Eigenvalues and eigenvectors of the second derivative

Explicit formulas for eigenvalues and eigenvectors of the second derivative with different boundary conditions are provided both for the continuous and discrete cases. In the discrete case, the standard central difference approximation of the second derivative is used on a uniform grid.
These formulas are used to derive the expressions for eigenfunctions of Laplacian in case of separation of variables, as well as to find eigenvalues and eigenvectors of multidimensional discrete Laplacian on a regular grid, which is presented as a Kronecker sum of discrete Laplacians in one-dimension.
The continuous case.
The index j represents the jth eigenvalue or eigenvector and runs from 1 to formula_1. Assuming the equation is defined on the domain formula_2, the following are the eigenvalues and normalized eigenvectors. The eigenvalues are ordered in descending order.
Periodic boundary conditions.
(That is: formula_8 is a simple eigenvalue and all further eigenvalues are given by formula_9, formula_10, each with multiplicity 2).
The discrete case.
Notation: The index j represents the jth eigenvalue or eigenvector. The index i represents the ith component of an eigenvector. Both i and j go from 1 to n, where the matrix is size n x n. Eigenvectors are normalized. The eigenvalues are ordered in descending order.
Derivation of Eigenvalues and Eigenvectors in the Discrete Case.
Dirichlet case.
In the 1D discrete case with Dirichlet boundary conditions, we are solving
Rearranging terms, we get
Now let formula_28. Also, assuming formula_29, we can scale eigenvectors by any nonzero scalar, so scale formula_30 so that formula_31.
Then we find the recurrence
Considering formula_35 as an indeterminate,
where formula_37 is the kth Chebyshev polynomial of the 2nd kind.
Since formula_38, we get that
It is clear that the eigenvalues of our problem will be the zeros of the nth Chebyshev polynomial of the second kind, with the relation formula_28.
These zeros are well known and are:
Plugging these into the formula for formula_42,
And using a trig formula to simplify, we find
Neumann case.
In the Neumann case, we are solving
In the standard discretization, we introduce formula_47 and formula_48 and define
The boundary conditions are then equivalent to
If we make a change of variables,
we can derive the following:
with formula_53 being the boundary conditions.
This is precisely the Dirichlet formula with formula_54 interior grid points and grid spacing formula_55. Similar to what we saw in the above, assuming formula_56, we get
This gives us formula_54 eigenvalues and there are formula_59. If we drop the assumption that formula_56, we find there is also a solution with formula_61 and this corresponds to eigenvalue formula_8.
Relabeling the indices in the formula above and combining with the zero eigenvalue, we obtain,
Dirichlet-Neumann Case.
For the Dirichlet-Neumann case, we are solving
where formula_65
We need to introduce auxiliary variables formula_66
Consider the recurrence
Also, we know formula_68 and assuming formula_69, we can scale formula_70 so that formula_71
We can also write
Taking the correct combination of these three equations, we can obtain
And thus our new recurrence will solve our eigenvalue problem when
Solving for formula_42 we get
Our new recurrence gives
where formula_79 again is the kth Chebyshev polynomial of the 2nd kind.
And combining with our Neumann boundary condition, we have
A well-known formula relates the Chebyshev polynomials of the first kind, formula_81, to those of the second kind by
Thus our eigenvalues solve
The zeros of this polynomial are also known to be
And thus
Note that there are 2n + 1 of these values, but only the first n + 1 are unique. The (n + 1)th value gives us the zero vector as an eigenvector with eigenvalue 0, which is trivial. This can be seen by returning to the original recurrence. So we consider only the first n of these values to be the n eigenvalues of the Dirichlet - Neumann problem.