Note on 2D Schrödinger operators with $\delta$-interactions on angles and crossing lines

In this note we sharpen the lower bound previously obtained by Lobanov et al [LLP10] for the spectrum of the 2D Schrödinger operator with a $\delta$-interaction supported on a planar angle. Using the same method we obtain the lower bound on the spectrum of the 2D Schrödinger operator with a $\delta$-interaction supported on crossing straight lines. The latter operators arise in the three-body quantum problem with $\delta$-interactions between particles.