Does no-level-crossing theorem (aka avoided crossing) always hold in perturbation theory?

Question

In perturbation, J.J. Sakurai Modern Quantum Mechanics Second Edition page 310 stated a no-level-crossing theorem stated that

"a pair of energy levels connected by perturbation do not cross as strength of perturbation is varied".

However, when I googled Avoided crossing, it stated that

"two eigenvalues of an Hermitian matrix representing a quantum observable and depending on $N$ continuous real parameters cannot become equal in value ("cross") except on a manifold of $N-2$ dimensions".

Thus, suppose a finite number of eigenstates $|n\rangle$ with $E_n$ dependent on $n+2$ parameters. Can the level crossing happen with a weak perturbation?

Answer 1

Firstly the N-2 condition is simply a constraint that gives you a number of equations without enough degrees of freedom to solve them. By having a manifold that can support more degrees of freedom this is indeed possible and is exactly what happens when a crossing is analytically continued into the complex plane as a function of the strength parameter of the perturbation. This is the area of physics which is described by non-Hermitian Hamiltonians with exceptional points. At it's easiest you will find descriptions of Hermitian level crossings (known as diabolic points) in the real plane being "split" into a pairs of complex crossings separated by square root branch points (called exceptional points) in the complex plane. You should take a look into reference [1] for more detail. A special case of this exists for Parity-Time symmetric systems.

[1] - https://arxiv.org/abs/1210.7536 - The physics of exceptional points By Dieter Heiss