Partial Overlaps of Many-Body Wavefunctions

Description

Eigenstates are mutually orthogonal when integrated over the whole space. One can however ask “how” orthogonal states are, by looking at the overlap computed over a fraction of space (for example, particles confined to one side of the system).

Expectation: all pairs of states are mutually orthogonal, but some are more orthogonal than others.

Formally, one can start from

$$ PO_{ab} (s) = \int_0^{L-s} dx \psi^\dagger_a \psi_b $$

with $s$ between $0$ and $L$.

For which systems can this partial overlap be computed exactly? For integrable systems?

Is this a smarter way to distinguish localized systems from delocalized ones?

Is this a way to define a physically relevant “distance” between wavefunctions? (for example, $dPO/ds$ around $s = 0$, or any other measure really).

This is complementary to entanglement entropy, but in a sense much more physical.

Applications

relaxation dynamics in locally quenched problems
better understanding of physical operator matrix elements