- Physical Review Journals
All JournalsPhysics Magazine

Physical Review Research

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Letter
  • Open Access

Families of isospectral and isoscattering quantum graphs

Pavel Kurasov1, Omer Farooq2, Michał Ławniczak2, Szymon Bauch2, Mats-Erik Pistol3, Matthew de Courcy-Ireland1,4, and Leszek Sirko2,*

  • *Contact author: sirko@ifpan.edu.pl
PDF

Phys. Rev. Research 7, L022071 – Published 20 June, 2025

DOI: https://doi.org/10.1103/6yk9-17y3

Abstract

A concept of germ graphs and the M-function formalism are employed to construct large families of isospectral and isoscattering graphs. This approach represents a complete departure from the original approach pioneered by Sunada, where isospectral graphs are obtained as quotients of a certain large symmetric graph. Using the M-function formalism and the symmetries of the graph itself we construct isospectral and isoscattering pairs. In our approach isospectral pairs do not need to be embedded into a larger symmetric graph as in Sunada's approach. We demonstrate that the introduced formalism can also be extended to graphs with dissipation. The theoretical predictions are validated experimentally using microwave networks emulating open quantum graphs with dissipation.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (56)

  1. L. Pauling, The diamagnetic anisotropy of aromatic molecules, J. Chem. Phys. 4, 673 (1936).
  2. P. Exner and R. Gawlista, Band spectra of rectangular graph superlattices, Phys. Rev. B 53, 7275 (1996).
  3. T. Kottos and U. Smilansky, Quantum chaos on graphs, Phys. Rev. Lett. 79, 4794 (1997).
  4. T. Kottos and U. Smilansky, Chaotic scattering on graphs, Phys. Rev. Lett. 85, 968 (2000).
  5. O. Hul, S. Bauch, P. Pakoński, N. Savytskyy, K. Życzkowski, and L. Sirko, Experimental simulation of quantum graphs by microwave networks, Phys. Rev. E 69, 056205 (2004).
  6. M. Białous, V. Yunko, S. Bauch, M. Ławniczak, B. Dietz, and L. Sirko, Power spectrum analysis and missing level statistics of microwave graphs with violated time reversal invariance, Phys. Rev. Lett. 117, 144101 (2016).
  7. M. Ławniczak, J. Lipovský, and L. Sirko, Non-Weyl microwave graphs, Phys. Rev. Lett. 122, 140503 (2019).
  8. V. Kostrykin and R. Schrader, A random necklace model, Waves Random Media 14, S75 (2004).
  9. R. Blümel, Y. Dabaghian, and R. V. Jensen, Explicitly solvable cases of one-dimensional quantum chaos, Phys. Rev. Lett. 88, 044101 (2002).
  10. M.-E. Pistol, Generating isospectral but not isomorphic quantum graphs, arXiv:2104.12885.
  11. O. Hul, M. Ławniczak, S. Bauch, A. Sawicki, M. Kuś, and L. Sirko, Are scattering properties of graphs uniquely connected to their shapes? Phys. Rev. Lett. 109, 040402 (2012).
  12. S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics, Adv. Phys. 55, 527 (2006).
  13. G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs (American Mathematical Society, Rhode Island, 2013).
  14. Z. Pluhař and H. A. Weidenmüller, Universal chaotic scattering on quantum graphs, Phys. Rev. Lett. 110, 034101 (2013).
  15. L. Chen, S. M. Anlage, and Y. V. Fyodorov, Statistics of complex Wigner time delays as a counter of S-matrix poles: Theory and experiment, Phys. Rev. Lett. 127, 204101 (2021).
  16. C.-Z. Wang, R. Kononchuk, U. Kuhl, and T. Kottos, Loss-induced violation of the fundamental transmittance-asymmetry bound in nonlinear complex wave systems, Phys. Rev. Lett. 131, 123801 (2023).
  17. P. Kurasov, Spectral Geometry of Graphs (Birkhäuser, Berlin, 2024).
  18. D. Kowal, U. Sivan, O. Entin-Wohlman, and Y. Imry, Transmission through multiply-connected wire systems, Phys. Rev. B 42, 9009 (1990).
  19. Y. Imry, Introduction to Mesoscopic Physics (Oxford University Press, Oxford, 1997).
  20. P. P. Rohde, A. Schreiber, M. Stefanak, I. Jex, and C. Silberhorn, Multi-walker discrete time quantum walks on arbitrary graphs, their properties and their photonic implementation, New J. Phys. 13, 013001 (2011).
  21. F. Elster, S. Barkhofen, T. Nitsche, J. Novotny, A. Gabris, I. Jex, and C. Silberhorn, Quantum walk coherences on a dynamical percolation graph, Sci. Rep. 5, 13495 (2015).
  22. K. A. Dick, K. Deppert, M. W. Larsson, T. Märtensson, W. Seifert, L. R. Wallenberg, and L. Samuelson, Synthesis of branched 'nanotrees' by controlled seeding of multiple branching events, Nat. Mater. 3, 380 (2004).
  23. K. Heo, E. Cho, J. E. Yang, M. H. Kim, M. Lee, B. Y. Lee, S. G. Kwon, M. S. Lee, M. H. Jo, H. J. Choi, T. Hyeon, and S. Hong, Large-scale assembly of silicon nanowire network-based devices using conventional microfabrication facilities, Nano Lett. 8, 4523 (2008).
  24. J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Natl. Acad. Sci. USA 51, 542 (1964).
  25. M. Kac, Can one hear the shape of a drum? Am. Math. Mon. 73, 1 (1966).
  26. T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. 121, 169 (1985).
