Preface
Sturm--Liouville theory is actually a generalization for the infinite dimensional case of famous eigenvalue/eigenvector problems for finite square matrices that we discussed in Part I of this tutorial. Although a Sturm--Liouville problem can be formulated in operator form as L[ y ] = λy similar to the matrix eigenvalue problem Ax = λx, where the operator L is an unbounded differential operator and y is a smooth function.
The corresponding theory that is known as Sturm--Liouville theory originated in two articles by the French mathematician of German descent Jacques Charles François Sturm (1803--1855).- Sturm, J.C.F., "Mémoire sur les Équations différentielles linéaires du second ordre", Journal de Mathématiques Pures et Appliquées, 1836, Vol. 1, pp. 106-186. [sept. 28, 1833]
- Sturm, , J.C.F., "Mémoire sur une classe des d'Équations à différences partielles". Journal de Mathématiques Pures et Appliquées, 1836, Vol. 1, pp. 373--444.
- Sturm & Liouville, "Démonstration d'un théorème de M. Cauchy relatif aux racines imaginaires des équations". Journal de Mathématiques Pures et Appliquées, 1836, Vol. 1, pp. 278--289.
- Sturm & Liouville, "Note sur un théorème de M. Cauchy relatif aux racines des équations simultanées", Comptes rendus de l'Académie des Sciences (English: Proceedings of the Academy of Sciences), 1837, Vol. 4, pp. 720--739.
-
Sturm & Liouville,
"Extrait d'un Mémoire sur le développement des fonctions en séries dont les
différents termes sont assujettis à satisfaire à une même équation
différentielle lindaire, contenant un paramètre variable",
Journal de Mathématiques Pures et Appliquées, 1837, Vol 2,
pp. 220--233.
Comptes rendus de l'Académie des Sciences ((English: Proceedings of the Academy of sciences), 1837, Vol. 4, pp. 675--677.
Sturm--Liouville Problems
\begin{equation} \label{EqSturm.3}
L\left[ x, \texttt{D} \right] = -\texttt{D}\,p(x)\,\texttt{D} + q(x)\,\texttt{I} , \qquad \texttt{D} =
\frac{\text d}{{\text d}x} , \quad \texttt{I} = \texttt{D}^0 ,
\end{equation}
where I is the identity operator, and p(x) and q(x) are given continuous
functions on some interval [𝑎, b]. Since p(x) follows the derivative operator, it
should be a differentiable function. The linear self-adjoint operator \eqref{EqSturm.3} is referred to as the
Sturm--Liouville operator.
Correspondingly, the Sturm--Liouville theory is about a real second-order homogeneous linear self-adjoint differential equation with a parameter λ of the form
\begin{equation} \label{EqSturm.1}
\frac{\text d}{{\text d}x} \left[ p(x)\,\frac{{\text d}y}{{\text d}x}
\right] - q(x)\, y + \lambda \,\rho (x)\,y(x) =0 , \qquad a < x < b ,
\end{equation}
subject to the homogeneous boundary conditions
\begin{equation} \label{EqSturm.2}
\alpha_0 y(a) - \alpha_1 y'(a) =0 , \qquad \beta_0 y(b ) + \beta_1 y' (b) =0 , \qquad |\alpha_0 | +
|\alpha_1 | \ne 0 \quad\mbox{and} \quad |\beta_0 | + |\beta_1 | \ne 0.
\end{equation}
Let us consider an arbitrary second order differential equation
\[
M\left[ x, \texttt{D} \right] u = a(x)\,u'' + b(x)\, u' + c(x)\, u = 0 ,
\]
with some given continuous functions 𝑎(x), b(x), and c(x). We
denote by p(x) an integrating factor
\[
p(x) = \exp \left\{ \int \frac{b(x)}{a(x)}\,{\text d} x \right\}
\]
Multiplying the given equation by p(x)/𝑎(x), we obtain
\[
L \left[ x, \texttt{D} \right] u = \frac{p(x)}{a(x)}\,M\left[ x, \texttt{D} \right] u = p(x)\, u'' + p'
(x)\,u' + \frac{p(x)}{a(x)}\,c(x)\,u = \frac{\text d}{{\text d}x} \left( p(x)\,u' \right) - q(x)\, u = 0,
\]
where \( q(x) = - p(x)\,c(x)/a(x) . \) So L is a Sturm--Liouville operator.
Upon introducing boundary operators
\begin{equation} \label{EqSturm.4}
B_a y = \alpha_0 y(a) - \alpha_1 y'(a) \qquad\mbox{and}\qquad B_{b} y = \beta_0 y(b) + \beta_1 y' (b) ,
\end{equation}
we can rewrite this Sturm--Liouville problem in operator form:
\begin{equation} \label{EqSturm.5}
L\left[ x, \texttt{D} \right] y = \rho\,\lambda y , \qquad B_a y = 0, \quad B_{b} y = 0.
\end{equation}
This boundary value problem has an obvious solution---the identically zero function. Since we are not after such
a trivial solution, we need something more. The Sturm--Liouville problem (S-L, for short) consists of two
parts: the first part is about finding values of parameter λ for which the problem has a nontrivial
solution (not identically zero); such values are called eigenvalues. The second part includes
determination of nontrivial solutions that are called eigenfunctions. Note that a Sturm--Liouville
problem may have other constraints, not necessarily formulated above.
