The overall aim of this dissertation is to investigate some problems in spectral anal-
ysis for self-adjoint and non self-adjoint operators which arise in different contexts of
physics.
In the first part of this thesis we study the problem of localisation of complex eigen-
values of non Hermitian perturbations of self-adjoint operators realised by means of
complex potentials. In particular, we focus our attention on two different operators.
The first one is the Laplacian defined on the real half line. The other is a second order
two dimensional operator which arises in the context of the physics of materials, in
particular from the study of the Hamiltonian of a double layer graphene. For both we
provide Keller-type estimates on the localisation of complex eigenvalues.
The existence of trapped waves solutions for a set of equations describing the dynamics
of a stratified two layers fluid of different densities, confined in a ocean channel of fixed
width and varying depth and subject to rotation is studied in the second chapter. The
existence of these solutions is then recovered by proving the existence of points in the
point spectrum of a two dimensional operator pencil. We prove that, under some
smallness assumptions on the difference between the two fluid densities and some
geometric assumption of the channel’s shape, the problem has positive solution.
The last part of this dissertation focuses on existence of particular Wronskian type
of solutions for the KdV equation of the type of complex complexitons. We study
these solutions both from a dynamical point of view when seen evolving in time, and
also for fixed values of time if regarded as potentials for a spectral problem for the
Schr ̈odinger operator.