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

We study the inverse spectral problem for Bessel type operators with potential $v(x): H_\kappa = -\partial^2_x + \frac{k}{x^2} + v(x)$. The potential is assumed smooth in (0,R) and with an asymptotic expansion in powers and logarithms as $x \to 0^+, v(x) = O(x^\alpha), \alpha \gt -2$. Specifically we show that the coefficients of the asymptotic expansion of the potential are spectrally determined. This is achieved by computing the expansion of the trace of the resolvent of this operator which is spectrally determined and elaborating the relation of the expansion of the resolvent with that of the potential, through the singular asymptotics lemma.

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Published 22 February 2023