There are two kinds of Sturm--Liouville problems. One is called classical or regular when p(x) > 0 and 1/p(x) > 0 for all points from the closed interval x∈[𝑎, b]. These assumptions are necessary to render the theory as simple as possible while retaining considerable generality. It turns out that these conditions are valid in many problems that we will consider shortly. If p(𝑎) = 0, or p(b) = 0, or p(𝑎) = 0 = p(b), the Sturm--Liouville problem is said to be singular. If conditions on coefficients p(x), q(x), and ρ(x) are held to make the Sturm--Liouville problem regular, but the interval is unbounded, then such problem is also referred to as singular.
Joseph Liouville was a French mathematician known for his work in analysis, differential geometry, and number theory and for his discovery of transcendental numbers---i.e., numbers that are not the roots of algebraic equations having rational coefficients. He was also influential as a journal editor and teacher. Joseph founded the Journal de Mathématiques Pures et Appliquées which retains its high reputation up to today.
In physics and other applications, many problems arise in the form of boundary value problems involving second order ordinary differential equations, written in self-adjoint form:
\[
L\left[ x, \texttt{D} \right] = -\texttt{D} \left( p(x)\,\texttt{D} \right) + q(x)\, \texttt{I} \qquad
\mbox{or}\qquad L\left[ x, \texttt{D} \right] y = -\frac{\text d}{{\text d}x} \left(
p(x)\,\frac{{\text d}y}{{\text d}x}\right) + q(x) \, y ,
\]
where \( \texttt{D} = {\text d}/{\text d}x \) is
the derivative operator and \( \texttt{I} = \texttt{D}^0 \) is the identity
operator. Sturm was the first who generalized the well known
matrix problem for eigenvalues/eigenvectors to the linear differential
operators by considering the eigenvalue problem \eqref{EqSturm.5}
for functions y(x) subject to some boundary conditions. Now this problem is referred to as the
Sturm--Liouville problem.
Here p(x), q(x), and ρ(x) > 0 are specified continuous functions at
the outset, which is usually some interval (finite or not) of real axis ℝ.
Example 1: Let us consider the Sturm--Liouville problem with periodic boundary conditions
\[
y'' + \lambda \,y =0 , \qquad y(0) = y(T) .
\tag{1.1}
\]
Solutions of the differential equation \( y'' + \lambda \,y =0 \) depend on the
sign of parameter λ. If λ = -μ2 is negative, the equation has two exponential
linearly independent solutions \( y_1 = e^{\mu x} \quad\mbox{and}\quad y_2 = e^{-\mu x}
\) that also sometimes can be written through hyperbolic sine and cosine functions. These
functions cannot be periodic, so negative λ is not an eigenvalue.
If λ = 0, the general solution of the differential equation \( y'' =0 \) is a linear function \( y = a + b\,x , \) which could be periodic only when b = 0. So zero is an eigenvalue corresponding to an eigenfunction which is a constant function in this case.
If λ = μ2 is positive, then the general solution becomes
\[
y = a\,\cos (\mu x) + b\,\sin (\mu x) ,
\]
with some constants 𝑎 and b. This function is periodic with period T if and only if
μ is proportional to 2π/T; so we get eigenvalues
\[
\mu_n = \frac{2n\pi}{T} \qquad \Longrightarrow \qquad \lambda_n = \left( \frac{2n\pi}{T} \right)^2 , \quad
n= 1,2,3,\ldots .
\]
The corresponding linearly independent eigenfunctions are
\[
\cos \frac{2n\pi x}{T} \qquad\mbox{and} \qquad \sin \frac{2n\pi x}{T} , \quad n=1,2,3,\ldots .
\]
Therefore, the eigenspace is two-dimensional and the eigenfunction can be written in vector form:
\[
\phi_k (x) = \left[ \begin{array}{c} \cos \left( \frac{k\pi x}{\ell} \right) \\
\sin \left( \frac{k\pi x}{\ell} \right) \end{array} \right] , \qquad k=1,2,\ldots .
\tag{1.2}
\]
We can include λ = 0 into the general formula and claim that Sturm--Liouville problem (1.2) has a
sequence of nonnegative eigenvalues
\[
\lambda_n = \left( \frac{2n\pi}{T} \right)^2 , \qquad n=0,1,2,3,\ldots ;
\tag{1.3}
\]
with corresponding eigenfunctions
\[
\phi_n (x) = a_n \cos \left( \frac{2n\pi x}{T} \right) + b_n \sin \left( \frac{2n\pi x}{T} \right) ,
\]
with some real constants 𝑎n, bn. These arbitrary constants indicate
that the eigenfunction is two-dimensional. Of course, you can organize it in one-dimensional array upon
introducing positive and negative indices:
\[
\phi_n (x) = \cos \left( \frac{2n\pi x}{T} \right) , \quad n=0,1,2,\ldots ; \qquad \phi_{-n} (x) = \sin
\left( \frac{2n\pi x}{T} \right) , \quad n=1, 2, 3 , \ldots .
\tag{1.4}
\]
Then setting 𝑎-n = bn, we utilize a one-dimensional list.
■
Example 2: Let us consider the one-dimensional Schrödinger equation
\[
- \frac{\hbar}{2m}\,\psi'' (x) = E\, \psi (x) , \qquad x \in (0, \ell ) ,
\tag{2.1}
\]
and the boundary conditions
\[
\psi (0) = 0 \qquad \psi (\ell ) = 0 .
\tag{2.2}
\]
We rewrite its Sturm--Liouville problem in the standard way:
\[
y'' + \lambda \,y =0 , \qquad y(0) = 0, \quad y(\ell ) =0 .
\tag{2.3}
\]
When a function is specified at the boundary, then the corresponding boundary conditions are named after G. Dirichlet.
Solutions of the differential equation \( y'' + \lambda \,y =0 \) depend on the
sign of parameter λ. If λ = -μ2 is negative, the equation has two exponential
linearly independent solutions \( y_1 = e^{\mu x} \quad\mbox{and}\quad y_2 = e^{-\mu x} .
