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We study the inverse spectral problem for Bessel type operators withpotential (v(x)): (H_\kappa=-\partial_x^2+\frac{k}{x^2}+v(x)). The potential isassumed smooth in ((0,R)) and with an asymptotic expansion in powers andlogarithms as (x\rightarrow 0^+, v(x)=O(x^\alpha), \alpha >-2). Specifically weshow that the coefficients of the asymptotic expansion of the potential arespectrally determined. This is achieved by computing the expansion of the traceof the resolvent of this operator which is spectrally determined andelaborating the relation of the expansion of the resolvent with that of thepotential, through the singular asymptotics lemma.

Spectrally determined singularities in a potential with an inverse square initial term