The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is to construct the matrices for each subdomain, letting the points on the boundaries be redundant for each matrix.
In the simplest one dimensional case with 2 subintervals, using the normal ordering, the last row of the left matrix and the first row of the right matrix will both describe the operator at the point shared between the two subintervals. We average the rows with some sort of weighting that maximizes the accuracy and add the two columns both corresponding to the redundant point.
The problem is, the only boundary conditions this specifies between the two subintervals is continuity. For an operator of order 2, for example, we would also need to specify the continuity of the first derivative to maintain uniqueness.
My question is how to specify the continuity of the first derivative in solving the eigenvalue problem.
My first idea was to add another row and a column of zeros, the row being the difference in the derivatives from the left and right at the boundary point. Then to prescribe that this difference is zero, I would solve the generalized eigenvalue problem with mass matrix identity except for the difference in derivatives, which is given weight zero.
Using the simplest numerics with a 2nd order centered difference and a first order backward (resp. forward) difference at the left (resp. right) of the boundary point, the eigenvalues I calculate reflect the eigenvalues of the subintervals considered separately.
If someone has any ideas or a good reference to some literature that isn't completely geared towards finite element methods (I'm working on a different method altogether, and identities involving the weak form are not helpful), that would be much appreciated.
A couple recent thoughts. Any linear scalar function on a vector space is a member of its dual, ie. is a row vector (using the general convention that vectors are columns, covectors are rows). For a given constraint, call this vector $u$. Define the operator $B = v u^T $, where v is one in the first index and zero elsewhere. If we wish the solve the eigenvalues of $A$ that are subject to this constraint, we need only find the simultaneous eigenvalues
$A x = \lambda x$
$(A - B) x = \lambda x$
The problem with this is that it does not "mimic appropriately" the continuous case I wish to approximate. In an underdetermined Sturm-Liouville problem, any real number is an eigenvalue, so it will simply be luck that the above system has a solution for finite dimensional matrices.
Another thought is, since I know the projection onto the null space is $I - v v^T$, perhaps there is some way to restrict my eigenvalue search for members of that subspace.