\)
Then the general solution becomes
\[
y(x) = c_1 e^{\mu x} + c_2 e^{-\mu x} ,
\]
with some constants c1 and c2. From the first boundary condition, we
have
\[
y(0) = c_1 + c_2 =0 \qquad \Longrightarrow \qquad c_1 = - c_2 .
\]
The boundary condition at x = ℓ dictates
\[
y(\ell ) = c_1 \left( e^{\mu \ell} - e^{-\mu \ell} \right) = 2\,c_1 \sinh (\mu \ell) =0 .
\]
Since sine hyperbolic can be zero only at the origin, we conclude that negative value of λ cannot be
an eigenvalue.
If λ =0, the general solution of the differential equation \( y'' =0 \) is a linear function:
\[
y(x) = c_1 + c_2 x .
\]
From boundary conditions, it follows
\[
y(0) = c_1 =0 \qquad\mbox{and} \qquad y(\ell ) = c_2 \ell =0 .
\]
So for λ =0 there is no nontrivial solution.
For positive λ = μ2, we get two linearly independent solutions sin (μx) and cos (μx), so their linear combination gives us the general solution.
\[
y(x) = c_1 \cos (\mu x) + c_2 \sin (\mu x) .
\]
To satisfy the boundary conditions, we have
\[
y(0) = c_1 = 0 \qquad \mbox{and} \qquad y(\ell ) = c_2 \sin (\mu\ell ) =0 .
\]
in order for function y to be other than zero, we must choose μ as
\[
\mu = \sqrt{\lambda} = \frac{n\pi}{\ell} , \qquad n=1,2,3,\ldots .
\]
Therefore, we get the eigenvalues
\[
\lambda_n = \left( \frac{n\pi}{\ell} \right)^2 \qquad \Longrightarrow \qquad E_n = \frac{1}{2m} \left(
\frac{\hbar \pi n}{\ell} \right)^2 , \qquad n=1,2,3,\ldots ;
\]
and corresponding eigenfunctions:
\[
\psi_n (x) = \sin \left( \frac{n\pi x}{\ell} \right) , \qquad n=1,2,3,\ldots .
\]
■
Example 3: We consider a similar problem, but with Neumann boundary conditions:
\[
y'' + \lambda \,y =0 , \qquad y'(0) = 0, \quad y'(\ell ) =0 .
\]
It could be shown similarly to the previous example that negative λ are not possible. So we need to
consider only two cases: λ = 0 and λ = μ² > 0. The former gives us an eigenvalue
λ = 0 to which corresponds a constant as an eigenfunction. To the latter corresponds the general
solution
\[
y(x) = c_1 \cos (\mu x) + c_2 \sin (\mu x) \qquad \Longrightarrow \qquad y' (x) = \mu \left( -c_1 \sin
(\mu x) + c_2 \cos (\mu x) \right) .
\]
The boundary conditions dictate c2 = 0 and \( \sin (\mu \ell ) =0 . \)
Thus, we get eigenvalues and corresponding eigenfunctions
\[
\lambda_n = \left( \frac{n\pi}{\ell} \right)^2 , \qquad y_n (x) = \cos\left( \frac{n\pi x}{\ell} \right) ,
\qquad n=0,1,2,\ldots .
\]
■
Example 4: There are two known Sturm--Liouville problems with mixed boundary conditions when on one end we have the Dirichlet condition while on the other end we have the Neumann condition. So we start with one of them:
\[
y'' + \lambda \,y =0 , \qquad y(0) = 0, \quad y'(\ell ) =0 .
\]
The general solution of the equation above subject to the Dirichlet boundary condition at the left end
x = 0 is
\[
y(x) = c\,\sin \left( \sqrt{\lambda}\,x \right) ,
\]
with some constant c. It is clear from our previous discussion that eigenvalues must be positive in
this case. The Neumann condition at the right end dictates
\[
y' (\ell ) = c\,\sqrt{\lambda}\,\cos \left( \sqrt{\lambda}\,\ell \right) =0.
\]
When a product is zero then at least one multiple must be zero. Obviously, c cannot be zero,
otherwise we will have a trivial solution. Since λ > 0 according to assumption, we get the
condition
\[
\cos \left( \sqrt{\lambda}\,\ell \right) = 0 \qquad \Longrightarrow \qquad \sqrt{\lambda}\,\ell =
\frac{\pi}{2} + n\,\pi , \quad n=0,1,2,\ldots .
\]
Therefore, we get the eigenvalues and corresponding eigenfunctions
\[
\lambda_n = \left( \frac{\pi \left( 1 + 2n \right)}{2\ell} \right)^2 \qquad\mbox{and} \qquad y_n = \sin
\frac{x\pi \left( 1 + 2n \right)}{2\ell} , \qquad n=0,1,2,\ldots .
\]
Now we consider a similar Sturm--Liouville problem:
\[
y'' + \lambda \,y =0 , \qquad y'(0) = 0, \quad y(\ell ) =0 .
\]
The general solution of the above equation with Neumann boundary condition at left end x = 0 is
\[
y(x) = c\,\cos \left( \sqrt{\lambda}\,x \right) ,
\]
with some constant c. The Dirichlet boundary condition on the other end x = ℓ requires
\[
\cos \left( \sqrt{\lambda}\,\ell \right) = 0 \qquad \Longrightarrow \qquad \sqrt{\lambda}\,\ell =
\frac{\pi}{2} + n\,\pi , \quad n=0,1,2,\ldots .
