Optimal bound on the number of resonance strings

Determine whether the number of distinct strings of resonances generated by a one-dimensional potential with singularities in the class \(\mathcal V\) is at most \(N\), and prove that \(N\) is the optimal bound without imposing restrictions on the singularity parameters.

Background

For potentials in the class $\mathcal V$ with $N+1$ singular points, Theorem 1.8 establishes an upper bound of $N(N+1)/2$ on the number of discontinuities of the logarithmic indicator derivative $\ell_V'$, and hence on the number of resonance strings. The paper notes that this improves an earlier $2^N-1$ bound but does not attain the sharper bound $N$ known under additional parameter assumptions in related work.

The authors report that the bound $N$ has been verified directly for all parameter relationships through $N=6$, motivating the conjecture that $N$ is the true optimal bound in both the present setting and the analogous setting of semiclassical resonances for multiple delta-function potentials.