Inverse-Spectral-Theory

In the previous part we have constructed our two fundamental solutions $y_1$ and $y_2$ to the ODE $$ -y_j^{\prime\prime} + q(x)y_j = \lambda y_j, $$ with initial data $$ y_1(0) = y_2^\prime(0) = 1,\quad y_1^\prime(0) = y_2(0) = 0. $$ The goal of this section is to establish useful facts about the $y_j$, for we need to use them in the later sections to establish meaningful results. These facts come in the form of estimates, knowledge about the (partial) derivatives of the $y_j$, as well as analyticity properties. ...

A while ago, while I was doing some readings, I found a result that worked for ‘‘an open dense set of nonlinearities’’. Putting aside what this really means, I found out that to establish this fact, one had to effectively reconstruct a differential operator from its eigenvalues. I found it rather fascinating, and took a shallow dive into this topic. Unfortunately, a lot of these were not getting into my head, and so I volunteered to do a talk series in my university’s unofficial mathematics Discord chat. But to do so would require me to prepare notes, which I did, but they were not to my standard, and so I am redoing them, this time in an online format here, so that I can finally do the talks and finally learn inverse spectral theory. ...

The problem that will concern us for the next two parts will be the initial value problem $$\begin{cases} \tag{\#} -y^{\prime\prime} + q(x)y = \lambda y \text{ for $x\in[0,1]$,} \\ y(0) = a, y^{\prime}(0) = b. \end{cases}$$ Here $q(x)$ lies in $L^2_\mathbb{C}([0,1])$, which we will recall to be the space of complex-valued square-integrable functions on $[0,1]$. Also, $\lambda, a, b$ are complex numbers. In the future, we will only concern ourselves with real-valued $q$, but it is very convenient to work with complex-valued $q$ now, as it will let us avail ourselves of complex-analytic tools. ...