Bimodality of eigenvectors Consider an eigenvalue / eigenvector problem for a matrix $A$:
$$\sum_j A_{ij} x_j = \lambda x_i$$
Under what conditions is $x_i$ unimodal in $i$? That is, up to a multiplicative constant, is $x_{i+1} \ge x_i$ for the first values of $i$, and then $x_{i+1} \le x_i$ for later values? Or is $x_i$ bimodal? How many peaks does it have?
Obviously I don't expect answers to such broad questions. This is "reference-request" question, or a terminology question. I need to know if this type of problem has been considered, perhaps for very specific matrices, and what are the relevant keywords (if there are any) to search in google scholar. Are there results in the literature related to this?
 A: I suspect, as you do, that this question only has a nice or simple answer for very particular cases.  Here are the cases that I can think of.
Wigner studied a class of random matrices and showed that the eigenvalues follow a particulr distribution.  It looks like the corresponding eigenvalue distribution has also been studied in [1] below, though I haven't read the paper.  This is a very restricted case and hence I expect it to have very rigorous results.
There is only one case where I have personally read any papers that address this issue, and even then it is addressed only obliquely.  The case in question is the Frobenius-Perron eigenvector of non-negative matrices.  The results are only fully trustworthy to a physicist when the matrix satisfies some minimal amount of randomness or well-connectedness, and to a mathematician they would lack sufficient rigor.
It has been a long time since I've read any of these papers, but the basic idea is that you can consider $A^n \sim \lambda^n x v^T$ (where $\lambda$, $x$, and $v$ are the Frobenius-Perron eigenvalue and right and left Frobenius-Perron eigenvectors, respectively), then relate various properties of $A$ to $\lambda$, $x$, and $v$.
The results are mainly of interest to the complex networks community, because complex networks can be represented by an adjacency matrix whose spectrum predicts important properties of dynamics occurring on the network.  [2] and [3] below are a couple of relevant papers of this type, and the references may give more.
There is an associated field of finding community structure in networks by looking at the spectrum of a matrix.  My recollection is that this is more general but may be slightly less rigorous than the last paragraph.  It is more general in the sense that the results will apply There are quite a few papers on this subject, since community structure was a hot topic for most of the last 15 years.  [4] is an example of such a paper, and the same author has written quite a lot more on the subject.
None of this comes anywhere close to answering your general question, but this post is too long for a comment!
[1] https://arxiv.org/abs/1102.0057
[2] https://arxiv.org/abs/0705.4503
[3] https://arxiv.org/abs/0902.1465
[4] https://arxiv.org/abs/physics/0605087
