A tridiagonal matrix is a matrix which only has elements on three diagonals. So for $\alpha, \beta, \gamma \in \mathbb{C}$ consider the bi-infinite tridiagonal Laurent operator $T$ with $\beta $ on the diagonal given by.
\begin{pmatrix} \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \dots & \alpha & \beta & \gamma & 0 & 0 & 0 & \dots \\ \dots & 0 & \alpha & \beta & \gamma & 0 & 0 & \dots\\ \dots & 0 & 0 & \alpha & \beta & \gamma & 0 & \dots\\ \dots & 0 & 0 & 0 & \alpha & \beta & \gamma & \dots \\ \dots & \dots & \dots & \dots & \dots & \dots & \dots & \dots \\ \end{pmatrix}
General theory tells us that $T$ is invertible if and only the symbol curve (which is in this case is an ellipsis) given by $ \{ z \in \mathbb{T} \mid \frac{\alpha}{z} + \beta + z \gamma \} $ does not enclose $0$.
In finite dimensions, we can consider the periodic version $T_n$. For example $T_5$ looks as follows:
\begin{pmatrix} \beta & \gamma & 0 & 0 & \alpha \\ \alpha & \beta & \gamma & 0 & 0 \\ 0 & \alpha & \beta & \gamma & 0 \\ 0 & 0 & \alpha & \beta & \gamma \\ \gamma & 0 & 0 & \alpha & \beta \\ \end{pmatrix}
Now, when $n$ tends to $\infty$ then the spectra of the periodic versions $\sigma(T_n)$ approximate the spectrum of the Laurent operator $T$. In the sense that in Haussdorff distance it holds that
$$ \sigma(T_n) \to \sigma(T). $$ This can for example be seen by Fourier Transform.
Now, suppose that I add finitely many diagonal terms. I.e. the matrix $K = \text{diag}(K_{1,1}, K_{2,2} , \dots, K_{m,m})$.
Then it is a Theorem (see https://link.springer.com/article/10.1007/BF01275512, Corollary 1.3) that $$ lim_{n \to \infty} \left( \sigma(T_n + P_n K P_n) \cup \sigma(T) \right) \to \sigma(T+K). $$ Here $P_n$ is the projection onto the $n$ sites of $T_n$.
But in general it is conjectured (https://link.springer.com/article/10.1007/BF01275512, Conjecture 7.3) that $$ lim_{n \to \infty} \left( \sigma(T_n + P_n K P_n) \right) \to \sigma(T+K). $$ Meaning that the finite system can also approximate the symbol curve as was the case without the pertubation. What is the status of this conjecture? And in the case $m=1$ where the pertubation is only a single entry, how could one go about proving it?
The reason I am interested is given on page 18 in here https://arxiv.org/pdf/2206.09879.pdf, where one can see that these question can help answer to what extend the spectra of some infinite volume open quantum systems are approximated by their finite volume periodic counterparts.
