${\mathsf B} {\mathsf b}(\overline{u}) = 0$. Therefore $\overline{u} \in {\mathcal D}(S)$. Using the operator $S$, the last two displayed equalities can be written as
\[
Su = \lambda u \quad \text{and} \quad S\overline{u} = \lambda \overline{u}.
\]
Next we calculate
\[
\int_a^b Su \, \overline{u} \, w\, dx = \lambda \int_a^b u \, \overline{u} \, w \,dx
\]
and
\[
\int_a^b u \, S\overline{u} \, w\, dx = \overline{\lambda} \int_a^b u \, \overline{u} \, w\, dx.
\]
Since by IV.2 we have
\[
\int_a^b (Su)\, v\, w\, dx = \int_a^b u\, (Sv)\, w \, dx,
\]
the last three equalities imply
\[
\lambda \int_a^b u \, \overline{u} \, w \,dx
= \overline{\lambda} \int_a^b u \, \overline{u} \, w \, dx
\]
and consequently
\[
\bigl(\lambda  \overline{\lambda} \bigr) \int_a^b u \, \overline{u} \, w\, dx = 0.
\]
Since $u \, \overline{u} = u^2 \geq 0$, and for at least one $x_0 \in [a,b]$ we have $u^2(x_0) \gt 0$, and $w(x_0) \gt 0$ we have $\int_a^b u^2 w\, dx > 0$. Therefore the last displayed equality implies
\[
\lambda = \overline{\lambda}
\]
that is $\lambda$ is a real number.