- Operator - a series of "operations" that act upon a function, sort of like a set of instructions. This can be anything from adding and subtracting certain parts of the function to taking the derivative, or integral. This maps a function from one vector space to another, and is not necessarily one to one.
- Hermitian Operator - an operator that is self adjoint, which means that if you put the operator into matrix form to act upon an equation, that matrix is the same when you take the conjugate transpose of it.
- Conjugate Transpose - Taking the transpose of a matrix, which means to switch the indices from (1,2) to (2,1) for the entire matrix. Then conjugate any terms that have imaginary components, which means switch the signs of the imaginary terms.

Sturm Liouville Form

Given a second order differential operator where D

^{2}, and D represent partial derivatives acting on some variable
We will first have this operator act on the following equation where the operator takes the place of the second order derivative acting on y where lambda is an eigenvalue, explained in a separate section.

giving us the following

We can then rearrange this by dividing through by a

_{2}(x)
Now we multiply through by e to the integral of a

_{1}over a_{2}and replace the operator with just y'' and y' for simplicity
Now we are going to once again regroup this into

by taking advantage of the product rule for taking derivatives, which makes the above two parts equivalent. Now we are going to define the parts of this equation by functions r, q, and p.

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