They are attempting to describe physical reality with a single wave equation, and have constructed one with a lot of interesting properties, and their goal is t

They are attempting to describe physical reality with a single wave equation, and have constructed one with a lot of interesting properties, and their goal is to break the hamiltonian down into independent components to describe physical systems. The system is a finite-dimensioned Hilbert space (but it's large, like, almost infinite, but being finite is an important feature according to the paper) - and each dimension can be described by an energy spectrum, it's that energy which gives rise to physical reality via fields (and the properties that emerge from those fields). They propose basically a needle-in-the-haystack approach to finding mathematical solutions to the individual energy dimensions, I guess.