Spectral theorem for normal operators is a classical result in functional analysis. There is, however, no satisfactory general spectral theory for non-normal operators. In my talk I will discuss a case study motivated by probabilistic applications: non-local operators with sufficiently regular, translation-invariant kernels (Lévy operators) in a half-line. These operators admit an explicit generalised eigenfunction expansion. However, the generalised eigenfunctions exhibit a rapid, exponential growth at infinity, which makes it necessary to work with a peculiar class of test functions instead of the usual space of compactly supported continuous functions. Complex-analytic methods play a key role in this development.