Numerical estimates of the essential spectra of quantum graphs with delta-interactions at vertices

ABSTRACT

In this paper, we consider periodic metric graphs embedded in ${\mathbb{R}}^n$, equipped by Schrödinger operators with bounded potentials q, and $\mathit{\delta }$-type vertex conditions. Graphs are periodic with respect to a group $\mathbb{G}$ isomorphic to ${\mathbb{Z}}^m$. Applying the limit operators method, we give a formula for the essential spectra of associated unbounded operators consisting of a union of the spectra of the limit operators defined by the potential q. We apply this formula and the spectral parameter power series (SPPS) method for the analysis of the essential spectral of Schrödinger operators with potentials q of the form $q=q_0+q_1$, where $q_0$ is a periodic potential and $q_1$ is a slowly oscillating at infinity potential. The conjunction of both methods lead to an effective technique that can be used for performing numerical analysis as well. Several numerical examples demonstrate the effectiveness of our approach.

KEYWORDS:

• Periodic quantum graphs
• limit operators method
• spectral parameter power series (SPPS) method
• dispersion equation
• slowly oscillating at infinity potential

AMS SUBJECT CLASSIFICATIONS:

• 34B45
• 34K28
• 34L16
• 81Q10
• 81Q35
• 81U30

Acknowledgements

VBF acknowledges to SNI program and SIP-IPN. VSR acknowledges to SNI program and CONACyT.

Notes

No potential conflict of interest was reported by the authors.

To the memory of Professor Vasilii V. Zhikov.

Additional information

Funding

This work was supported by the SNI program and SIP-IPN [project number 20170312], and SNI program and CONACyT [project number CB-2012-179872-F].