Edge switching transformations of quantum graphs

Abstract

Discussed here are the effects of basics graph transformations on the spectra of associated quantum graphs. In particular it is shown that under an edge switch the spectrum of the transformed Schrödinger operator is interlaced with that of the original one. By implication, under edge swap the spectra before and after the transformation, denoted by $\{E_n{\}}_{n=1}^{\infty }$ and $\{{\tilde{E}}_n{\}}_{n=1}^{\infty }$ correspondingly, are level-2 interlaced, so that $E_{n-2}\le {\tilde{E}}_n\le E_{n+2}$. The proofs are guided by considerations of the quantum graphs' discrete analogs.

References in corpus (1)

• A remark on Krein's resolvent formula and boundary conditions

Cited by in corpus (2)

• The edge switch transformation in microwave networks
• The Statistics of Spectral Shifts due to Finite Rank Perturbations