Degeneracy despite Interaction

Physics Asked by Emroz Khan on November 27, 2020

Lifting of degeneracies of a composite system through interactions among its constituents is a generic theme of all physical systems – from splitting of $s-$orbital energies when two hydrogen atoms are brought nearby to the frequency splitting of propagating modes in optical waveguides. Through this post I am looking for an exception to this rule:

Is there any entity in physics (say real particles, or may be abstract ones) where two identical copies of those entities when brought closer and closer to each other, and are allowed to interact, instead of showing a monotonically increasing energy gap with decreasing interaction length, they show sinusoidal variations and at times degenerate energy levels?

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