Estimate for change in eigenvalues of truncated half line operator

Question

Let $(t_n)_{n=1}^\infty\subset (0,\infty)$ be an infinite, discrete set of points. I use those to define a Sturm Liouville operator $H$ on the half line $[0,\infty)$. It acts on $H^2(0,\infty)$ as $$Hf=-\frac{d^2}{dx^2}f$$ with the Neumann condition $f'=0$ at $x=0$, and I also impose certain boundary conditions at all points $t_n$ to make it self-adjoint. Here's an example just to make things concrete: $$f(t_n^+)=-f(t_n^-), f'(t_n^+)=\pi f'(t_n^-)$$ I now choose some point $t_n$ and define a "truncated" operator on $H^2(t_n,\infty)$ using the same differential expression and boundary conditions at $t_{n+1},t_{n+2}...$ etc, and at $t_n$ itself I impose the boundary condition $f'(t_n)=0$. Denote the new operator by $T$.

I want to try and compare the spectra of $H$ and $T$. I can show that in terms of essential spectrum, $T$ and $H$ should be exactly the same. But it is of course possible for $H$ and $T$ to have some different eigenvalues (which you cannot see through the essential spectrum). I am interest in estimating how different these eigenvalues might be. I don't have a concrete goal in mind, but for instance a good bound on the maximal difference between eigenvalues of $H$ and $T$ (which probably depends on $t_n$ and the boundary conditions etc.) would be nice.

Is anyone familiar with a work where something like this is done? Here are some thoughts I have about this:

1. Since $H$ and $T$ have drastically different domains, we cannot try to estimate the norm of their difference, and in fact not even the norm of their difference of resolvents (at least not in a way which is immediately obvious to me).
2. I have a "feeling" that $H$ and $T$ should have the same (finite!) number of eigenvalues, which might make these estimates simpler. The (very not rigorous) intuition behind this is that we can think of both $H$ and $T$ as truncation of some larger operator on the full real line, and due to bulk boundary correspondence, the number of eigenvalues (edge modes) which appear due to this truncation is equal to the Chern number.
3. Another (again, not rigorous) feeling I have is that it might be useful to consider the quadratic forms which define the operators $H$ and $T$. While $H$ and $T$ are "incomparable" in the usual ways, their associated quadratic forms are very similar to one another. If the truncation boundary condition at $t_n$ would be Dirichlet ($f=0$), then you could even in some sense say that the domain of the quadratic form for $T$ is contained in that of $H$ (and you could possibly use this for some spectral estimates). Even though that's not the case, it still feels like something can be done in this direction.

Anyway, any suggestions and references for things like this would be highly appreciated.

Thanks in advance!