\]
Therefore, we get the eigenvalues and corresponding eigenfunctions
\[
\lambda_n = \left( \frac{\pi \left( 1 + 2n \right)}{2\ell} \right)^2 \qquad\mbox{and} \qquad y_n = \cos
\frac{x\pi \left( 1 + 2n \right)}{2\ell} , \qquad n=0,1,2,\ldots .
\]
■
Example 5: Consider the Sturm--Liouville problem
\[
y'' + \lambda \,y =0 , \qquad y(0) = 0 , \quad y(1) +y'(1 ) =0 .
\tag{5.1}
\]
Assuming λ positive, the general solution of the given differential equation becomes
\[
y(x) = c_1 \cos \left( \sqrt{\lambda} x \right) + c_2 \sin \left( \sqrt{\lambda} x \right) .
\]
From the boundary condition y(0) = 0, it follows that c1 = 0. The boundary
condition at x = 1 is satisfied if
\[
c_2 \sin \left( \sqrt{\lambda} \right) + c_2 \sqrt{\lambda}\,\cos \left( \sqrt{\lambda} \right) =0.
\]
Choosing c2 ≠ 0, we see that the last equation is equivalent to
\[
\tan \sqrt{\lambda} = - \sqrt{\lambda} .
\tag{5.2}
\]
The transcendent equation (5.2) could be solved only by a numerical procedure such as Newton's method.
Nevertheless, if we let \( t = \sqrt{\lambda} , \) then we see from their graphs
that there exists an infinite number of roots of \( \tan t = -t . \)
Plot[{Tan[t], -t}, {t, 0, 12}, PlotStyle -> Thick]
\[
t_n \,\to \frac{\pi}{2} + n\pi \qquad \Longrightarrow \qquad \lambda_n = \left( \frac{\pi}{2} + n\pi
\right)^2 \qquad\mbox{as} \quad n \to \infty .
\]
The corresponding eigenfunctions (up to arbitrary multiple) are
\[
\phi_n (x) = \sin \left( \sqrt{\lambda_n} x\right) \to \sin \left( \frac{\pi \left( 1 + 2n \right) x}{2}
\right) \qquad\mbox{as} \quad n \to \infty .
\]
■
Three Spaces of Functions
To prepare the ground for the Sturm–Liouville theory we survey basic function spaces. To be more specific, we denote by 𝔏¹([𝑎, b], ρ) the set of (Lebesgue) integrable functions of the finite norm
\[
\| f(x) \|_1 = \int_a^b |f(x)|\,\rho (x)\,{\text d} x < \infty .
\]
To simplify the
notation, sometimes this space is also denoted as 𝔏([𝑎, b], ρ) by dropping "1".
So a sequence of functions { fn(x) } converges in mean (or 𝔏¹) to
f(x) iff
\[
\| f_n (x) - f(x) \|_1 = \int_a^b \left\vert f_n (x) - f(x) \right\vert {\text d} x \,\to 0 \qquad \mbox{
as} \quad n\to\infty .
\]
This space includes discontinuous functions.
Let f be a real- or complex-valued function defined on [𝑎, b] except, possibly, for
finitely many points. f is called piecewise continuous on [𝑎, b] if it has at
most
finitely many points of discontinuity, and if at any such point f admits left and right
limits (such a discontinuity is called a jump (or step) discontinuity).
Our next space, C([𝑎, b]), is also a Banach space; it consists of all real-valued
continuous functions on the closed interval [𝑎, b] with the norm
\[
\| f(x) \|_{\infty} = \max_{x\in [a, b]} \left\vert f(x) \right\vert .
\]
A sequence { fn(x) } is said to converge uniformly to f(x) iff
for any positive ε, there exists an integer N ∈ ℕ such that
\[
| f(x) - f_n (x)| < \varepsilon \qquad \mbox{ for all} \quad x \in [a,b] \quad\mbox{and} \quad n > N.
\]
|
We plot a neibohood of a function in uniform norm with Mathematica:
p1 = Plot[x^2, {x, -1, 1}, PlotStyle -> {Black, Thickness[0.01]}];
p2 = Plot[{(x^2) - 1, (x^2) + 1}, {x, -1, 1}, PlotStyle -> Gray, Filling -> {1 -> {2}} ]; p3 = Plot[(l^2 + Sqrt[Abs[l]]*Sin[10*l^2]), {l, -1, 1}, PlotStyle -> {Thick, Blue}]; Show[p1, p2, p3, PlotRange -> All] |
|
| Approximation in uniform norm. | Mathematica code |
Finally, we consider a set of all real- or complex-valued functions on interval [𝑎, b] for which the following inner product is finite
\[
\langle f(x) , g(x) \rangle = \langle f(x) \,|\, g(x) \rangle = \int_q^b \overline{f(x)} g(x)\,\rho
(x)\,{\text d} x = \int_q^b f(x)^{\ast} g(x)\,\rho (x)\,{\text d} x .
\]
There is no standard notation for inner product: in mathematics, it is common to separate functions f
and g by a comma, while in physics, they are usually separated by a vertical line. We utilize both
notations in hope to confuse the reader. Also, a complex conjugate \( \overline{f(x)} \)
of the function f(x) is usually denoted in mathematics by overline. Physicists prefer to
use asterisk for complex conjugate.
A complete vector space (if every Cauchy sequence of functions has a limit in 𝔏² norm) with inner product is known as a Hilbert space. The norm in this space is defined via its inner product
\[
\| f(x) \|_2 = \langle f(x) , f(x) \rangle^{1/2} = \left( \int_a^b \left\vert f(x) \right\vert^2 \rho (x)
\,{\text d} x \right)^{1/2} .
\]
This space is usually denoted by 𝔏²([𝑎, b], ρ).
