Quantum backflow in the presence of defects

A physical quantity that is positive in classical physics can become negative in quantum physics, but it may be bounded. Quantum inequalities are lower bounds on averages of these physical quantities. In the case of
energy densities of a quantum field, it is called a quantum energy inequality. In the case of the probability current density of right-moving states, it is called the quantum backflow effect.

This thesis is concerned with various aspects of the quantum backflow effect in the presence of defects. The backflow effect states that a particle moving towards a reference point with positive momentum may have the probability of being found at the right of the reference point decreased with time. Defects represent a way of implementing generalised point interactions without necessarily having an explicit potential function to be added to the Hamiltonian of a physical system and are described by sewing conditions defined at the defect location. Starting from the Dirac delta-distribution, which can be regarded as a potential function but also as a point defect, we extend the analysis to the jump-defect, a discontinuous and purely transmitting integrable defect allowing conservation of total energy and momentum. In this thesis, we will examine how the backflow is affected in the presence of different defects giving special focus on the jump-defect, which does not have a backscattering contribution to the backflow constant and makes our analysis compatible with conservation laws. Beyond the Schrödinger equation, we will introduce and analyse backflow with defects in the Dirac equation, which takes into account the spin contribution to the probability current. The existence of bound states are shown to be relevant for the bounds on backflow, and numerical results will support that. Furthermore, we will investigate how the backflow constant in the presence of defects differs from the interaction-free situation.