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International Journal of Pure & Applied Mathematical Research

SPECTRAL ANALYSIS OF STURM–LIOUVILLE DIFFERENTIAL OPERATORS ON METRIC GRAPHS: EIGENVALUE DISTRIBUTION AND BOUNDARY CONDITIONS

Mr. S. M. Vadnere, Dr. A. B. Jadhav

Mr. S. M. Vadnere, Department of Mathematics, Shivneri Mahavidyalaya Shirur Anantpal, Maharashtra, India

Dr. A. B. Jadhav, Department of Mathematics,D.S.M. College, Parbhani, Maharashtra, India

DOI : https://doi.org/10.5281/zenodo.18243287 Page No : 01-10

Published Online : 2025-12-20

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ABSTRACT

This research explores the spectral properties of Sturm–Liouville and Schrödinger differential operators defined on metric and quantum graphs, with the aim of understanding how network structure influences operator behavior. By extending classical one-dimensional spectral theory to graph-based domains, the study analyzes the role of graph topology, edge lengths, and vertex coupling conditions in shaping eigenvalue distributions, spectral gaps, and resonance phenomena. The findings demonstrate that spectral characteristics are not determined by operator form alone but arise from the combined effects of geometry and boundary interactions at vertices. Particular emphasis is placed on the role of vertex coupling as a controllable mechanism for spectral manipulation without altering local potentials. From a practical standpoint, the results provide insights relevant to quantum transport networks, waveguide systems, and engineered structures where geometry-driven control is essential. The study also highlights implications for inverse spectral problems, where partial spectral data may reveal structural information about the underlying graph. The study provides the valuable insights through the balanced theoretical and applied understanding of differential operators on graphs and their relevance to contemporary spectral analysis.

Keywords: Sturm–Liouville operators; Schrödinger operators; spectral theory; metric graphs; quantum graphs; eigenvalue analysis; boundary conditions

References

Band, R., Berkolaiko, G., & Joyner, W. (2016). Quasi-isospectrality and eigenvalue asymptotics on quantum graphs. Journal of Physics A: Mathematical and Theoretical, 49(41), 415201. https://doi.org/10.1088/1751-8113/49/41/415201
Berkolaiko, G., & Exner, P. (2009). Resonances in quantum graphs with general coupling conditions. Journal of Physics A: Mathematical and Theoretical, 42(42), 425201. https://doi.org/10.1088/1751-8113/42/42/425201
Berkolaiko, G., &Kuchment, P. (2016). Introduction to quantum graphs. American Mathematical Society.https://doi.org/10.1090/surv/186
Bondarenko, N. P. (2020). Spectral data characterization for the Sturm–Liouville operator on the star-shaped graph. arXiv. https://arxiv.org/abs/2009.02522
Cacciapuoti, C., & Exner, P. (2017). Nontrivial edge coupling from a Dirichlet network squeezing: The case of a bent waveguide. Journal of Physics A: Mathematical and Theoretical, 50(49), 495302.
Cacciapuoti, C., Finco, D., &Noja, D. (2015). Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph. Physical Review E, 91(1), 013206.
https://doi.org/10.1103/PhysRevE.91.013206
Carlson, R., & Fulling, S. A. (2017). Eigenvalue asymptotics for quantum graphs with general vertex conditions. Journal of Mathematical Physics, 58(8), 083507.
Chernyshenko, A., &Pivovarchik, V. (2021). Cospectral quantum graphs. arXiv. https://arxiv.org/abs/2112.14235
Exner, P., & Keating, J. P. (2008). Quantum graphs: An introduction and a brief survey. In Analysis on graphs and its applications (pp. 1–33). Cambridge University Press. https://doi.org/10.1017/CBO9780511623813.002
Exner, P., &Lipovský, J. (2019). Equivalence of resolvent and scattering resonances on quantum graphs. Proceedings of the Royal Society A, 475(2221), 20180647.https://doi.org/10.1098/rspa.2018.0647
Gnutzmann, S., & Smilansky, U. (2018). Quantum graphs: Applications to quantum chaos and universal spectral statistics. Advances in Physics, 55(5–6), 527–625.
https://doi.org/10.1080/00018730600908025
Kuchment, P. (2002). Graph models for waves in thin structures. Waves in Random Media, 12(4), R1–R24. https://doi.org/10.1088/0959-7174/12/4/201
Kuchment, P. (2004). Quantum graphs: I. Some basic structures. Waves in Random Media, 14(1), S107–S128. https://doi.org/10.1088/0959-7174/14/1/014
Kurasov, P., & Nowaczyk, M. (2015). Inverse spectral problem for quantum graphs. Journal of Physics A: Mathematical and Theoretical, 48(43), 435203.
Mugnolo, D., & Nicaise, S. (2018). Well-posedness and spectral properties of diffusion equations on networks. ESAIM: Mathematical Modelling and Numerical Analysis, 52(3), 1077–1115.https://doi.org/10.1051/m2an/2018004
Post, O. (2012). Spectral analysis on graph-like spaces. Springer. https://doi.org/10.1007/978-3-642-23898-6