Example 6: Let us consider two functions
\[
f(x) = x\,\cos x , \qquad\mbox{and} \qquad g(x) = x^2 \sin x , \qquad x \in [-\pi , \pi ].
\]
The uniform norm of f(x) is easy to determine because it attains maximum values at endpoints,
so
\[
\| f(x) \|_{\infty} = \max |x\,\cos x | = \pi .
\]
To determine the uniform norm of g(x), we use Mathematica:
FindMaximum[x^2 *Sin[x], {x, Pi}]
{3.9453, {x -> 2.28893}}
Therefore,
\[
\| g(x) \|_{\infty} = \max |x^2 \sin x | \approx 2.28893 .
\]
The mean norms of these functions are found with Mathematica:
Integrate[Abs[x*Cos[x]], {x, -Pi, Pi}]
2 \[Pi]
and
Integrate[Abs[x^2*Sin[x]], {x, -Pi, Pi}]
2 (-4 + \[Pi]^2)
\[
\| f(x) \|_{1} = \int_{-\pi}^{\pi} \left\vert x\,\cos x \right\vert {\text d} x = 2 \pi \approx 6.28319 ;
\]
\[
\| g(x) \|_{1} = \int_{-\pi}^{\pi} \left\vert x^2 \sin x \right\vert {\text d} x = 2 \left( \pi^2 -4
\right) \approx 11.7392.
\]
Their mean square norms are
Sqrt[Integrate[(x*Cos[x])^2, {x, -Pi, Pi}]
Sqrt[1/6 \[Pi] (3 + 2 \[Pi]^2)]
and
Sqrt[Integrate[(x^2*Sin[x])^2, {x, -Pi, Pi}]]
Sqrt[(3 \[Pi])/2 - \[Pi]^3 + \[Pi]^5/5]
\[
\| f(x) \|_{2} = \left( \int_{-\pi}^{\pi} \left( x\,\cos x \right)^2 {\text d} x \right)^{1/2} =
\sqrt{\frac{\pi}{6} \left( 3 + 2 \pi^2 \right)} \approx 3.45054 ;
\]
\[
\| g(x) \|_{2} = \left( \int_{-\pi}^{\pi} \left( x^2 \sin x \right)^2 {\text d} x \right)^{1/2} =
\sqrt{\frac{3\pi}{2} - \pi^3 + \frac{\pi^5}{5}} \approx 5.90847 ,
\]
Example 7: Let us start with unbounded interval [0, ∞) and set weight function to be 1. The function
\[
\chi_{[\alpha , \beta ]} (x) = \begin{cases}
1, & \ x \in [\alpha , \beta ], \\
0, & \ \mbox{ otherwise},
\end{cases}
\]
is called the characteristic function of the interval [α, β].
The sequence of functions \( \displaystyle \phi_n = \chi_{[n , n+1 ]} (x) , \quad n=1,2,3,\ldots , \) converges pointwise to zero on [0, ∞), but \( \displaystyle \| \phi_n \| = 1, \) so this sequence does not converge in norm to zero.
On the other hand, consider the interval [0, 1], and let {[𝑎n , bn]} be a sequence of intervals such that each x ∈ [0, 1] belongs, and also does not belong, to infinitely many intervals [𝑎n , bn], and such that bn − 𝑎n = 2 −k(n), where {k(n)} is a nondecreasing sequence satisfying \( \displaystyle \lim_{n\to\infty} k(n) = \infty . \) Since
\[
\| \chi_{[a_n , b_n} \|^2_2 = \int_0^1 \left[ \chi_{[a_n , b_n ]} (x) \right]^2 {\text d} x = 2^{-k(n)}
\to 0 ,
\]
it follows that the sequence { χ[𝑎n, bn] } tends to zero
function in the mean. However,
{ χ[𝑎n, bn] } does not converge at any point of [0, 1]
since for a fixed 0 ≤ x0 ≤ 1, the sequence { χ[𝑎n,
bn] } attains infinitely many times the value 0 and also infinitely many times the
value 1.
End of Example 7
Properties of Eigenvalues and Eigenfuncnctions
Many properties of solutions are based on the following identity.
Lagrange identity:
For any two twice continuously differentiable on a closed interval [𝑎, b] functions
u(x) and v(x), the following identity holds
\[
v\,L\left[ x, \texttt{D} \right] u - u\,L\left[ x, \texttt{D} \right] v = \frac{\text d}{{\text d}x}
\left[ p(x) \left( v\, \frac{{\text d}u}{{\text d}x} - u \,\frac{{\text d}v}{{\text d}x}\right) \right] ,
\]
where L is the self-adjoint differential operator \eqref{EqSturm.3}.
Theorem 1:
Properties of the regular Sturm--Liouville problem \eqref{EqSturm.1},
\eqref{EqSturm.2}.
Suppose that functions p(x), p'(x), q(x), ρ(x) are continuous on [𝑎, b], and also p(x) and ρ(x) are positive. Then the Sturm–Liouville problem \eqref{EqSturm.5} has the following properties.
Suppose that functions p(x), p'(x), q(x), ρ(x) are continuous on [𝑎, b], and also p(x) and ρ(x) are positive. Then the Sturm–Liouville problem \eqref{EqSturm.5} has the following properties.
- There exists an infinite number of real eigenvalues that can be arranged in increasing order λ1 < λ2 < λ3 < … λn < … such that λn →∞ as n→∞.
- For each eigenvalue there is only one eigenfunction (up until nonzero multiple).
- Eigenfunctions corresponding to different eigenvalues are linearly independent.
