Kato's theorem and ultralong-range Rydberg molecules

We consider non-adiabatic coupling in the "trilobite"-like long-range Rydberg molecules created by perturbing degenerate high-$\ell$ Rydberg states with a ground-state atom. Due to the flexibility granted by the high Rydberg level density, the avoided crossings between relevant potential energy curves can become extremely narrow, leading to highly singular non-adiabatic coupling. We find that the gap between the trilobite potential curve and neighboring "butterfly" or "dragonfly" potential curves can even vanish, as in a conical intersection, if the gap closes at an internuclear distance which matches a node of the $s$-wave radial wave function. This is an unanticipated outcome of Kato's theorem.