The bulk-boundary correspondence, which in topological insulators describes the relationship between the bulk invariant computed for a system with periodic boundary conditions and the number of boundary states in the corresponding system with open boundary conditions, is well-known and important for predicting the behavior of these systems. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena, but in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions now differs fundamentally from the spectrum of the system with open boundary conditions. It is thus no longer possible to use the Bloch Hamiltonian to predict the appearance of boundary states.

Integral to understanding the behavior of these systems, is to understand how the boundary states behave. This is what is studied in the accompanying papers, Biorthogonal bulk-boundary correspondence in non-Hermitian systems, Non-Hermitian extensions of higher-order topological phases and their biorthogonal bulk-boundary correspondence, and Phase transitions and generalized biorthogonal trace polarization in non-Hermitian systems, of this thesis, where also a new kind of biorthogonal bulk-boundary correspondence is developed.

The aim of this licentiate thesis is to give the background necessary to understand the accompanying papers. It is divided into two parts. The first part describes the well-established theory of boundary states in a certain class of Hermitian systems for which there exist exact solutions that are straightforward to analyze, which then are generalized to the non-Hermitian case in the accompanying papers. The second part gives some background to non-Hermitian systems, the unusual phenomena that occur in them, and an introduction to biorthogonal quantum mechanics and why it is necessary to redefine the inner product one uses when calculating quantum mechanical probabilities.