-
The set of eigenfunctions { ϕn } corresponding to the set of eigenvalues is
orthogonal with respect to the weight function ρ(x) on the interval x ∈ [𝑎,
b]:
\[ \left\langle \phi_n , \phi_k \right\rangle = \int_0^{\ell} \phi_n (x)\,\phi_k (x)\,\rho (x)\,{\text d}x = 0 \qquad \mbox{for} \quad n\ne k. \]
Let ϕ(x) and ψ(x) be two eigenfunctions corresponding to two distinct
eigenvalues λ and μ, respectively. Then using the Lagrange identity, we have
\begin{align*}
0 &= \left\langle L\left[ x, \texttt{D} \right] \phi , \psi \right\rangle -
\left\langle \phi , L\left[ x, \texttt{D} \right] \psi \right\rangle
\\
&= -\lambda \langle \rho\,\phi , \psi \rangle + \mu \langle \phi, \rho\,\psi \rangle
\\
&= \left( \mu - \lambda \right) \int_a^b {\text d} x\,\rho (x)\,\phi (x)^{\ast} \psi (x) .
\end{align*}
Since we have assumed that μ ≠ λ, it follows that
\[
\int_a^b {\text d} x\,\rho (x)\,\phi (x)^{\ast} \psi (x) = 0.
\]
Let ϕ(x) be an eigenfunction corresponding to eigenvalue λ, and &psi(x) be
an eigenfunction corresponding to eigenvalue μ, with μ ≠ λ. Then we have
\[
L \left[ x, \texttt{D} \right] \phi (x) = \lambda\,\rho (x)\,\phi (x) \qquad\mbox{and} \qquad L \left[ x,
\texttt{D} \right] \psi (x) = \lambda\,\rho (x)\,\psi (x) .
\]
Using Lagrange's identity, we get
\[
\int_a^b \left( \psi\,L \left[ x, \texttt{D} \right] \phi (x) \,\vert \,\phi\, L \left[ x, \texttt{D}
\right] \psi (x) \right) {\text d}x = \left[ p(x) \left( \psi\, \frac{{\text d}\phi}{{\text d}x} - \phi
\,\frac{{\text d}\psi}{{\text d}x}\right) \right]_{x=1}^{x=b} = 0.
\]
On the other hand, we have
\[
\int_a^b \left( \psi\,L \left[ x, \texttt{D} \right] \phi (x) \,\vert \,\phi\, L \left[ x, \texttt{D}
\right] \psi (x) \right) {\text d}x = \int_a^b \left( \psi\,\lambda\,\rho\phi - \phi\,\mu \,\rho\,\psi
\right) {\text d}x = \left( \lambda - \mu \right) \int_a^b \psi\,\phi\,\rho\,{\text d}x = \left( \lambda -
\mu \right) \left\langle \psi , \psi \right\rangle .
\]
Since λ ≠ μ, we obtain orthogonality of eigenfunctions ψ and φ.
Theorem 2:
If in addition to conditions of the previous statement, q(x) ≥ 0 and all coefficients
α0, α1, β0, β1 ≥ 0, then all
eigenvalues of Sturm--Liouville problem \eqref{EqSturm.5} are not negative.
When zero is an eigenvalue, we usually start labeling the eigenvalues at 0 rather than at 1 for convenience. That is we label the eigenvalues 0 = λ0 < λ1 < λ2 < ··· . .
Sturm’s Comparison Theorem:
For i = 1,2, let ui(x) be a nontrivial solution of the differential equation
\( \displaystyle \left( p_i(x)\, u'_i \right)' + q_i u_i = 0 \) on α ≤
x ≤ β. Suppose further that the coefficients are continuous and for x ∈ [α,
β]
Sturm’s proofs of course do not meet the standards of modern rigor. They meet
the standards of his time, and are in fact correct in method and can without too
much trouble be made rigorous. The first efforts to do this are due to Maxime Bôcher
in a series of papers in the Bulletin of the AMS and are also contained in
his book. Bôcher remarks that “the work of Sturm may, however, be
made perfectly rigorous without serious trouble and with no real modification of
method”. The conditions placed on the coefficients were to make them piecewise
continuous. Bôcher used Riccati equation
techniques in some of his proofs; it should be noted
that Sturm mentions the Riccati equation, but does not employ it in his proofs.
Riccati equation techniques in variational theory go back at least to Legendre who
in 1786 gave a flawed proof of his necessary condition for a minimizer of an integral
functional. A correct proof of Legendre’s condition using Riccati equations can be
found in Bolza’s 1904 lecture notes. Bolza attributes this proof to Weierstrass.
\( q_2 (x) \ge q_1 (x) , \quad \) with \( \quad q_2 (x_0 ) > q_1 (x_0 ) \quad \) for some x0, \( p_2 (x) \le p_1 (x) . \quad \)
Then if α, β are two consecutive zeros of u1(x), the open interval (α, β) will contain at least one zero of u2(x).- Bôcher. M., The theorems of oscillation of Sturm and Klein, Bull. Amer. Math. Soc. 4 (1897–1898), 295–313, 365–376.
- Bôcher, M., Leçons sur les méethodes de Sturm dans la théorie des équations différentielles linéaires, et leurs déeveloppements modernes, Gauthier-Villars, Paris, 1917.
- Bolza, O., Lectures on the Calculus of Variations, Dover, New York, 1961.
Sturm’s Separation Theorem:
If u1(x), u2(x) are two linearly independent solutions of
the differential equation \( \displaystyle \left( p(x)\, u' \right)' + q\, u_i = 0 \)
and α, β are two consecutive zeros of u1(x), then
u2(x) has a zero on the open interval (α, β).
Theorem 5: Fredholm alternative.
