Majorana states near an impurity in the Kitayev infinite and semi-infinite model

For an infinite Kitaev chain with an impurity described by a deltalike potential, we analytically prove that two overlapping Majorana bound states in a topologically trivial phase in the case of a small superconducting gap exist under the condition $V_0=2\Delta$, where $V_0$ is the value of the potential and $\Delta$ is the superconducting order parameter. For a semi-infinite Kitaev chain with an impurity in the case of a small gap, we prove that there are two overlapping Majorana bound states in the trivial phase and one Majorana bound state in the topological phase and that the Majorana bound state in the latter case is stable under changes in the model parameters. We find explicit analytic expressions for the corresponding wave functions in all cases.