Abstract:

We obtain exact formulas for the Kolmogorov widths of the inverse Laplace operator on a star graph with $m\ge2$ edges, Robin conditions at the outer vertices and a $\delta$-condition at the centre. The spectrum splits into a symmetric and a standard Sturm--Liouville branch, we derive their characteristic equations, prove that their roots interlace, and write the widths out in residue classes modulo $m$. We establish asymptotics of the widths that are uniform on compact parameter sets, and analyse the critical case, in which the quadratic form loses strict positivity, including the widths on the orthogonal complement of the kernel.