Suppose that we have a regular Sturm–Liouville problem \eqref{EqSturm.5}. Then either the homogeneous
boundary value problem
\[
\frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}y}{{\text d}x} \right) - q(x)\,y(x) +
\lambda\,\rho (x)\,y(x) = 0 ,
\]
\[
\alpha_0 y(a) - \alpha_1 y' (a) = 0 , \qquad \beta_0 y(b) + \beta_1 y' (b) = 0
\]
has a nonzero solution (when λ is an eigenvalue), or the nonhomogeneous boundary value problem
\[
\frac{\text d}{{\text d}x} \left( p(x)\, \frac{{\text d}y}{{\text d}x} \right) - q(x)\,y(x) +
\lambda\,\rho (x)\,y(x) = f(x) ,
\]
\[
\alpha_0 y(a) - \alpha_1 y' (a) = 0 , \qquad \beta_0 y(b) + \beta_1 y' (b) = 0
\]
has a unique solution for any continuous function f(x) on the interval [𝑎, b].
Eigenfunction Expansions
Now we are going to take advantage of considering the differential operator \eqref{EqSturm.3} in the Hilbert space 𝔏²([𝑎, b], ρ). First, using integration by parts, it is not hard to show that the linear operator \eqref{EqSturm.3} is self-adjoint:
\[
\left\langle L \left[ x, \texttt{D} \right] u \,\vert \,v(x) \right\rangle = \int_a^b \left( \frac{{\text
d}}{{\text d}x}\,p(x)\, \frac{{\text d}u}{{\text d}x} - q(x)\,u(x) \right) v(x)\,\rho (x)\,{\text d}x =
\left\langle u\,\vert L \left[ x, \texttt{D} \right] v \right\rangle .
\]
However, the most important is the ability to compute the eigenfunction decomposition (which is actually the
spectral decomposition according to the differential operator L) of a wide class of functions. That is,
for any function f(x) ∈ 𝔏²([𝑎, b], ρ) satisfying the
prescribed boundary conditions, we wish to synthesize it as
\begin{equation} \label{EqSturm.6}
f(x) = \sum_k c_k \phi_k (x) ,
\end{equation}
where ϕk(x) are eigenfunctions. We wish to find out whether we can represent a
function in this way, and if so, we wish to calculate ck
(and of course we would want to know if the sum converges).
Although this topic will be considered in detail in a dedicated section. we
outline the main ideas.
Assuming that series \eqref{EqSturm.6} converges, we multiply it by an eigenfunction ρ(x) ϕn(x) and integrate with respect to x from 𝑎 to b. This yields
\[
\langle f(x)\,|\,\phi_n (x) \rangle = \int_a^b \overline{f(x)} \phi_n (x)\,\rho (x)\, {\text d} x = \sum_k
c_k \int_a^b \overline{\phi_k (x)} \phi_n (x)\,\rho (x)\, {\text d} x = \sum_k c_k \langle \phi_k (x) ,
\phi_n (x) \rangle .
\]
Using the orthogonal property of eigenfunctions
\begin{equation} \label{EqSturm.7}
\langle \phi_k (x) , \phi_n (x) \rangle = \int_a^b \overline{\phi_k (x)} \phi_n (x)\,\rho (x)\, {\text d} x
= \begin{cases}
0, & \ \mbox{ if} \quad n \ne k , \\
\| \phi_n (x) \|^2_2 , & \ \mbox{ for} \quad n=k ;
\end{cases}
\end{equation}
we obtain
\[
\langle f(x)\,|\,\phi_n (x) \rangle = c_n \| \phi_n (x) \|_2^2 \qquad \Longleftrightarrow \qquad \int_a^b
f(x)^{\ast} \phi_n (x)\,\rho (x)\,{\text d}x = c_n \int_a^b \left\vert \phi_n (x) \right\vert^2 \rho (x)\,
{\text d} x .
\]
Therefore, we find
\begin{equation} \label{EqSturm.8}
c_n = \frac{\langle f(x)\,|\,\phi_n (x) \rangle}{\| \phi_n (x) \|_2^2} .
\end{equation}
and the Fourier series \eqref{EqSturm.6} becomes
\begin{equation} \label{EqSturm.9}
f(x) = \sum_k \frac{\langle f(x)\,|\,\phi_k (x) \rangle}{\| \phi_k (x) \|_2^2} \,\phi_k (x) .
\end{equation}
These Fourier coefficients \eqref{EqSturm.8} satisfy the so-called Bessel inequality:
\[
\sum_{k\ge 1} \left\vert c_k \right\vert^2 \le \left\langle f , f
\right\rangle .
\]
A set of orthogonal functions {
φk } in 𝔏² is called complete in the closed
interval [𝑎,b]
whenever the vanishing of inner products with all the orthogonal functions implies the member of
𝔏²([𝑎, b], ρ) is equal to zero almost everywhere in the domain.
The term complete was introduced by the famous Russian mathematician
Vladimir Steklov
(1864--1926). A set of functions { φk } in 𝔏² is complete if and only if the Parseval identity holds:
\begin{equation} \label{EqSturm.10}
\sum_{k\ge 1} \left\vert c_k \right\vert^2 = \left\langle f, f \right\rangle = \int_a^b \left\vert f(x)
\right\vert^2 {\text d}x = \| f(x) \|^2_2 .
\end{equation}
The identity above was stated by the famous French mathematician Marc-Antoine Parseval (1755--1836) in 1799.
The main reason to study Sturn--Liouville problems is that their eigenfunctions provide a basis for expansion for certain class of functions. This tells us that basis functions have close relationships with linear operators, and this basis is orthogonal for self-adjoint second order differential operators. In other words, solutions of differential equations can be approximated more efficiently by means of better basis functions. For example, a periodic function or solution is expressed more efficiently by periodic basis functions than by polynomials like the Taylor series.
