Periodic Functions

1. The problem statement, all variables and given/known data
What kind of conditions do eigenvalues impose to ensure periodicity?
Is it plausible to say that irrational multiples of eigenvalues imply no harmonic oscillations, if so why?

2. Relevant equations



3. The attempt at a solution

 

[tex]

\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -b & 0 \\ 0 & -b & 0 & -c \\ 0 & 0 & -c & 0\end{array}\right]

[/tex]

For any fixed a, I want to find b and c in terms of a such that lambda_i / lambda_ j is a rational number for every i,j=1,2,3,4 .

See, before I was working with

[tex]


\left[\begin{array}{cccc}0 & -a & 0 & 0 \\ -a & 0 & -a & 0 \\ 0 & -a & 0 & -a \\ 0 & 0 & -a & 0\end{array}\right]


[/tex]

But that gave me irrational e-values if I put it into Maple and I don't want that.

 

c and b must be in terms of a.

But forget that for now, how would I show that the eigenvalues are irrational for n>3 ? That's why I asked this guy in this thread https://www.physicsforums.com/showthread.php?t=224954&page=3

how to find an equation for the eigenvalues.