On the Negative Spectrum of One-Dimensional Schrödinger Operators on Quantum Trees with Point Interactions

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-23-2025-162
Published: Jul 16, 2025
Cite as:
M. Fazeel Anwar, Muhammad Usman, Hassaan Hafeez Rana, Ahsan Ulhaq, On the Negative Spectrum of One-Dimensional Schrödinger Operators on Quantum Trees with Point Interactions, Int. J. Anal. Appl., 23 (2025), 162.

Main Article Content

M. Fazeel Anwar, Muhammad Usman, Hassaan Hafeez Rana, Ahsan Ulhaq

Abstract

We present an explicit algorithm to determine the number of negative eigenvalues of Schrödinger operators on rooted quantum trees equipped with delta or delta-prime vertex interactions. We employ the methods of [Behrndt and Luger [5]], and the structure of trees to generate a sequence which has the same number of negative elements as the original Laplace operator. We show that the number of negative eigenvalues of the Schrödinger operators with delta interactions equals the number of negative terms in this sequence, while for delta-prime interactions, it reduces to the number of negative interaction strengths.

Article Details

References

  1. M.F. Anwar, M. Usman, M.D. Zia, Perturbation Determinant and Levinson’s Formula for Schrödinger Operators with 1-d General Point Interaction, Anal. Math. Phys. 14 (2024), 57. https://doi.org/10.1007/s13324-024-00922-1.
  2. S. Albeverio, L. Nizhnik, On the Number of Negative Eigenvalues of a One-Dimensional Schrödinger Operator with Point Interactions, Lett. Math. Phys. 65 (2003), 27–35. https://doi.org/10.1023/a:1027396004785.
  3. T. Aktosun, M. Klaus, R. Weder, Small-energy Analysis for the Self-adjoint Matrix Schrödinger Operator on the Half Line, J. Math. Phys. 52 (2011), 102101. https://doi.org/10.1063/1.3640029.
  4. G. Berkolaiko, P. Kuchment, Introduction to Quantum Graphs, American Mathematical Society, Providence, Rhode Island, 2013. https://doi.org/10.1090/surv/186.
  5. J. Behrndt, A. Luger, On the Number of Negative Eigenvalues of the Laplacian on a Metric Graph, J. Phys. Math. Theor. 43 (2010), 474006. https://doi.org/10.1088/1751-8113/43/47/474006.
  6. N. Goloschapova, L. Oridoroga, On the Negative Spectrum of One-dimensional Schrödinger Operators with Point Interactions, Integral Equ. Oper. Theory 67 (2010), 1–14. https://doi.org/10.1007/s00020-010-1759-x.
  7. P. Exner, P. Šeba, Free Quantum Motion on a Branching Graph, Reports Math. Phys. 28 (1989), 7–26. https://doi.org/10.1016/0034-4877(89)90023-2.
  8. A. Figotin, P. Kuchment, Spectral Properties of Classical Waves in High-contrast Periodic Media, SIAM J. Appl. Math. 58 (1998), 683–702. https://doi.org/10.1137/s0036139996297249.
  9. S. Gnutzmann, U. Smilansky, Quantum Graphs: Applications to Quantum Chaos and Universal Spectral Statistics, Adv. Phys. 55 (2006), 527–625. https://doi.org/10.1080/00018730600908042.
  10. M. Harmer, Inverse Scattering on Matrices with Boundary Conditions, J. Phys. Math. Gen. 38 (2005), 4875–4885. https://doi.org/10.1088/0305-4470/38/22/012.
  11. M.S. Harmer, Inverse Scattering for the Matrix Schrödinger Operator and Schrödinger Operator on Graphs with General Self-adjoint Boundary Conditions, ANZIAM J. 44 (2002), 161–168. https://doi.org/10.1017/s1446181100008014.
  12. M.S. Harmer, The Matrix Schrödinger Operator and Schrödinger Operator on Graphs, Thesis. University of Auckland, New Zealand, (2004).
  13. V. Kostrykin, R. Schrader, Kirchhoff’s Rule for Quantum Wires, J. Phys. A: Math. Gen. 32 (1999), 595–630. https://doi.org/10.1088/0305-4470/32/4/006.
  14. T. Kottos, U. Smilansky, Quantum Chaos on Graphs, Phys. Rev. Lett. 79 (1997), 4794–4797. https://doi.org/10.1103/physrevlett.79.4794.
  15. P. Kuchment, Quantum Graphs: I. Some Basic Structures, Waves Rand. Media 14 (2004), S107–S128. https://doi.org/10.1088/0959-7174/14/1/014.
  16. P. Kuchment, Quantum Graphs: Ii. Some Spectral Properties of Quantum and Combinatorial Graphs, J. Phys. A: Math. Gen. 38 (2005), 4887–4900. https://doi.org/10.1088/0305-4470/38/22/013.
  17. P. Kurasov, Quantum Graphs: Spectral Theory and Inverse Problems, in Press.
  18. M.M. Malamud, Certain Classes of Extension of a Lacunary Hermitian Operator, Ukr. Math. J. 44 (1992), 190–204. https://doi.org/10.1007/bf01061743.
  19. O. Post, Spectral Analysis on Graph-like Spaces, Springer, Berlin, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-23840-6.
  20. U. Smilansky, Quantum Chaos on Discrete Graphs, J. Phys. A: Math. Theor. 40 (2007), F621–F630. https://doi.org/10.1088/1751-8113/40/27/F07.