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Let $\tilde{\mathcal H}$ be a Hilbert space, and let $L(\tilde{\mathcal H})$ denote the corresponding space of linear operators. By fixing a basis, we can, via Fourier transform, identify an important sub algebra by taking only those operators which commute with shifting the index of a basis vector by an arbitrary amount: $$\mathcal A\simeq C(S^1)\subset L(\tilde{\mathcal H})$$ If $\tilde{\mathcal H}$ is the configuration space of a bulk, one-dimensional quantum system, then this is the algebra of translationally-invariant bulk observables. When I say Toeplitz extension, I mean as given in the sense of Prodan and Schulz-Baldes's Bulk and Boundary Invariants for Complex Topological Insulators:

Definition (Toeplitz extension) The Toeplitz extension $T(\mathcal A)$ is the following: letting $\mathcal H\subset \tilde{\mathcal H}$ denote the subspace spanned by all basis vectors with non-negative index (i.e. representing the configuration space of the same exact quantum system, but now with boundary), then, letting $K(\mathcal H)$ denote the algebra of compact observables, we can express the Toeplitz extension as an extension of $\mathcal A$ by this compact algebra, that is, we can define the Toeplitz extension by the following short-exact sequence: $$0\to K(\mathcal H)\to T(\mathcal A)\xrightarrow{} \mathcal A\to 0$$ Where the first algebra homomorphism is just a set inclusion, and the second algebra homomorphism is uniquely specified by sending the shift operator to its bulk counterpart.

As a condensed matter physicist, I am interested in the following problem: given an infinitely-extended one-dimensional quantum system, completely homogenous so that its Hamiltonian lies in the sub algebra $\mathcal A$ (and thus is easily exactly soluble), and if I chop it into halves, what can I rigorously say about the spectral data of the remaining pieces, especially the form of the eigenvectors, using the maps of the Toeplitz extension?

Put more precisely, suppose I start with a self-adjoint operator $H\in T(\mathcal A)$ representing my quantum system with boundary. If I know the eigenvectors of the infinitely-extended version of my system, (i.e. I have diagonalized the image $j(H)$ under the second Toeplitz map $j$), then does the theory around the Toeplitz extension tell me anything immediate/insightful about the eigenvectors of $H$ itself, which

  • takes advantage of the fact that the algebra which is being extended is $C(S^1)$, and
  • saves one from imposing boundary conditions on the eigenvectors of $j(H)$, or, alternatively, says something rigorous about why this process works?
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