Solving the following problem would help better understand a stabilization scheme for a quantum computer, originally developed at Yale and currently pursued by Amazon. Consider an infinite tridiagonal matrix $A$, with diagonal entries (indexed by integer $q$) $-2q(2q+1)$ and off diagonal entries $\pm (2q+1) \alpha^2$, with $\alpha>0$ a real parameter. Around $q=0$, the matrix looks like
$$ A = \left(\begin{array}{ccccccccc} \ddots & \ddots\\ \ddots & -30 & 5\alpha^{2} & 0 & 0 & 0 & 0 & 0\\ & -3\alpha^{2} & -12 & 3\alpha^{2} & 0 & 0 & 0 & 0\\ & 0 & -\alpha^{2} & -2 & \alpha^{2} & 0 & 0 & 0\\ & 0 & 0 & \alpha^{2} & 0 & -\alpha^{2} & 0 & 0\\ & 0 & 0 & 0 & 3\alpha^{2} & -6 & -3\alpha^{2} & 0\\ & 0 & 0 & 0 & 0 & 5\alpha^{2} & -20 & -5\alpha^{2}\\ & 0 & 0 & 0 & 0 & 0 & 7\alpha^{2} & -42 & \ddots\\ & & & & & & & \ddots & \ddots \end{array}\right). $$
I'd like to find its eigenvalues and eigenvectors for nonzero $\alpha$. Solving $Af = \lambda f$ with function $f$ and eigenvalue $\lambda$ yielding the following recursion relation for the coefficients $c_q/(2q+1)$ of $f$: $$ \alpha^{2}c_{q+1}-\alpha^{2}c_{q-1}+\left(\frac{\lambda}{2q+1}-2q\right)c_{q}=0 $$ From https://arxiv.org/abs/1312.2017, the (unique) zero-eigenvalue eigenvector $f_0$ with entries $c_q=(-1)^q I_q(\alpha^2)$, where $I_q$ is the modified Bessel function of the first kind. The above recurrence relation at $\lambda=0$ simplifies to a recursion property of the Bessel functions (DLMF, Eq. 10.6.1), and one can show that
$$ Af_0 = 0. $$
I do not know the nonzero eigenvalues or their eigenvectors. Continued fractions can be used to find the eigenvalues, but I'm looking for a deeper understanding of both eigenvalues and vectors. Here are some paths forward:
One can use a Fourier trick to recast the matrix as a differential operator acting on the space of functions $f$ of angle $\phi \in [0,2\pi)$, i.e., $f(\phi+2\pi)=f(\phi)$: $$ \partial_{\phi}^{2}+\left(\alpha^{2}\sin\phi-i\frac{1}{2}\right)\partial_{\phi}-\frac{1}{2}i\alpha^{2}\sin\phi. $$ Via a change of variables, this is a Whitaker-Hill equation that has complex coefficients and is damped, $$ \partial_{\phi}^{2}-\frac{1}{2}i\partial_{\phi}-\frac{\alpha^{4}}{4}\sin^{2}\phi+\frac{\alpha^{2}}{2}\cos\phi-i\frac{\alpha^{2}}{4}\sin\phi. $$ I haven't found this particular combination of features in the literature.
Numerically, the transpose of $A$ has the same eigenvalues. Recasting that as a differential equation yields the operator $$ \partial_{\phi}^{2}+\left(\alpha^{2}\sin\phi-i\frac{1}{2}\right)\partial_{\phi} .$$ This doesn't have the same eigenfunctions, but its simpler form yields some hope for the eigenvalues. However, recall that this is acting on the space of functions $f(\phi+2\pi)=f(\phi)$, so not all changes of variables are allowed.