  Non-normalizable states are difficult to interpret in the orthodox quantum
formalism but often occur as solutions to physical constraints in quantum
gravity. We argue that pilot-wave theory gives a straightforward physical
interpretation of non-normalizable quantum states, as the theory requires only
a normalized density of configurations to generate statistical predictions. In
order to better understand such states, we conduct the first study of
non-normalizable solutions of the harmonic oscillator from a pilot-wave
perspective. We show that, contrary to intuitions from orthodox quantum
mechanics, the non-normalizable eigenstates and their superpositions are bound
states in the sense that the velocity field $v_y \to 0$ at large $\pm y$. We
argue that defining a physically meaningful equilibrium density for such states
requires a new notion of equilibrium, named pilot-wave equilibrium, which is a
generalisation of the notion of quantum equilibrium. We define a new
$H$-function $H_{pw}$, and prove that a density in pilot-wave equilibrium
minimises $H_{pw}$, is equivariant, and remains in equilibrium with time. We
prove an $H$-theorem for the coarse-grained $H_{pw}$, under assumptions similar
to those for relaxation to quantum equilibrium. We give an explanation of the
emergence of quantization in pilot-wave theory in terms of instability of
non-normalizable states due to perturbations and environmental interactions.
Lastly, we discuss applications in quantum field theory and quantum gravity,
and implications for pilot-wave theory and quantum foundations in general.
