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We study properties of the recently established refined topological recursionfor some simple spectral curves associated to quadratic differentials. We proveexplicit formulas for the free energy and Voros coefficients of thecorresponding quantum curves, and conjecture expressions for all other (smooth)genus zero degree two curves. The results can be written in terms ofBridgeland's notion of refined BPS structure associated to the same initialdata, together with a quantum correction to the central charge. Thecorresponding "invariants" appear to be new, but their interpretation in termsof Donaldson-Thomas theory or QFT is not entirely clear.

Refined BPS structures and topological recursion - the Weber and Whittaker curves