First, a quick remark: I'm a mathematician, now working on some problems coming from physics (in particular Ising models on quasiperiodic chains). A few things I find rather mysterious. I would appreciate your help.
For the purpose of generality, let's consider the following Ising model on a chain of $N$ nodes.
$$H_N = - \sum_{i = 1}^N J_i\sigma_i^{(x)}\sigma_{i+1}^{(x)} - \sum_{i = 1}^N\sigma_i^{(z)},$$
with $J_i$ depending on the node $i$ (we assume no particular order for generality), and $\sigma_i^{(x),(z)}$ the Pauli matrices. By Jordan-Wigner, we can consider the corresponding Fermionic operator given by
$$\widehat{H}_N = \sum_{i,j}\left[c_i^{\dagger}A_{ij}c_j + \frac{1}{2}\left(c_i^{\dagger}B_{ij}c_j^{\dagger}+ H.c.\right)\right],$$
where $c_i$, $1\leq i \leq N$ are anticommuting Fermionic operators and $\left\{A_{ij}\right\},\left\{B_{ij}\right\}$, $1\leq i, j \leq N$ are the elements of appropriately chosen matrices $A, B$, which depend on $\left\{J_i\right\}_{1\leq i \leq N}$.
We use periodic boundary conditions. Now we can extend $\widehat{H}_N$ to a lattice of infinite size, by gluing the unit cell of size $N$ infinitely many times. Let us call this new extension $\tilde{H}_N$. Now the questions:
1) What is $H.c.$?
2) I am interested in the thermodynamic limit $N\rightarrow\infty$. Is it obvious whether the sequence of operators $\left\{\tilde{H}_N\right\}$ converges, say in strong operator topology, to some well-defined operator $\tilde{H}$ as $N\rightarrow\infty$?
Let me motivate the second question: For a certain sequence $\left\{J_i\right\}$, constructed deterministically with certain properties (so-called quasi-periodic sequence), I believe I can say something about what Physicists call the "energy-spectrum in the thermodynamic limit". I'm interested to know whether this energy spectrum is the spectrum (in the usual functional-analytic sense) of some operator $\tilde{H}$.
Thanks for any help!