Cite this Article as : S. M. Vadnere, & A. B. Jadhav (2025), “Spectral Analysis of Sturm–Liouville Differential Operators on Metric Graphs: Eigenvalue Distribution and Boundary Conditions”, International Journal of Pure & Applied Mathematical Research, Vol. 9, Issue 2, 2025, pp 1-10, DOI: https://doi.org/10.5281/zenodo.18243287

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References

Band, R., Berkolaiko, G., & Joyner, W. (2016). Quasi-isospectrality and eigenvalue asymptotics on quantum graphs. Journal of Physics A: Mathematical and Theoretical, 49(41), 415201. https://doi.org/10.1088/1751-8113/49/41/415201
Berkolaiko, G., & Exner, P. (2009). Resonances in quantum graphs with general coupling conditions. Journal of Physics A: Mathematical and Theoretical, 42(42), 425201. https://doi.org/10.1088/1751-8113/42/42/425201
Berkolaiko, G., &Kuchment, P. (2016). Introduction to quantum graphs. American Mathematical Society.https://doi.org/10.1090/surv/186
Bondarenko, N. P. (2020). Spectral data characterization for the Sturm–Liouville operator on the star-shaped graph. arXiv. https://arxiv.org/abs/2009.02522
Cacciapuoti, C., & Exner, P. (2017). Nontrivial edge coupling from a Dirichlet network squeezing: The case of a bent waveguide. Journal of Physics A: Mathematical and Theoretical, 50(49), 495302.
Cacciapuoti, C., Finco, D., &Noja, D. (2015). Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph. Physical Review E, 91(1), 013206.
https://doi.org/10.1103/PhysRevE.91.013206
Carlson, R., & Fulling, S. A. (2017). Eigenvalue asymptotics for quantum graphs with general vertex conditions. Journal of Mathematical Physics, 58(8), 083507.
Chernyshenko, A., &Pivovarchik, V. (2021). Cospectral quantum graphs. arXiv. https://arxiv.org/abs/2112.14235
Exner, P., & Keating, J. P. (2008). Quantum graphs: An introduction and a brief survey. In Analysis on graphs and its applications (pp. 1–33). Cambridge University Press. https://doi.org/10.1017/CBO9780511623813.002
Exner, P., &Lipovský, J. (2019). Equivalence of resolvent and scattering resonances on quantum graphs. Proceedings of the Royal Society A, 475(2221), 20180647.https://doi.org/10.1098/rspa.2018.0647
Gnutzmann, S., & Smilansky, U. (2018). Quantum graphs: Applications to quantum chaos and universal spectral statistics. Advances in Physics, 55(5–6), 527–625.
https://doi.org/10.1080/00018730600908025
Kuchment, P. (2002). Graph models for waves in thin structures. Waves in Random Media, 12(4), R1–R24. https://doi.org/10.1088/0959-7174/12/4/201
Kuchment, P. (2004). Quantum graphs: I. Some basic structures. Waves in Random Media, 14(1), S107–S128. https://doi.org/10.1088/0959-7174/14/1/014
Kurasov, P., & Nowaczyk, M. (2015). Inverse spectral problem for quantum graphs. Journal of Physics A: Mathematical and Theoretical, 48(43), 435203.
Mugnolo, D., & Nicaise, S. (2018). Well-posedness and spectral properties of diffusion equations on networks. ESAIM: Mathematical Modelling and Numerical Analysis, 52(3), 1077–1115.https://doi.org/10.1051/m2an/2018004
Post, O. (2012). Spectral analysis on graph-like spaces. Springer. https://doi.org/10.1007/978-3-642-23898-6

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SPECTRAL ANALYSIS OF STURM–LIOUVILLE DIFFERENTIAL OPERATORS ON METRIC GRAPHS: EIGENVALUE DISTRIBUTION AND BOUNDARY CONDITIONS

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