Sturm-Liouville theory extends
the concept of Hermitian matrix eigenvalue problems
to linear second-order ordinary differential equations.
It states that, given suitable boundary conditions,
any such equation can be rewritten using the Sturm-Liouville operator,
and that the corresponding eigenvalue problem,
known as a Sturm-Liouville problem,
will give real eigenvalues and a complete set of eigenfunctions.
Consider the most general form of a second-order linear
differential operator , where , , and
are real functions of and are nonzero for all :
Analogously to matrices,
we now define its adjoint operator as follows:
What is , given the above definition of ?
We start from the inner product :
The newly-formed operator on must be ,
but there is an additional boundary term.
To fix this, we demand that
and that , leaving:
Let us look at the expression for we just found,
with the restriction in mind:
So is self-adjoint, i.e. is the same as !
Indeed, every such second-order linear operator is self-adjoint
if it satisfies the constraints and .
But what if ?
Let us multiply by an unknown
and divide by :
We now demand that the derivative of the unknown satisfies:
Taking the indefinite integral of this differential equation
yields an expression for :
We define an additional function
based on the last term of shown above:
When rewritten using and ,
the modified operator looks like this:
This is the self-adjoint form from earlier!
So even if , any second-order linear operator
with can easily be made self-adjoint.
The resulting general form is called the Sturm-Liouville operator ,
for nonzero :
Still subject to the constraint
such that .
An eigenvalue problem of
is called a Sturm-Liouville problem (SLP).
The goal is to find the eigenvalues
and corresponding eigenfunctions that fulfill:
Where is a real weight function satisfying for .
By convention, the trivial solution is not valid.
Some authors have the opposite sign for and/or .
In our derivation of above,
we imposed the constraint to ensure that
Consequently, to have a valid SLP,
the boundary conditions (BCs) on must be such that,
for any two (possibly identical) eigenfunctions and , we have:
There are many boundary conditions that satisfy this requirement.
Some notable ones are listed non-exhaustively below.
Verify for yourself that these work:
- Dirichlet BCs:
- Neumann BCs:
- Robin BCs: with
- Periodic BCs: , , and
- Legendre “BCs”:
If this is fulfilled, Sturm-Liouville theory gives us
useful information about and .
By definition, the following must be satisfied
for two arbitrary eigenfunctions and :
We multiply each by the other eigenfunction,
subtract the results, and integrate:
The operator is self-adjoint of course,
so the first two terms vanish, leaving us with:
When , we get ,
so the equation is only satisfied if ,
meaning the eigenvalue is real for any .
When , then
may or may not be zero depending on the degeneracy.
If there is no degeneracy, then ,
meaning , i.e. the eigenfunctions are orthogonal.
In case of degeneracy, manual orthogonalization is needed,
which is guaranteed to be doable using the Gram-Schmidt method.
In conclusion, an SLP has real eigenvalues
and orthogonal eigenfunctions: for all , :
When solving a differential eigenvalue problem,
knowing that all eigenvalues are real is a huge simplification,
so it is always worth checking whether you are dealing with an SLP.
Another useful fact:
it turns out that SLPs always have an infinite number of discrete eigenvalues.
Furthermore, there always exists a lowest eigenvalue ,
called the ground state.
Not only are an SLP’s eigenfunctions orthogonal,
they also form a complete basis, meaning any well-behaved
can be expanded as a generalized Fourier series with coefficients :
This series converges faster if satisfies the same BCs as ;
in that case the expansion is also valid for the inclusive interval .
To find an expression for the coefficients ,
we multiply the above generalized Fourier series by
and integrate it to get inner products on both sides:
Because the eigenfunctions of an SLP are mutually orthogonal,
the summation disappears:
After isolating this for , we see that
the coefficients are given by the projection of the target
function onto the normalized eigenfunctions :
As a final remark, we can see something interesting
by rearranging the generalized Fourier series
after inserting the expression for :
Upon closer inspection, the parenthesized summation
must be the Dirac delta function
for the integral to work out.
In fact, this is the underlying requirement for completeness:
- O. Bang,
Applied mathematics for physicists: lecture notes, 2019,