Explicit tight bounds on the stably recoverable information for the inverse source problem

For the inverse source problem with the two-dimensional Helmholtz equation, the singular values of the source-to-near-field operator reveal a sharp frequency cut-off in the stably recoverable information on the source. We prove and numerically validate an explicit, tight lower bound B- for the spectral location B of this cut-off. We also conjecture, justify and support numerically a tight upper bound B+ for the cut-off. The bounds are expressed in terms of zeros of Bessel functions of the first and second kind. Finally, we propose our near-field upper bound B+ as an improvement of a commonly used upper bound on the spectral cutoff for the source-to-far-field operator.