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PreprintCC BY 4.0arXiv

A General Method for Extending Isoscattering Non-Isospectral Quantum Graphs

  • Omer Farooq
  • Michal Lawniczak
  • Leszek Sirko
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Abstract

The present paper sets out a general method for extending the isoscattering, non-isospectral quantum graphs. The methodology is illustrated by considering two graphs, denoted by Γ1 and Γ2, which have the same total metric length but different topologies. It is demonstrated that, despite the fact that their scattering matrices are identical, their eigenvalues, denoted by kn, and Euler characteristics differ from one another. The next step in the process is to extend both graphs by affixing the same star graph to each of them. Utilising the Titchmarsh-Weyl M-function formalism, it is proven that the newly constructed, extended graphs are also isoscattering and non-isospectral.

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Cite

BibTeXRIS
BibTeX
@misc{farooq2026general,
  title = {{A General Method for Extending Isoscattering Non-Isospectral Quantum Graphs}},
  author = {Omer Farooq and Michal Lawniczak and Leszek Sirko},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13349},
  eprint = {2610.13349},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13349},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - A General Method for Extending Isoscattering Non-Isospectral Quantum Graphs
AU  - Omer Farooq
AU  - Michal Lawniczak
AU  - Leszek Sirko
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13349
UR  - https://arxiv.org/abs/2610.13349
AB  - The present paper sets out a general method for extending the isoscattering, non-isospectral quantum graphs. The methodology is illustrated by considering two graphs, denoted by Γ_1 and Γ_2, which have the same total metric length but different topologies. It is demonstrated that, despite the fact that their scattering matrices are identical, their eigenvalues, denoted by k_n, and Euler characteristics differ from one another. The next step in the process is to extend both graphs by affixing the same star graph to each of them. Utilising the Titchmarsh-Weyl M-function formalism, it is proven that the newly constructed, extended graphs are also isoscattering and non-isospectral.
ER  -