 Solution

Let $U$ be the quasi-local unitary carrying the ground space of $H_0$ to that of a topologically equivalent $H$. DefineUnitary conjugation preserves self-adjointness and the Majorana algebra. Quasi-locality spreads each endpoint operator only into an exponentially decaying tail, and the two operators preserve the ground space because $c_1,c_{2M}$ preserve the ground space of $H_0$.

Diagonalize the quadratic Hamiltonian asThe hint gives a real expansion $\gamma ={\sum }_j{g}_j{\chi }_j$. Any coefficient along a pair with ${ϵ}_j>0$ would create or remove a positive-energy quasiparticle and send some ground state outside the two-lowest-state subspace. Ground-space preservation therefore forces ${\gamma }_1$ and ${\gamma }_{2M}$ to have support only in the zero-energy Majorana subspace. Hencein the ideal infinite-chain limit, with only exponentially small finite-size corrections. They are the exponentially localized endpoint Majorana zero modes throughout the topological phase.