For example, the linear operator \( L\left[ y \right] = y'' + \lambda \,y \) determines the basis functions. When λ < 0, the basis functions will be
\[
\left\{ t^m e^{-n \sqrt{|\lambda |}\, t} , \ t^m e^{n \sqrt{|\lambda |}\, t}
\right\}, \qquad m > 0, \quad n\ge 1.
\]
When λ = 0, it is represented by the set of basis functions
\[
\left\{ t^{2n} \ | \ n \ge 0\right\} .
\]
When λ > 0, the basis functions
will be
\[
\left\{ t^m \cos \left( n\sqrt{\lambda}\,x \right) , \ t^m \sin
\left( n\sqrt{\lambda}\,x \right)
\right\}, \qquad m ≥ 0, \quad n\ge 1.
\]
Therefore, we expect that the basis of periodic functions
\[
\left\{ \cos \left( n\pi\,x \right) , \ \sin
\left( n\pi\,x \right)
\right\}, \qquad n\ge 0
\]
is more suitable for approximation of periodic functions than Taylor's series.
Example 7: Let us find eigenvalues and eigenfunctions for the regular Sturm--Liouville problem
\[
\frac{\text d}{{\text d}x} \left( e^x \frac{{\text d}y}{{\text d}x} \right) + \lambda\,e^x y = 0 , \qquad
y(0) = y(1) = 0 .
\]
■
Example 8: ■
Spectral Decomposition
Decomposition of complicated systems into its constituent parts is one of science’s most powerful strategies for analysis and understanding. Large-scale systems with linearly coupled components. Spectral decomposition—splitting a linear operator into independent modes of simple behavior—is greatly appreciated in mathematical physics. For example, a wave dynamics is usually captured by superposition of simple modes. Quantum mechanics and statistical mechanics identify the energy eigenvalues of Hamiltonians as the basic objects in thermodynamics: transitions among the energy eigenstates yield heat and work. The eigenvalue spectrum reveals itself most directly in other kinds of spectra, such as the frequency spectra of light emitted by the gases that permeate the galactic filaments of our universe.
Spectral decomposition often allows a problem to be simplified by approximations that use only the dominant contributing modes. Indeed, human-face recognition can be efficiently accomplished using a small basis of “eigenfaces”. Certainly, there are many applications that highlight the importance of decomposition. In this section, we concentrate our attention on application of decomposition theory to differential equations generated by a linear second order self-adjoint operator subject to boundary conditions. Now it is known that depending on boundary conditions, the corresponding boundary value problem may have discrete spectum (set of eigenvalues) or continuous spectrum or their combination.
When solving an inhomogeneous differential equation
\[
L \left[ x, \texttt{D} \right] y = f ,
\]
with a differential operatoe L, we expand the input function and the unknown solution into the series
over eigenfunctions:
\[
f(x) = \sum_n f_n \phi_n (x) , \qquad y(x) = \sum_n c_n \phi_n (x) .
\]
Upon substituting these series expansions into the differential equation, we reduce this problem to simple
multiplication problem:
\[
L \left[ x, \texttt{D} \right] \sum_n c_n \phi_n (x) = \sum_n c_n L \left[ x, \texttt{D} \right] \phi_n (x)
= \sum_n c_n \lambda_n \,\phi_n (x) = \sum_n f_n \phi_n (x)
\]
because ϕn(x) are eigenfunctions. Assuming that all infinite series above
converge, we obtain
\[
c_n \lambda_n = f_n \qquad \Longrightarrow \qquad c_n = \frac{f_n}{\lambda_n} .
\]
- Li H. and Torney D., A complete system of orthogonal step functions, Proceedings of The American mathematical Society, Vol. 132, No 12, pp. 3491--3504.
- Lützen, J., Sturm and Liouville's Work on Ordinary Linear Differential Equations. The Emergence of Sturm-Liouville Theory, online.
- Titchmarsh, E.C., Eigenfunction expansions associated with second-order differential equations I, Clarendon Press, Oxford, 1962.
- Whittaker, E.T. and Watson, G.N., Modern analysis, Cambridge University Press, 1950
- Zettl, A., Computing continuous spectrum, in Trends and Developments in Ordinary Differential Equations, 393–406, Y. Alavi and P. Hsieh editors, World Scientific, 1994.
- Zettl, A., Sturm-Liouville problems, in Spectral Theory and Computational Methods of Sturm-Liouville problems, 1–104, Lecture Notes in Pure and Applied Mathematics 191, Marcel Dekker, Inc., New York, 1997.
Some utube references:
- Li H. and Torney D., A complete system of orthogonal step functions, Proceedings of The American mathematical Society, Vol. 132, No 12, pp. 3491--3504.
- Lützen, J., Sturm and Liouville's Work on Ordinary Linear Differential Equations. The Emergence of Sturm-Liouville Theory, online.
- Titchmarsh, E.C., Eigenfunction expansions associated with second-order differential equations I, Clarendon Press, Oxford, 1962.
- Whittaker, E.T. and Watson, G.N., Modern analysis, Cambridge University Press, 1950
- Zettl, A., Computing continuous spectrum, in Trends and Developments in Ordinary Differential Equations, 393–406, Y. Alavi and P. Hsieh editors, World Scientific, 1994.
- Zettl, A., Sturm-Liouville problems, in Spectral Theory and Computational Methods of Sturm-Liouville problems, 1–104, Lecture Notes in Pure and Applied Mathematics 191, Marcel Dekker, Inc., New York, 1997.
Some utube references: