Update page 'Partial Overlaps of Many Body Wavefunctions'

Partial-Overlaps-of-Many-Body-Wavefunctions.md

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+# 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