  27. C. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math. 110, 1 (1992).
  28. C. Gordon, D. Webb, and S. Wolpert, One cannot hear the shape of a drum, Bull. Am. Math. Soc. 27, 134 (1992).
  29. S. Sridhar and A. Kudrolli, Experiments on not “hearing the shape” of drums, Phys. Rev. Lett. 72, 2175 (1994).
  30. A. Dhar, D. M. Rao, U. Shankar, and S. Sridhar, Isospectrality in chaotic billiards, Phys. Rev. E 68, 026208 (2003).
  31. B. Gutkin and U. Smilansky, Can one hear the shape of a graph? J. Phys. A: Math. Gen. 34, 6061 (2001).
  32. P. Kurasov and M. Nowaczyk, Inverse spectral problem for quantum graphs, J. Phys. A: Math. Gen. 38, 4901 (2005).
  33. J. S. Fabila-Carrasco, F. Lledó, and O. Post, A geometric construction of isospectral magnetic graphs, Anal. Math. Phys. 13, 64 (2023).
  34. O. Parzanchevski and R. Band, Linear representations and isospectrality with boundary conditions, J. Geom. Anal. 20, 439 (2010).
  35. R. Band, O. Parzanchevski, and G. Ben-Shach, The isospectral fruits of representation theory: Quantum graphs and drums, J. Phys. A: Math. Theor. 42, 175202 (2009).
  36. R. Band, A. Sawicki, and U. Smilansky, Scattering from isospectral quantum graphs, J. Phys. A: Math. Theor. 43, 415201 (2010).
  37. R. Band, A. Sawicki, and U. Smilansky, Note on the role of symmetry in scattering from isospectral graphs and drums, Acta Phys. Pol. A 120, A-149 (2011).
  38. M. Ławniczak, A. Sawicki, S. Bauch, M. Kuś, and L. Sirko, Resonances and poles in isoscattering microwave networks and graphs, Phys. Rev. E 89, 032911 (2014).
  39. D. Mugnolo and V. Pivovarchik, Distinguishing cospectral quantum graphs by scattering, J. Phys. A: Math. Theor. 56, 095201 (2023).
  40. M. Ławniczak, A. Sawicki, M. Białous, and L. Sirko, Isoscattering strings of concatenating graphs and networks, Sci. Rep. 11, 1575 (2021).
  41. S. Avdonin and P. Kurasov, Inverse problems for quantum trees, Inverse Probl. Imaging 2, 1 (2008).
  42. J. Harrison, Quantizing graphs, one way or two? Rev. Math. Phys. 36, 2460001 (2024).
  43. P. Kurasov and J. Muller, n-Laplacians on metric graphs and almost periodic functions: I, Ann. Henri Poincaré 22, 121 (2021).
  44. T. J. Christiansen and M. Zworski, A mathematical formulation of the Mahaux–Weidenmüller formula for the scattering matrix, J. Phys. A: Math. Theor. 42, 415202 (2009).
  45. Y. V. Fyodorov and D. V. Savin, Statistics of impedance, local density of states, and reflection in quantum chaotic systems with absorption, JETP Lett. 80, 725 (2004).
  46. D. V. Savin, H.-J. Sommers, and Y. V. Fyodorov, Universal statistics of the local Green's function in wave chaotic systems with absorption, JETP Lett. 82, 544 (2005).
  47. See Supplemental Material at http://link.aps.org/supplemental/10.1103/6yk9-17y3 for analytical calculations of the M functions for the graphs described in the text of the Letter.
  48. O. Hul, P. Šeba, and L. Sirko, Departure of some parameter-dependent spectral statistics of irregular quantum graphs from random matrix theory predictions, Phys. Rev. E 79, 066204 (2009).
  49. B. Dietz, V. Yunko, M. Białous, S. Bauch, M. Ławniczak, and L. Sirko, Nonuniversality in the spectral properties of time-reversal-invariant microwave networks and quantum graphs, Phys. Rev. E 95, 052202 (2017).
  50. M. Ławniczak, B. Tiggelen, and L. Sirko, Experimental investigation of distributions of the off-diagonal elements of the scattering matrix and Wigner's matrix for networks with broken time reversal invariance, Phys. Rev. E 102, 052214 (2020).
  51. L. Chen, T. Kottos, and S. M. Anlage, Perfect absorption in complex scattering systems with or without hidden symmetries, Nat. Commun. 11, 5826 (2020).
  52. M. Allgaier, S. Gehler, S. Barkhofen, H.-J. Stöckmann, and U. Kuhl, Spectral properties of microwave graphs with local absorption, Phys. Rev. E 89, 022925 (2014).
  53. M. Ławniczak and L. Sirko, Investigation of the diagonal elements of the Wigner's reaction matrix for networks with violated time reversal invariance, Sci. Rep. 9, 5630 (2019).
  54. M. Ławniczak, A. Akhshani, O. Farooq, M. Białous, S. Bauch, B. Dietz, and L. Sirko, Distributions of the Wigner reaction matrix for microwave networks with symplectic symmetry in the presence of absorption, Phys. Rev. E 107, 024203 (2023).
  55. A. Rehemanjiang, M. Allgaier, C. H. Joyner, S. Müller, M. Sieber, U. Kuhl, and H.-J. Stöckmann, Microwave realization of the Gaussian symplectic ensemble, Phys. Rev. Lett. 117, 064101 (2016).
  56. J. Lu, J. Che, X. Zhang, and B. Dietz, Experimental study of quantum chaos in microwave Dirac billiards, Phys. Rev. E 102, 022309 (2020).

Outline

Information

Phys. Rev. Research 7, L022071– Published 20 June, 2025
  • Received 9 December 2024
  • Revised 25 February 2025
  • Accepted 14 May 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation