## On Eigen(patch(ix))values - (RWLA,MCT,GT,AM Part VIII)

So in the continuation of this series, I have been thinking long and hard about the curious property of the existence of eigen(patch(ix))values that I have talked about in a previous post. I began to question whether such eigen(patch(ix))values are limited to a finite set (much as in finite matrices) or whether there was some other fundamental insight, like, if 1 is an eigen(patch(ix))value, then all elements of are too (or all of minus a finite set). In my latest attempt to understand this, the question comes down to, using the "star" operator, whether

has discrete values of or, "what values can lambda take for the equation to be true," in direct analogy with eigenvalues when we're dealing with discrete matrices. I am not using yet "integral transform notation" because this development seemed more intuitive to me, and thus I'm also limiting the treatment to "surfaces" that are smooth and defined on , like I first thought of them. Thus, the above equation translates to:

and, if we recall our construction of the patch (or patchix if we relax the assumption that integrating with respect to x is 1) :

where are constants. It is very tempting to divide as

must hold provided . So we have excluded an eigen(patch(ix))value right from the start, which is interesting.

We can systematically write the derivatives of , as we're going to need them if we follow the algorithm I delineated in one of my previous posts (NB: we assume a finite number of derivatives or periodic ones, or infinite derivatives such that the subsequent sums we'll write are convergent):

provided, as before, . We want to calculate the constants , to see if they are restricted in some way by a formula, and we do this by integrating by parts as we did in a previous post to obtain the cool "pasquali series." Thus, we have that if , the tabular method gives:

and so,

if we remember the alternating sign of the multiplications, and we are allowed some leeway in notation. Ultimately, this last bit means: .

Since we have already explicitly written the derivatives of , the derivatives can be written as and respectively.

We have then:

Since we aim to solve for , multiplying by makes things easier, and also we must rearrange all elements with in them, so we get:

Subtracting both sides the common term and factoring the constant we endeavor to solve for, we get:

or

A similar argument for suggests

where the new constants introduced emphasizes the expectation that the sums converge. Plugging in the one into the other we get:

and now we seem to have additional restrictions on lambda: and . Furthermore, the constant drops out of the equation, suggesting these constants can be anything we can imagine (all of without restriction), but then we have the constraint:

which is extraordinarily similar to its analogue in finite matrix or linear algebra contexts. Expanding suggests:

which we can solve by the quadratic equation of course, as:

So not only is not equal to a few values, it is incredibly restricted to two of them.

So here's a sort of conjecture, and a plan for the proof. The allowable values of is equal to the number of x terms (or ) carries. We have already shown the base case, we need only show the induction step, that it works for and terms.